diff --git a/-9AyT4oBgHgl3EQfqfj3/content/tmp_files/2301.00546v1.pdf.txt b/-9AyT4oBgHgl3EQfqfj3/content/tmp_files/2301.00546v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f9a4649f8caa9458a01853873c97dc0c84d20ea1 --- /dev/null +++ b/-9AyT4oBgHgl3EQfqfj3/content/tmp_files/2301.00546v1.pdf.txt @@ -0,0 +1,1349 @@ +arXiv:2301.00546v1 [cond-mat.mes-hall] 2 Jan 2023 +Tunable caging of excitation in decorated Lieb-ladder geometry with long range +connectivity +Atanu Nandy∗ +Department of Physics, Acharya Prafulla Chandra College, +New Barrackpore, Kolkata West Bengal-700 131, India +Controlled Aharonov-Bohm caging of wave train is reported in a quasi-one dimensional version +of Lieb geometry with next nearest neighbor hopping integral within the tight-binding framework. +This longer wavelength fluctuation is considered by incorporating periodic, quasi-periodic or fractal +kind of geometry inside the skeleton of the original network. +This invites exotic eigenspectrum +displaying a distribution of flat band states. Also a subtle modulation of external magnetic flux +leads to a comprehensive control over those non-resonant modes. Real space renormalization group +method provides us an exact analytical prescription for the study of such tunable imprisonment of +excitation. The non-trivial tunability of external agent is important as well as challenging in the +context of experimental perspective. +Keywords: Caging, flat band, interferometer, renormalization. +I. +INTRODUCTION +Recent exciting headway in experimental condensed +matter physics helps us to emulate several quantum +mechanical phenomena in a quite tunable environment. +This unprecedented advancement in fabrication tech- +nique provides a scope for direct visualization of different +theoretically proposed phenomena like localization of ex- +citation in low dimensional networks [1, 2]. That is why +creation of so called artificial systems for the simulation +of complex many-body systems containing additional de- +gree of freedom has grabbed considerable scientific im- +pact [3]. Moreover, scientific communities have already +addressed the celebration of sixty years of the pioneer- +ing work of Anderson [4]. +The absence of diffusion of +wave packet in the random disorder environment is well +known. In fact this now becomes a general prescription +in diverse topics of condensed matter physics starting +from optical lattice of ultra cold atoms [5] to the acous- +tics, wave guide arrays [6] or in micro cavities having +exciton-polaritons [7]. Unlike the case of Anderson lo- +calization (AL), the concept of compact localized states +(CLS) [8]-[15] in several one or two dimensional periodic +or non-periodic structures has attracted the spot light of +fundamental research. The journey started nearly thirty +years ago approximately from Sutherland [16]. +This unconventional non-diffusive progress of wave has +generated significant attention because of its contribu- +tion to various novel physical phenomena in strongly +correlated system, such as unconventional Anderson +localization [17, 18], +Hall ferromagnetism [19, 20], +high-temperature superconductivity [21], and superflu- +idity [22], to name a few. Moreover, this study has kept +scientists intrigued since it offers a suitable platform to +investigate several phenomena that are linked with the +information of quantum physics together with the topo- +∗Electronic address: atanunandy1989@gmail.com +logical effect including fractional quantum hall effect [23] +and flat band ferromagnetism [24]. For these CLS, the +diminishing envelope of the wave train beyond finite size +characteristics trapping cell implies extremely low group +velocity due to the divergent effective mass tensor. This +means that the particle behaves like a super heavy such +that it cannot move. The vanishing curvature of the E−k +plot corresponding to such momentum independent self- +localized states are generally caused by the destructive +nature of the quantum interference occurred by multiple +quantum dots and the local spatial symmetries involved +with the underlying structure. Hence these are also called +as flat band states. +In general, occurrence of dispersionless flat band can +be classified into two categories depending on their sta- +bility with respect to the application of magnetic pertur- +bation. In particular, the type of geometries discussed +by Mielke [25] and Tasaki [19] cannot contain flat bands +for finite magnetic flux. Whereas, the other type of lat- +tices e.g., Lieb lattice [26], there exists macroscopically +degenerate flat band even in the presence of flux. In fact, +the non dispersive band is completely insensitive to the +applied external perturbation. As it is well known that +the inherent topology of the line-centered square lattice +(also known as the Lieb lattice) induces interesting spec- +tral properties such as the macroscopically degenerated +zero-energy flat band, the Dirac cone in the low- energy +spectrum, and the typical Hofstadter-type spectrum in +a magnetic field. Moreover, Lieb geometry is one of the +most prominent candidate useful for magnetism. +The +spectral divergence of the zero-energy flat band provides +that platform. +In this manuscript, inspired by all the experimental re- +alizations of Aharaonov-Bohm caging, we study a quasi- +one dimensional Lieb-ladder network within the tight- +binding formalism. +The phenomenon of imprisonment +of wave train is studied when the next nearest neighbor +(NNN) connection term is added to the Hamiltonian. In- +teresting modulation of self-trapping of excitation is also +studied in details when the NNN connectivity is ‘dec- + +2 +orated’ by either magnetic flux or some quasi-periodic, +fractal kind of objects. +As a second motivation we have analyzed an Aharonov- +Bohm interferometer model made in the form of a quasi- +one dimensional Lieb geometry to study the flux con- +trolled localization aspects. +It is needless to mention +that this flux controlled caging is a subset of widely +used phenomena Aharonov-Bohm caging [27] and this +has been experimentally verified in recent times [1, 2]. +However, when an electron traverses a closed loop that +traps a finite magnetic flux Φ, its wave function picks +up a phase factor. +This simple sentence is at the the +core of the pioneering Aharonov-Bohm (AB) effect [28]- +[32] which has led to a substantial research in the stan- +dard AB interferometry that dominated the fundamental +physics, both theoretical and experimental perspective, +in the mesoscopic scale over the past few decades [33]- +[35]. It is to be noted that the current experiments by +Yamamoto et al. [36] has stimulated more experiments on +quantum transmission in AB interferometers [37]. Also +the previously mentioned theoretical model studies have +also played an important part in studying the elemen- +tary characteristics of the electronic states and coherent +conductance in quantum networks in the mesoscopic di- +mensions [35]. The recent advancement in the fabrication +and lithography processes have opened up the possibility +to make a tailor-made geometry with the aid of quan- +tum dots (QD) or Bose–Einstein condensates (BEC). It is +needless to mention that this has provoked a substantial +content of theoretical research even in model quantum +networks with a complex topological character [38, 39]. +In this article, highly motivated by the ongoing sce- +nario of theory and experiments in AB interferometry, +we investigate the spectral and the transmission prop- +erties of a model quantum network in which diamond +shaped Aharonov-Bohm interferometers are arranged in +the form of a quasi-one dimensional Lieb ladder geom- +etry. +Such diamond-based interferometer models have +previously been analyzed as the minimal prototypes of +bipartite networks having nodes with different coordi- +nation numbers, and representing a family of itinerant +geometrically frustrated electronic systems [40]. There +are other studies which include the problem of imprison- +ment of excitation under the influence of spin-orbit inter- +action [41], a flux-induced semiconducting behavior [42], +quantum level engineering for AB cages [43] or, as models +of spin filters [44]. +In what follows we demonstrate our findings. Sec. II +discusses the basic quasi-one dimensional Lieb ladder net- +work in respect of energy band and transmittivity. In +Sec. III we have incorporated a next nearest neighbor +connectivity by inserting a rhombic loop inside the unit +cell and discussed the flux sensitive localization. +Af- +ter that in Sec. IV the NNN hopping is decorated by +a quasiperiodic Fibonacci geometry and the distribution +of self-localized states has been studied. Sec. V demon- +strates the self-similar pattern of compact localized states +as a function of magnetic flux. In Sec. VI we have stud- +ied the Lieb Aharonov-Bohm interferometer model in re- +spect of its electronic eigenspectrum. Finally in Sec. VII +we draw our conclusions. +II. +MODEL SYSTEM AND HAMILTONIAN +We start our demonstration from the Fig. 1(a) where +a quasi-one dimensional version of the Lieb geometry is +shown. We make a distinction between the sites (blue col- +ored dots marked as A site and red colored dots marked +as B sites) based on their coordination numbers. The +(a) +A +B +x +y +(b) +ε +τ +γ +ξ +FIG. 1: (Color online) (a) A quasi-one dimensional Lieb lad- +der network with endless axial span and (b) the effective two- +arm ladder with renormalized parameters. +array is modeled by the standard tight-binding Hamilto- +nian written in the Wannier basis, viz., +H = +� +j +ǫjc† +jcj + +� +⟨jk⟩ +[tjkc† +jck + h.c.] +(1) +where the first term bears the potential information of +the respective quantum dot location and the second one +indicates the kinetic signature between two neighboring +lattice sites. The on-site potential of the respective sites +are marked as ǫA and ǫB and the nearest neighbor overlap +parameter can be assigned as t. +Without any loss of +generality, numerically the site potentials are taken as +uniform (equal to zero) and the nearest neighbor hopping +is also same (equal to unity) everywhere. By virtue of real +space renormalization group (RSRG) technique one can +easily eliminate the amplitude of an appropriate subset +of nodes to caste the original system into an effective +two-strand ladder system with renormalized parameters +as cited in the Fig. 1(b). The decimation method can be +easily implemented with the help of difference equation, +the discretized form of the Schr¨odinger’s equation, viz., +(E − ǫj)ψj = +� +k +tjkψk +(2) +This decimation provides the renormalized uniform two- +leg ladder network with different parameters. After this +renormalization procedure, all the atomic sites carry +identical on-site energy ¯ǫ and the intra-arm hopping τ. +The inter-arm vertical connectivity is marked as γ as +cited in the Fig. 1(b). This decimation produces a next +nearest neighbor hopping, denoted by ξ, which generates +overlap between the wave functions of the two diagonally + +3 +opposite atomic sites. The detailed forms of those pa- +rameters are given by, +¯ǫ = ǫ + 2t2(E − ǫ1) +δ +τ = t2(E − ǫ1) +δ +γ = 2t2t1 +δ +ξ = t2t1 +δ +(3) +where ǫ1 = ǫ + t2/(E − ǫ), t1 = t2/(E − ǫ) and δ = [(E − +ǫ1)2 − t2 +1]. With the above renormalized parameters and +by virtue of RSRG approach, one can trivially compute +the electronic density of states (DOS) ρ(E) for this quasi- +one dimensional Lieb strip as a function of the energy of +the incoming projectile by using the standard expression, +viz., +ρ(E) = − +� 1 +Nπ +� +Im[T rG(E)] +(4) +Here G(E) = [E−H +i∆]−1 is the usual green’s function +and ∆ is the imaginary part of the energy, reasonably +small enough, added for the numerical evaluation of DOS. +N denotes the total number of atomic sites present in the +system and ‘Tr’ is the trace of the green’s function. +A. +Density of eigenstates and transport +In Fig. 2(a) the variation of DOS is presented as a func- +tion of energy where we see the presence of the absolutely +continuous Bloch bands populated by extended eigen- +functions. We have checked that for any energy belong- +ing to the resonant band, the overlap parameter keeps +on non-decaying behavior and that is a signature of the +state being delocalized. At the band center (E = 0), the +central spike confirms the existence of momentum inde- +pendent flat band state which is an inherent signature of +the Lieb geometry. The spectral divergence correspond- +ing to the zero energy mode comes from the vanishing +group velocity of the wave packet as ρ ∝ +� +v−1 +g dk. With +the aid of difference equation one can obtain the distribu- +tion of amplitude for such self-localized eigenstate. The +non-vanishing amplitudes are pinned at the intermediate +sites as shown in Fig. 2(b) and one such characteristic +trapping island is isolated from the other by a distinct +physical boundary formed by the sites with zero ampli- +tude as a result of destructive quantum interference. The +dispersionless nature of the central band is responsible for +anomalous behavior in the transport and optical prop- +erties. The construction of this state definitely resem- +bles the essence of a molecular state which is spatially +quenched within a finite size cluster of atomic sites. The +analogous wave function does not present any evolution +(a) +(b) +0 +0 +0 +0 ++1 +−1 +0 +0 +0 +0 +−1 ++1 +−1 ++1 +0 +0 +0 +0 +(c) +�4 +�2 +0 +2 +4 +0.0 +0.1 +0.2 +0.3 +0.4 +E +T �E� +FIG. 2: (Color online) (a) Plot of density of eigenstates as +a function of energy E for quasi-one dimensional Lieb-ladder +geometry, (b) denotes the amplitude distribution profile for +E = 0 and (c)variation of transmittance with energy. +dynamics beyond the trapping cell. Extremely low mo- +bility of the wave train is the key factor for the disper- +sionless signature of the state. But here we should point +out that since the compact localized state, thus formed, +lies inside the continuum zone of extended states, here +the hopping integral never dies out for E = 0. Hence, +one should observe non-zero transport for that particu- +lar mode. The localization character can be prominently +viewed in presence of any perturbation when the spec- +trum shows central gap around E = 0, if any. +To corroborate the above findings related to the spec- +tral landscape we now present a precise discussion to +elucidate the electronic transmission characteristics for +this quasi-one dimensional system. For this analysis we +have considered a finite-sized underlying network. Now +the ladder-like system needs to be clamped in between +two pairs of semi-infinite periodic leads with the corre- +sponding parameters. One can then adopt the standard +green’s function approach [45, 46] and compute the same +for the composite system (lead-system-lead). The trans- +mission probability [47]-[51] can be written in terms of +this green’s function including the self-energy term as, +τij = T r[ΓiGr +i ΓjGa +i ] +(5) + +1.0 +0.8 +0.6 +Q +0.4 +0.2 +0.0 +-2 +-3 +-1 +0 +2 +3 +1 +E4 +Here the terms Γi and Γj respectively denote the con- +nection of the network with the i-th and j-th leads and +G’s are the retarded and advanced Green’s functions of +the system. The result is demonstrated in the Fig. 2(c). +It describes a wide resonant window for which we have +obtained ballistic transport. +The existence of Bloch- +like eigenfunctions for this wide range of Fermi energy +is solely responsible for this high transmission behavior. +The conducting nature of the spectral density is basically +reflected in this transmission plot. +B. +Band dispersion +To study the energy-momentum relation of this peri- +odic system we will cast the original Hamiltonian in terms +of wave vector k by virtue of the following expression, +H = +� +k +ψ† +kH(k)ψk +(6) +Using this relation, the Hamiltonian matrix in k-space +reads as, +H(k) = + + +ǫ +t +0 +t(1 + e−ika) +0 +t +ǫ +t +0 +0 +0 +t +ǫ +0 +t(1 + e−ika) +t(1 + eika) 0 +0 +ǫ +0 +0 +0 t(1 + eika) +0 +ǫ + + +(7) +The straightforward diagonalization of the above matrix +�Π +� Π +2 +0 +Π +2 +Π +�2 +�3 +�1 +0 +1 +2 +3 +ka +E +FIG. 3: (Color online) Band dispersion diagram of a quasi- +one dimensional Lieb-ladder network showing the central flat +band and other two pairs of dispersive bands. +reveals the entire band picture of the Lieb-ladder network +as presented in Fig. 3. It clearly shows one momentum +insensitive non-dispersive band at E = 0 with absolutely +zero curvature and two pairs of Bloch bands carrying +dispersive signature at E = ± +� +2(1 + cos ka) and E = +± +� +2(2 + cos ka). The central flat band state confirms +the existence of robust type of molecular state. +Φ +Φ +Φ +Φ +Φ +Φ +Φ +FIG. 4: (Color online) A quasi-one dimensional array of Lieb- +ladder geometry with next nearest neighbor (NNN) hopping +term incorporated by a diamond loop threaded by uniform +magnetic flux Φ. +III. +DIAMOND-LIEB NETWORK +In the previous description presented so far, the off- +diagonal element, i.e., the hopping parameter is taken to +be restricted within the nearest neighboring atomic sites +only within the tight-binding formulation. We now con- +sider the same quasi-one dimensional Lieb-ladder geom- +etry with next nearest neighbor (NNN) hopping integral +taken into consideration between the A types of sites as +cited in the Fig. 4. With the inclusion of longer range +connectivity the entire periodic geometry turns out to +be quasi-one dimensional Lieb ladder with a rhombic ge- +ometry embedded inside the skeleton. +This additional +overlap parameter introduces another closed loop within +each unit cell where the impact of application of magnetic +perturbation may be examined in details. +(a) +�2 +�1 +0 +1 +2 +�4 +�2 +0 +2 +4 +���0 +E +(b) +0 +0 +0 +0 +0 +0 +−1 +−1 ++1 ++1 +0 +0 +0 +0 ++1 ++1 +−1 +−1 +Φ +Φ +Φ +FIG. 5: (Color online) (a) Presentation of allowed eigenspec- +trum as a function of magnetic flux for diamond-Lieb net- +work and (b) amplitude profile corresponding to the energy +E = ǫ − 2t cos Θ. +Before presenting the numerical results and discussion +it is necessary to mention that uniform magnetic pertur- +bation may also be applied within each rhombic plaque- +tte. This can be feasible by an appropriate choice of the +gauge. This can introduce additional externally tunable +parameter which may lead to interesting band engineer- +ing. This flux tunable localization of excitation will be +discussed in the subsequent subsection. + +5 +A. +Allowed eigenspectrum as a function of flux +Now we analyze the impact of uniform magnetic per- +turbation on the sustainability of the self-localized states. +The magnetic flux is applied inside each embedded rhom- +bic plaquette. As a result of this application of magnetic +flux, the time reversal symmetry is broken (at least lo- +cally) along the arm of the rhombic plaquette. This is +considered by introducing a Peierls’ phase factor associ- +ated with the hopping integral, viz., t → teiΘ, where, +Θ = 2πΦ/4Φ0 and Φ0 = hc/e is termed as funda- +mental flux quantum. +The resultant nature of quan- +tum interference happened due to multiple quantum dots +is the ultimate determining factor for the sustainability +of the self-localized modes after applying the perturba- +tion. Here we have evaluated the allowed eigenspectrum +(Fig. 5(a)) with respect to the applied flux for this flux +included quasi-one dimensional diamond-Lieb geometry. +The spectrum is inevitably flux periodic. Multiple band +crossings, formation of several minibands and thus merg- +ing of each other are seen in this quasi-continuous pat- +tern. +Here we should give emphasis on a pertinent issue. +Fig. 5(b) shows a consistent demonstration of ampli- +tude profile (satisfying the difference equation) for en- +ergy E = ǫ − 2t cos Θ, ǫ being the uniform potential +energy everywhere. One non-vanishing cluster is again +isolated from the other by a physical barrier formed by +the sites with zero amplitude as a direct consequence of +phase cancellation at those nodes. This immediately tells +us that the incoming electron coming with this particu- +lar value of energy will be localized inside the trapping +island. But now the energy eigenvalue is sensible to the +applied flux which is an external agency. The central mo- +tivation behind the application of this external parameter +is that if possible, we may invite a comprehensive tun- +ability of such bound states solely by manipulating the +applied flux. We do not need to disturb any internal pa- +rameter of the system, instead one can, in principle, con- +trol the band engineering externally by a suitable choice +of flux. The external perturbation can be tuned contin- +uously satisfying the eigenvalue equation to control the +position of the caged state. +B. +Density of states profile +For the completeness of the analysis, we have com- +puted the variation of density of states profile as a func- +tion of energy of the incoming projectile for this quasi-one +dimensional lattice with longer wavelength fluctuation +using the standard green’s function technique both in +the absence and presence of external perturbation. The +variation with respect to the energy of the incoming pro- +jectile for different values of magnetic flux is shown in the +Fig. 6. The applied flux values are respectively Φ = 0, +Φ = Φ0/4 and Φ = Φ0/2. All the variations are plot- +ted for system size N = 753. As it is evident from the +plots that there are different absolutely continuous sub- +bands populated by extended kind of eigenfunctions. The +existence of such dispersive modes is expected because +of the inherent translational periodicity of the geometry. +We have examined that for any mode belonging to the +continuum zones the hopping integral shows oscillatory +behavior which confirms the signature of the resonant +modes. It is needless to say that the intricate nature of +the DOS is highly sensitive on the external perturbation. +Also the density of states plots as well as the allowed +eigenspectrum support the existence of flux dependent +caged state as discussed in the previous section. +C. +Band engineering +In presence of uniform magnetic flux one can easily ex- +press the Hamiltonian in the k-space language. The di- +agonalization of this matrix will give the band dispersion +as a function of flux. In this quasi-one dimensional dia- +mond Lieb geometry we have got that, there are two flux +independent dispersive bands E = ± +� +2(1 + cos ka) and +three other flux sensible resonant bands. Therefore we +should highlight a very pertinent issue here. For the last +three flux dependent bands, one can easily control the +group velocity of the wave train as well as the effective +mass (equivalently the mobility) of the particle by tuning +the external source of perturbation. This non-trivial ma- +nipulation of the internal parameters of the system with +the aid of flux makes this aspect of band engineering more +challenging as well as interesting indeed. +Before going to detailed discussion, it is important to +be noted that, when an electron moves around a closed +loop that traps a magnetic flux, the wave function picks +up a phase related to the magnetic vector potential, viz., +ψ = ψ0ei +� +A.dr. The magnetic flux here plays an equiva- +lent role as the wave vector [55]. One can thus think of a +k−Φ/Φ0 diagram which is equivalent to a typical kx−ky +diagram for electrons traveling in a two-dimensional pe- +riodic lattice. The “Brillouin zone” equivalents are ex- +pected to show up, across which variations of the group +velocity will take place. This is precisely shown in the +Fig. 7. In this plot, every contour presented corresponds +to a definite value (positive or negative) of the group +velocity of the wave packet. The red lines are the con- +tours with zero mobility. Hence they are the equivalents +of the boundaries of the Brillouin zone across which the +group velocity reverts its sign if one moves parallel to +the Φ-axis at any fixed value of the wave vector k, or vice +versa. This essentially signifies that, we can, in principle, +make an electron accelerate (or retard) without manipu- +lating its energy by changing the applied magnetic flux +only. The vanishing group velocity contours (marked by +red) indicate that the associated wavefunctions are self- +localized around finite size islands of atomic sites, making +the eigenmode a non-dispersive one. As the curvature of +the band is related to the mobility of the wave packet one +can conclude from the Fig. 7 that tuning of the curva- + +6 +(a) +�4 +�2 +0 +2 +4 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +E +Ρ +(b) +�4 +�2 +0 +2 +4 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +E +Ρ +(c) +�4 +�2 +0 +2 +4 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +E +Ρ +FIG. 6: (Color online) Variation of density of states ρ(E) as a function of energy E of the excitation. The external magnetic +flux values are respectively (a) Φ = 0, (b) Φ = Φ0/4 and (c) Φ = Φ0/2. +�2 +�1 +0 +1 +2 +�Π +� Π +2 +0 +Π +2 +Π +���0 +� +FIG. 7: (Color online) k − Φ diagram showing different group +velocity contours for electron moving in diamond embedded +Lieb geometry. The red lines mark the zero group velocity +of the wave packet. These red contours act as border lines +showing a continuous change of vg with respect to flux. +ture of the dispersive band is also possible with the help +of external perturbation. +IV. +LIEB LADDER WITH QUASIPERIODIC +NEXT NEAREST NEIGHBOR INTERACTION +In the previous case the amplitude for E = 0 will be +pinned at the top and down vertices of the diamond em- +bedded. From this standpoint we now decorate each arm +of the rhombic plaquette by a finite generation quasiperi- +oidic fibonacci kind of geometry with two different hop- +pings tx and ty respectively. The generation sequence for +this quasiperiodic structure follows the standard inflation +rule X → XY and Y → X. Based on this prescription +regarding the anisotropy in off-diagonal term, there ex- +ists three different types of atomic sites α (flanked by +two X-bonds), β (in between X − Y pair) and γ (in be- +tween Y − X pair). +Here we should mention that we +consider the generations with X type of bond at their +extremities, i.e., G2n+1, (n being integer). This is only +for convenience and does not alter the result Physics-wise +as we go for thermodynamic limit. +FIG. 8: (Color online) Distribution of self-localized modes +showing a typical three-subband pattern for large enough gen- +eration. +Hence if we start with a odd generation Fibonacci seg- +ment that decorates each arm of the diamond, then one +can decimate the chain n-times by employing the RSRG +method to get back the original diamond structure with +renormalized parameters. The recursive flows of the pa- +rameters are governed by the following equations, viz., +ǫα(n + 1) = ǫα(n) + t2 +x(n) +∆(n)[2E − (ǫβ(n) + ǫγ(n))] +ǫβ(n + 1) = ǫα(n) + (E − ǫβ(n))t2 +x(n) +∆(n) ++ +t2 +x(n) +(E − ǫβ(n)) +ǫγ(n + 1) = ǫγ(n) + (E − ǫγ(n))t2 +x(n) +∆(n) ++ +t2 +y(n) +(E − ǫβ(n)) +ǫC(n + 1) = ǫα(n) + 2t2 +x(n) +∆(n) [2E − (ǫβ(n) + ǫγ(n))] +tx(n + 1) = t2 +x(n)ty(n) +∆(n) +ty(n + 1) = +tx(n)ty(n) +(E − ǫβ(n)) +(8) +where ∆(n) = [(E − ǫβ(n))(E − ǫγ(n))] − t2 +y(n) +Obviously after decimation if we want to explore the +same compact localized state (at E = ǫ) in this renor- +malized lattice, then due to the iterative procedure, on- +site potential is now a complicated function of energy. + +4 +*** +*** +3 ++ ++ +n +2 +1 +0 +1 +1 +1 +1 +1 +1 +-3 +-2 +-1 +0 +2 +3 +1 +E7 +And if we now extract roots from the eigenvalue equa- +tion (E−ǫα) = 0, all the roots will produce a multifractal +distribution of the set of compact localized states. Obvi- +ously as we increase the generation of the fibonacci struc- +ture, in the thermodynamic limit, all the self-localized +modes exhibit a global three subband structure. The pat- +tern is already prominent in Fig. 8. Each subband can be +fine scanned in the energy scale to bring out the inherent +self-similarity and multifractality, the hallmark of the Fi- +bonacci quasicrystals [56]. The self-similarity of the spec- +trum have been checked by going over to higher enough +generations, though we refrain from showing these data +to save space here. +V. +LIEB LADDER WITH FRACTAL TYPE OF +LONG RANGE CONNECTION +FIG. 9: (Color online) An infinite array of Lieb strip with +long range connectivity decorated by fractal object. +We start this demonstration from the Fig. 9 where a fi- +nite generation of self-similar Vicsek geometry [57, 58] is +grafted inside the basic Lieb motif. The longer range con- +nection is here established through the aperiodic object. +Also a uniform magnetic flux Φ may be applied in each +small plaquette of the fractal structure. It should be ap- +preciated that while a Lieb geometry in its basic skeleton +is known to support a robust type of central self-localized +state, the inclusion of fractal structure of a finite genera- +tion in each unit cell disturbs the translational ordering +locally (though it is maintained on a global scale in the +horizontal direction) in the transverse direction. +This +non-trivial competitive scenario makes the conventional +methods of obtaining the self-localized states impossi- +ble to be implemented, especially in the thermodynamic +limit. We take the help of RSRG technique to bypass this +issue and present an analytical formalism from which one +can exactly determine the localized modes as a function +of external flux. Starting from a finite generation of scale +invariant fractal network, after suitable steps of decima- +tion [57, 58] one can produce a Lieb ladder geometry +with a diamond plaquette embedded into it (as discussed +in the previous discussion). The renormalized potential +of the top vertex of the diamond is now a complicated +function of energy and flux. Therefore straightforward +solving of the equation [E − ǫA(E, Φ)] = 0 gives us a in- +teresting distribution of compact localized states in the +E − Φ space. +This non-trivial distribution of eigenvalues as a func- +tion of flux may be considered an equivalent dispersion +relation since for an electron moving round a closed path, +FIG. 10: (Color online) Distribution of self-localized states +with applied flux. +the magnetic flux behaves the similar physical role as +that of the wave vector [55]. The distribution of eigen- +modes compose an interesting miniband-like structure as +a function of external perturbation. The competition be- +tween the global periodicity and the local fractal entity +has a crucial impact on this spectrum. We can continu- +ously engineer the magnetic flux to engineer the impris- +onment of wave train with high selectivity. +Moreover, +there are a number of inter-twined band overlap, and a +quite densely packed distribution of allowed modes, form- +ing quasi-continuous E − Φ band structure. Close obser- +vation of this eigenspectrum reveals the formation of in- +teresting variants of the Hofstadter butterflies [59]. The +spectral landscape is a quantum fractal, and encoding +the gaps with appropriate topological quantum numbers +remains an open problem for such deterministic fractals. +Before ending this section we should put emphasis on +a very pertinent point. +An aperiodic fractal object is +inserted in the unit cell of the periodic geometry. The +self-similar pattern of the fractal entity will have the in- +fluence on the spectrum. All such self-localized modes +are the consequences of destructive quantum interfer- +ence and the geometrical configuration of the underly- +ing system. For this class of energy eigenvalue, the spa- +tial span of the cluster of atomic sites containing non- +vanishing amplitudes increases with the generation of the +fractal geometry incorporated. Hence with an appropri- +ate choice of the RSRG index n, the onset of localization +and hence the spread of trapping island can be staggered, +in space. This tunable delay of the extent of localization +has already been studied for a wide varieties of fractal +geometries [57, 58, 60, 61]. This comprehensive discus- +sion regarding the manipulation of the geometry-induced +localization makes the phenomenon of Aharonov-Bohm +caging more interesting as well as challenging from the +experimental point of view. + +-1.00 +0.75 +-0.50 +-0.25 +0.25 +0.50 +0.75 +0.00 +1.00 +Φ/Φ +08 +(a) +(b) +1 +2 +N +1 +2 +N +Φ +FIG. 11: (Color online) (a) Schematic diagram of elementary +diamond-Lieb interferometer and (b) demonstrates the deco- +ration of basic unit. +VI. +DIAMOND-LIEB INTERFEROMETER +In this section we investigate the spectral character- +istics of a quantum network in which each arm of the +Lieb-ladder geometry is ‘decorated’ by diamond-shaped +Aharonov-Bohm (AB) interferometer [37]. Each elemen- +tary interferometer is pierced by a invariable magnetic +perturbation applied perpendicular to the plane of the in- +terferometer, and traps a flux Φ (in unit of Φ0 = hc/e). +This type of diamond based interferometers have been +formerly studied as the minimal prototypes of bipartite +structures having nodes with different coordination num- +bers, and representing a family of itinerant geometri- +cally frustrated electronic systems [52]-[54]. We refer to +Fig. 11(a). A standard diamond-Lieb AB interferome- +ter is shown pictorially there whereas Fig. 11(b) demon- +strates that each diamond loop can take a shape of a +quantum ring consisting of multiple lattice points. Each +arm of the diamond may be decorated by N number of +atomic scatterers between the vertices, such that the to- +tal number of single level quantum dots contained in a +single interferometer is 4(N + 1). An uniform magnetic +flux Φ may be allocated within each loop, and the elec- +tron hopping is restricted to take the non-vanishing value +for the nearest neighboring nodes only. +To study the systematic spectral analysis we take the +help of RSRG approach. Each elementary loop of the +interferometer is properly renormalized to transform it +into a simple diamond having just four sites. +Due to +this decimation process we will get three types of sites +A, B and C (respectively marked by black, red and blue +colored atomic sites in the Fig. 11(a)) with corresponding +parameters given by +˜ǫA = ǫ + 6tUN−1(x) +UN(x) +˜ǫB = ǫ + 4tUN−1(x) +UN(x) +˜ǫC = ǫ + 2tUN−1(x) +UN(x) +tF (B) = te±i(N+1)θ/UN(x) +(9) +Here, UN(x) is the N-th order Chebyshev polynomial of +second kind, and x = (E −ǫ)/2t. The ‘effective’ diamond +loops are then renormalized in a proper way (C types of +sites are being decimated out) such that we will get back +the Lieb ladder with renormalized on-site potential and +overlap integral respectively given by +˜ǫ4 = +˜ǫB + +4tFtB +(E − ǫC) +˜ǫ6 = +˜ǫA + +6tFtB +(E − ǫC) +˜t = +2tF tB +(E − ǫC) +(10) +We will now exploit all the above equations to extract the +detailed information about the electronic spectrum and +the nature of the eigenstates provided by such a model +interferometer. +A. +Spectral landscape and inverse participation +ratio +To analyze we first put N = 0 here so that the quan- +tum ring of elementary interferometer takes the form of a +diamond (Fig. 11(a)). The density of states with energy +for different values of magnetic flux enclosed within each +elementary interferometer is shown in the upper panel of +the Fig. 12. From the plot, we see that in absence of +magnetic flux the density of states reflects the periodic +nature of the geometry. It consists of absolutely contin- +uous zones populated by resonant eigenstates with sharp +spikes at E = 0 and ±2. +But here it is to be noted +that the localized character of those modes cannot be +distinctly revealed because of its position within the con- +tinuum of extended modes. But when we apply quarter +flux quantum the central localized mode becomes iso- +lated and prominent. It is also seen from the plots that +with the gradual increment of flux value the window of +resonant modes in the DOS profile shrinks along the en- +ergy scale and ultimately leads to extreme localization of +eigenstates for half flux quantum. +Actually, the effec- +tive overlap parameter between the two axial extremities +of the interferometer vanishes for this special flux value +and this makes the complete absence of resonant modes +to be possible. This is the basic physical background of +extreme localization of excitation. We should appreciate +that this typical flux induced localization of wave train +inside a charateristic trapping island is a subset of the +usual Aharonov-Bohm caging [27] +For the sake of completeness of the discussion related +to the spectral property of such quantum interferometer +model, we have also calculated the inverse participation +ratio (IPR) to certify the above density of states plots. +To formulate the localization of a normalized eigenstate +the inverse participation ratio is defined as +I = +L +� +i=1 +|ψi|4 +(11) + +9 +(a) +�4 +�2 +0 +2 +4 +0 +0.2 +0.4 +0.6 +0.8 +1 +E +Ρ +(b) +�4 +�2 +0 +2 +4 +0 +0.2 +0.4 +0.6 +0.8 +1 +E +Ρ +(c) +�4 +�2 +0 +2 +4 +0 +0.2 +0.4 +0.6 +0.8 +1 +E +Ρ +(d) +�4 +�2 +0 +2 +4 +0.005 +0.010 +0.015 +0.020 +0.025 +0.030 +0.035 +0.040 +E +IPR +(e) +�4 +�2 +0 +2 +4 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +E +IPR +(f) +�4 +�2 +0 +2 +4 +0.1 +0.2 +0.3 +0.4 +0.5 +E +IPR +FIG. 12: (Color online) (Upper panel) Variation of density of states ρ(E) as a function of energy E of the excitation and +(lower panel) indicates the variation of inversion participation ratio (IPR) wth energy. The external magnetic flux values are +respectively (a) Φ = 0, (b) Φ = Φ0/4 and (c) Φ = Φ0/2. +It is known that for an extended mode IPR goes as 1/L, +but it approaches to unity for a localized state. The lower +panel of Fig. 12 describes the variation of IPR with the +energy of the injected projectile for different flux values. +It is evident from the plots that the IPR supports the +spectral profile cited in the upper panel of Fig. 12. As +we see that with nominal strength of perturbation the +central gap opens up around E = 0, clearly indicating +the central localized mode. +The shrinking of resonant +window with the gradual increment of flux is also ap- +parent from the IPR plots. It is also interesting to ap- +preciate that for half flux quantum IPR plot (Fig. 12f) +also demonstrates the AB-caging leading to the extreme +localization of eigenstates. +B. +Flux dependent eigenspectrum +�2 +�1 +0 +1 +2 +�3 +�2 +�1 +0 +1 +2 +3 +���0 +E +FIG. 13: (Color online) Flux dependent allowed eigenspec- +trum for the diamond-Lieb AB-interferometer model. +The +pattern is flux periodic. +Fig. 13 represents the essential graphical variation of +allowed eigenspectrum for a diamond-Lieb AB interfer- +ometer with N = 0 with respect to the external magnetic +flux. With the increment of N, the number of scatterers +in each elementary interferometer, the spectrum will be +densely packed with several band crossings. The present +variation is seen to be flux periodic of periodicity equal +to one flux quantum. It is needless to say that the eigen- +spectrum is inevitably sensitive to the numerical values +of the parameters of the Hamiltonian. However, the pe- +riodicity retains for such spectrum after every single flux +quantum change of the external perturbation. +It is observed that there is a tendency of clustering of +the allowed eigenvalues towards the edges of the eigen- +spectrum as is clear from the above-mentioned diagram. +A number of band crossings are noticed and the spec- +trum cites kind of a zero band gap semiconductor like +behavior, mimicking Dirac point as observed in case of +graphene, at Φ/Φ0 = ±i, i being an integer including +zero. As we increase the complexity in each interferom- +eter by increasing N, the central gap gets consequently +filled up by more eigenstates, and the E −Φ contours get +more flattened up forming a quasi-continuous spectrum, +exotic in nature. The central eigenstate corresponding to +eigenvalue E = 0 is a robust kind of mode irrespective of +the application of perturbation, i.e., the existence of that +state is insensitive to the value of the external flux. More- +over, when the magnetic flux is set as Φ = (i+1/2)Φ0, we +observe a spectral collapse. In that case one can easily +identify the localization character of the central state. +Most importantly, it is evident from the spectral land- +scape that the it consists of a set of discrete points (eigen- +values) for half flux quantum. This is the canonical case +of extreme localization. For such special flux value the +vanishing overlap parameter makes the geometry equiv- +alent to discrete set of lattice points with zero connectiv- +ity between them. This makes the excitation to be caged + +10 +within the trapping island. Further it is to be noted that +this AB-caging [27] may happen for any value of N, the +number of eigenvalues in the discrete set depends on the +choice of N. +VII. +CLOSING REMARKS +A methodical analysis of the flux induced tunable +caging of excitation in a quasi-one dimensional Lieb net- +work with long range connectivity is reported in this +manuscript within the tight-binding framework. +With +the inclusion of second neighbor overlap integral in a dec- +orated way, external source of perturbation can act as an +important role for the selective caging of wave packet. +Flux dependent band engineering and hence the com- +prehensive control over the group velocity of the wave +train as well as the band curvature are studied in de- +tails. Decoration of the next nearest neighbor hopping +in certain quasiperiodic fashion or by some determinis- +tic fractal object is also demonstrated analytically. Real +space renormalization group approach provides us a suit- +able platform to obtain an exact prescription for the de- +termination of self-localized modes induced by destruc- +tive quantum interference effect. As we have seen that +in the quasiperiodic Fibonacci variation the distribution +of eigenstates shows a standard three-subband pattern +while in case of fractal entity countably infinite number +of localized modes cite an interesting quasi-continuous +distribution against flux. We have also critically studied +the spectral properties of a diamond Lieb interferome- +ter. The energy spectrum shows an exotic feature com- +prising extended, staggered and edge-localized eigenfunc- +tions. 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Scr. 96, 045802 (2021). + diff --git a/-9AyT4oBgHgl3EQfqfj3/content/tmp_files/load_file.txt b/-9AyT4oBgHgl3EQfqfj3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f63b7061d92c52e62ecb7f4c06638d2ffa97433f --- /dev/null +++ b/-9AyT4oBgHgl3EQfqfj3/content/tmp_files/load_file.txt @@ -0,0 +1,905 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf,len=904 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='00546v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='mes-hall] 2 Jan 2023 Tunable caging of excitation in decorated Lieb-ladder geometry with long range connectivity Atanu Nandy∗ Department of Physics, Acharya Prafulla Chandra College, New Barrackpore, Kolkata West Bengal-700 131, India Controlled Aharonov-Bohm caging of wave train is reported in a quasi-one dimensional version of Lieb geometry with next nearest neighbor hopping integral within the tight-binding framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' This longer wavelength fluctuation is considered by incorporating periodic, quasi-periodic or fractal kind of geometry inside the skeleton of the original network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' This invites exotic eigenspectrum displaying a distribution of flat band states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Also a subtle modulation of external magnetic flux leads to a comprehensive control over those non-resonant modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Real space renormalization group method provides us an exact analytical prescription for the study of such tunable imprisonment of excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The non-trivial tunability of external agent is important as well as challenging in the context of experimental perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Keywords: Caging, flat band, interferometer, renormalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' INTRODUCTION Recent exciting headway in experimental condensed matter physics helps us to emulate several quantum mechanical phenomena in a quite tunable environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' This unprecedented advancement in fabrication tech- nique provides a scope for direct visualization of different theoretically proposed phenomena like localization of ex- citation in low dimensional networks [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' That is why creation of so called artificial systems for the simulation of complex many-body systems containing additional de- gree of freedom has grabbed considerable scientific im- pact [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Moreover, scientific communities have already addressed the celebration of sixty years of the pioneer- ing work of Anderson [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The absence of diffusion of wave packet in the random disorder environment is well known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' In fact this now becomes a general prescription in diverse topics of condensed matter physics starting from optical lattice of ultra cold atoms [5] to the acous- tics, wave guide arrays [6] or in micro cavities having exciton-polaritons [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Unlike the case of Anderson lo- calization (AL), the concept of compact localized states (CLS) [8]-[15] in several one or two dimensional periodic or non-periodic structures has attracted the spot light of fundamental research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The journey started nearly thirty years ago approximately from Sutherland [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' This unconventional non-diffusive progress of wave has generated significant attention because of its contribu- tion to various novel physical phenomena in strongly correlated system, such as unconventional Anderson localization [17, 18], Hall ferromagnetism [19, 20], high-temperature superconductivity [21], and superflu- idity [22], to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Moreover, this study has kept scientists intrigued since it offers a suitable platform to investigate several phenomena that are linked with the information of quantum physics together with the topo- ∗Electronic address: atanunandy1989@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='com logical effect including fractional quantum hall effect [23] and flat band ferromagnetism [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' For these CLS, the diminishing envelope of the wave train beyond finite size characteristics trapping cell implies extremely low group velocity due to the divergent effective mass tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' This means that the particle behaves like a super heavy such that it cannot move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The vanishing curvature of the E−k plot corresponding to such momentum independent self- localized states are generally caused by the destructive nature of the quantum interference occurred by multiple quantum dots and the local spatial symmetries involved with the underlying structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Hence these are also called as flat band states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' In general, occurrence of dispersionless flat band can be classified into two categories depending on their sta- bility with respect to the application of magnetic pertur- bation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' In particular, the type of geometries discussed by Mielke [25] and Tasaki [19] cannot contain flat bands for finite magnetic flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Whereas, the other type of lat- tices e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=', Lieb lattice [26], there exists macroscopically degenerate flat band even in the presence of flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' In fact, the non dispersive band is completely insensitive to the applied external perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' As it is well known that the inherent topology of the line-centered square lattice (also known as the Lieb lattice) induces interesting spec- tral properties such as the macroscopically degenerated zero-energy flat band, the Dirac cone in the low- energy spectrum, and the typical Hofstadter-type spectrum in a magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Moreover, Lieb geometry is one of the most prominent candidate useful for magnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The spectral divergence of the zero-energy flat band provides that platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' In this manuscript, inspired by all the experimental re- alizations of Aharaonov-Bohm caging, we study a quasi- one dimensional Lieb-ladder network within the tight- binding formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The phenomenon of imprisonment of wave train is studied when the next nearest neighbor (NNN) connection term is added to the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' In- teresting modulation of self-trapping of excitation is also studied in details when the NNN connectivity is ‘dec- 2 orated’ by either magnetic flux or some quasi-periodic, fractal kind of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' As a second motivation we have analyzed an Aharonov- Bohm interferometer model made in the form of a quasi- one dimensional Lieb geometry to study the flux con- trolled localization aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' It is needless to mention that this flux controlled caging is a subset of widely used phenomena Aharonov-Bohm caging [27] and this has been experimentally verified in recent times [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' However, when an electron traverses a closed loop that traps a finite magnetic flux Φ, its wave function picks up a phase factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' This simple sentence is at the the core of the pioneering Aharonov-Bohm (AB) effect [28]- [32] which has led to a substantial research in the stan- dard AB interferometry that dominated the fundamental physics, both theoretical and experimental perspective, in the mesoscopic scale over the past few decades [33]- [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' It is to be noted that the current experiments by Yamamoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' [36] has stimulated more experiments on quantum transmission in AB interferometers [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Also the previously mentioned theoretical model studies have also played an important part in studying the elemen- tary characteristics of the electronic states and coherent conductance in quantum networks in the mesoscopic di- mensions [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The recent advancement in the fabrication and lithography processes have opened up the possibility to make a tailor-made geometry with the aid of quan- tum dots (QD) or Bose–Einstein condensates (BEC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' It is needless to mention that this has provoked a substantial content of theoretical research even in model quantum networks with a complex topological character [38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' In this article, highly motivated by the ongoing sce- nario of theory and experiments in AB interferometry, we investigate the spectral and the transmission prop- erties of a model quantum network in which diamond shaped Aharonov-Bohm interferometers are arranged in the form of a quasi-one dimensional Lieb ladder geom- etry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Such diamond-based interferometer models have previously been analyzed as the minimal prototypes of bipartite networks having nodes with different coordi- nation numbers, and representing a family of itinerant geometrically frustrated electronic systems [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' There are other studies which include the problem of imprison- ment of excitation under the influence of spin-orbit inter- action [41], a flux-induced semiconducting behavior [42], quantum level engineering for AB cages [43] or, as models of spin filters [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' In what follows we demonstrate our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' II discusses the basic quasi-one dimensional Lieb ladder net- work in respect of energy band and transmittivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' III we have incorporated a next nearest neighbor connectivity by inserting a rhombic loop inside the unit cell and discussed the flux sensitive localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Af- ter that in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' IV the NNN hopping is decorated by a quasiperiodic Fibonacci geometry and the distribution of self-localized states has been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' V demon- strates the self-similar pattern of compact localized states as a function of magnetic flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' VI we have stud- ied the Lieb Aharonov-Bohm interferometer model in re- spect of its electronic eigenspectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Finally in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' VII we draw our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' MODEL SYSTEM AND HAMILTONIAN We start our demonstration from the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 1(a) where a quasi-one dimensional version of the Lieb geometry is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' We make a distinction between the sites (blue col- ored dots marked as A site and red colored dots marked as B sites) based on their coordination numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The (a) A B x y (b) ε τ γ ξ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 1: (Color online) (a) A quasi-one dimensional Lieb lad- der network with endless axial span and (b) the effective two- arm ladder with renormalized parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' array is modeled by the standard tight-binding Hamilto- nian written in the Wannier basis, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=', H = � j ǫjc† jcj + � ⟨jk⟩ [tjkc† jck + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='] (1) where the first term bears the potential information of the respective quantum dot location and the second one indicates the kinetic signature between two neighboring lattice sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The on-site potential of the respective sites are marked as ǫA and ǫB and the nearest neighbor overlap parameter can be assigned as t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Without any loss of generality, numerically the site potentials are taken as uniform (equal to zero) and the nearest neighbor hopping is also same (equal to unity) everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' By virtue of real space renormalization group (RSRG) technique one can easily eliminate the amplitude of an appropriate subset of nodes to caste the original system into an effective two-strand ladder system with renormalized parameters as cited in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The decimation method can be easily implemented with the help of difference equation, the discretized form of the Schr¨odinger’s equation, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=', (E − ǫj)ψj = � k tjkψk (2) This decimation provides the renormalized uniform two- leg ladder network with different parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' After this renormalization procedure, all the atomic sites carry identical on-site energy ¯ǫ and the intra-arm hopping τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The inter-arm vertical connectivity is marked as γ as cited in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' This decimation produces a next nearest neighbor hopping, denoted by ξ, which generates overlap between the wave functions of the two diagonally 3 opposite atomic sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The detailed forms of those pa- rameters are given by, ¯ǫ = ǫ + 2t2(E − ǫ1) δ τ = t2(E − ǫ1) δ γ = 2t2t1 δ ξ = t2t1 δ (3) where ǫ1 = ǫ + t2/(E − ǫ), t1 = t2/(E − ǫ) and δ = [(E − ǫ1)2 − t2 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' With the above renormalized parameters and by virtue of RSRG approach, one can trivially compute the electronic density of states (DOS) ρ(E) for this quasi- one dimensional Lieb strip as a function of the energy of the incoming projectile by using the standard expression, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=', ρ(E) = − � 1 Nπ � Im[T rG(E)] (4) Here G(E) = [E−H +i∆]−1 is the usual green’s function and ∆ is the imaginary part of the energy, reasonably small enough, added for the numerical evaluation of DOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' N denotes the total number of atomic sites present in the system and ‘Tr’ is the trace of the green’s function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Density of eigenstates and transport In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 2(a) the variation of DOS is presented as a func- tion of energy where we see the presence of the absolutely continuous Bloch bands populated by extended eigen- functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' We have checked that for any energy belong- ing to the resonant band, the overlap parameter keeps on non-decaying behavior and that is a signature of the state being delocalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' At the band center (E = 0), the central spike confirms the existence of momentum inde- pendent flat band state which is an inherent signature of the Lieb geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The spectral divergence correspond- ing to the zero energy mode comes from the vanishing group velocity of the wave packet as ρ ∝ � v−1 g dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' With the aid of difference equation one can obtain the distribu- tion of amplitude for such self-localized eigenstate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The non-vanishing amplitudes are pinned at the intermediate sites as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 2(b) and one such characteristic trapping island is isolated from the other by a distinct physical boundary formed by the sites with zero ampli- tude as a result of destructive quantum interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The dispersionless nature of the central band is responsible for anomalous behavior in the transport and optical prop- erties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The construction of this state definitely resem- bles the essence of a molecular state which is spatially quenched within a finite size cluster of atomic sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The analogous wave function does not present any evolution (a) (b) 0 0 0 0 +1 −1 0 0 0 0 −1 +1 −1 +1 0 0 0 0 (c) �4 �2 0 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='4 E T �E� FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 2: (Color online) (a) Plot of density of eigenstates as a function of energy E for quasi-one dimensional Lieb-ladder geometry, (b) denotes the amplitude distribution profile for E = 0 and (c)variation of transmittance with energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' dynamics beyond the trapping cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Extremely low mo- bility of the wave train is the key factor for the disper- sionless signature of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' But here we should point out that since the compact localized state, thus formed, lies inside the continuum zone of extended states, here the hopping integral never dies out for E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Hence, one should observe non-zero transport for that particu- lar mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The localization character can be prominently viewed in presence of any perturbation when the spec- trum shows central gap around E = 0, if any.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' To corroborate the above findings related to the spec- tral landscape we now present a precise discussion to elucidate the electronic transmission characteristics for this quasi-one dimensional system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' For this analysis we have considered a finite-sized underlying network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Now the ladder-like system needs to be clamped in between two pairs of semi-infinite periodic leads with the corre- sponding parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' One can then adopt the standard green’s function approach [45, 46] and compute the same for the composite system (lead-system-lead).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The trans- mission probability [47]-[51] can be written in terms of this green’s function including the self-energy term as, τij = T r[ΓiGr i ΓjGa i ] (5) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='6 Q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='0 2 3 1 0 2 3 1 E4 Here the terms Γi and Γj respectively denote the con- nection of the network with the i-th and j-th leads and G’s are the retarded and advanced Green’s functions of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The result is demonstrated in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' It describes a wide resonant window for which we have obtained ballistic transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The existence of Bloch- like eigenfunctions for this wide range of Fermi energy is solely responsible for this high transmission behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The conducting nature of the spectral density is basically reflected in this transmission plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Band dispersion To study the energy-momentum relation of this peri- odic system we will cast the original Hamiltonian in terms of wave vector k by virtue of the following expression,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' H = � k ψ† kH(k)ψk (6) Using this relation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' the Hamiltonian matrix in k-space reads as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' H(k) = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 ǫ t 0 t(1 + e−ika) 0 t ǫ t 0 0 0 t ǫ 0 t(1 + e−ika) t(1 + eika) 0 0 ǫ 0 0 0 t(1 + eika) 0 ǫ \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb (7) The straightforward diagonalization of the above matrix �Π � Π 2 0 Π 2 Π �2 �3 �1 0 1 2 3 ka E FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 3: (Color online) Band dispersion diagram of a quasi- one dimensional Lieb-ladder network showing the central flat band and other two pairs of dispersive bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' reveals the entire band picture of the Lieb-ladder network as presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' It clearly shows one momentum insensitive non-dispersive band at E = 0 with absolutely zero curvature and two pairs of Bloch bands carrying dispersive signature at E = ± � 2(1 + cos ka) and E = ± � 2(2 + cos ka).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The central flat band state confirms the existence of robust type of molecular state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Φ Φ Φ Φ Φ Φ Φ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 4: (Color online) A quasi-one dimensional array of Lieb- ladder geometry with next nearest neighbor (NNN) hopping term incorporated by a diamond loop threaded by uniform magnetic flux Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' DIAMOND-LIEB NETWORK In the previous description presented so far, the off- diagonal element, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=', the hopping parameter is taken to be restricted within the nearest neighboring atomic sites only within the tight-binding formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' We now con- sider the same quasi-one dimensional Lieb-ladder geom- etry with next nearest neighbor (NNN) hopping integral taken into consideration between the A types of sites as cited in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' With the inclusion of longer range connectivity the entire periodic geometry turns out to be quasi-one dimensional Lieb ladder with a rhombic ge- ometry embedded inside the skeleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' This additional overlap parameter introduces another closed loop within each unit cell where the impact of application of magnetic perturbation may be examined in details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' (a) �2 �1 0 1 2 �4 �2 0 2 4 ���0 E (b) 0 0 0 0 0 0 −1 −1 +1 +1 0 0 0 0 +1 +1 −1 −1 Φ Φ Φ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 5: (Color online) (a) Presentation of allowed eigenspec- trum as a function of magnetic flux for diamond-Lieb net- work and (b) amplitude profile corresponding to the energy E = ǫ − 2t cos Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Before presenting the numerical results and discussion it is necessary to mention that uniform magnetic pertur- bation may also be applied within each rhombic plaque- tte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' This can be feasible by an appropriate choice of the gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' This can introduce additional externally tunable parameter which may lead to interesting band engineer- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' This flux tunable localization of excitation will be discussed in the subsequent subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 5 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Allowed eigenspectrum as a function of flux Now we analyze the impact of uniform magnetic per- turbation on the sustainability of the self-localized states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The magnetic flux is applied inside each embedded rhom- bic plaquette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' As a result of this application of magnetic flux, the time reversal symmetry is broken (at least lo- cally) along the arm of the rhombic plaquette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' This is considered by introducing a Peierls’ phase factor associ- ated with the hopping integral, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=', t → teiΘ, where, Θ = 2πΦ/4Φ0 and Φ0 = hc/e is termed as funda- mental flux quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The resultant nature of quan- tum interference happened due to multiple quantum dots is the ultimate determining factor for the sustainability of the self-localized modes after applying the perturba- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Here we have evaluated the allowed eigenspectrum (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 5(a)) with respect to the applied flux for this flux included quasi-one dimensional diamond-Lieb geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The spectrum is inevitably flux periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Multiple band crossings, formation of several minibands and thus merg- ing of each other are seen in this quasi-continuous pat- tern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Here we should give emphasis on a pertinent issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 5(b) shows a consistent demonstration of ampli- tude profile (satisfying the difference equation) for en- ergy E = ǫ − 2t cos Θ, ǫ being the uniform potential energy everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' One non-vanishing cluster is again isolated from the other by a physical barrier formed by the sites with zero amplitude as a direct consequence of phase cancellation at those nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' This immediately tells us that the incoming electron coming with this particu- lar value of energy will be localized inside the trapping island.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' But now the energy eigenvalue is sensible to the applied flux which is an external agency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The central mo- tivation behind the application of this external parameter is that if possible, we may invite a comprehensive tun- ability of such bound states solely by manipulating the applied flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' We do not need to disturb any internal pa- rameter of the system, instead one can, in principle, con- trol the band engineering externally by a suitable choice of flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The external perturbation can be tuned contin- uously satisfying the eigenvalue equation to control the position of the caged state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Density of states profile For the completeness of the analysis, we have com- puted the variation of density of states profile as a func- tion of energy of the incoming projectile for this quasi-one dimensional lattice with longer wavelength fluctuation using the standard green’s function technique both in the absence and presence of external perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The variation with respect to the energy of the incoming pro- jectile for different values of magnetic flux is shown in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The applied flux values are respectively Φ = 0, Φ = Φ0/4 and Φ = Φ0/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' All the variations are plot- ted for system size N = 753.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' As it is evident from the plots that there are different absolutely continuous sub- bands populated by extended kind of eigenfunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The existence of such dispersive modes is expected because of the inherent translational periodicity of the geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' We have examined that for any mode belonging to the continuum zones the hopping integral shows oscillatory behavior which confirms the signature of the resonant modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' It is needless to say that the intricate nature of the DOS is highly sensitive on the external perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Also the density of states plots as well as the allowed eigenspectrum support the existence of flux dependent caged state as discussed in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Band engineering In presence of uniform magnetic flux one can easily ex- press the Hamiltonian in the k-space language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The di- agonalization of this matrix will give the band dispersion as a function of flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' In this quasi-one dimensional dia- mond Lieb geometry we have got that, there are two flux independent dispersive bands E = ± � 2(1 + cos ka) and three other flux sensible resonant bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Therefore we should highlight a very pertinent issue here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' For the last three flux dependent bands, one can easily control the group velocity of the wave train as well as the effective mass (equivalently the mobility) of the particle by tuning the external source of perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' This non-trivial ma- nipulation of the internal parameters of the system with the aid of flux makes this aspect of band engineering more challenging as well as interesting indeed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Before going to detailed discussion, it is important to be noted that, when an electron moves around a closed loop that traps a magnetic flux, the wave function picks up a phase related to the magnetic vector potential, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=', ψ = ψ0ei � A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The magnetic flux here plays an equiva- lent role as the wave vector [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' One can thus think of a k−Φ/Φ0 diagram which is equivalent to a typical kx−ky diagram for electrons traveling in a two-dimensional pe- riodic lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The “Brillouin zone” equivalents are ex- pected to show up, across which variations of the group velocity will take place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' This is precisely shown in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' In this plot, every contour presented corresponds to a definite value (positive or negative) of the group velocity of the wave packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The red lines are the con- tours with zero mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Hence they are the equivalents of the boundaries of the Brillouin zone across which the group velocity reverts its sign if one moves parallel to the Φ-axis at any fixed value of the wave vector k, or vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' This essentially signifies that, we can, in principle, make an electron accelerate (or retard) without manipu- lating its energy by changing the applied magnetic flux only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The vanishing group velocity contours (marked by red) indicate that the associated wavefunctions are self- localized around finite size islands of atomic sites, making the eigenmode a non-dispersive one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' As the curvature of the band is related to the mobility of the wave packet one can conclude from the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 7 that tuning of the curva- 6 (a) �4 �2 0 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='0 E Ρ (b) �4 �2 0 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='0 E Ρ (c) �4 �2 0 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='0 E Ρ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 6: (Color online) Variation of density of states ρ(E) as a function of energy E of the excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The external magnetic flux values are respectively (a) Φ = 0, (b) Φ = Φ0/4 and (c) Φ = Φ0/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' �2 �1 0 1 2 �Π � Π 2 0 Π 2 Π ���0 � FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 7: (Color online) k − Φ diagram showing different group velocity contours for electron moving in diamond embedded Lieb geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The red lines mark the zero group velocity of the wave packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' These red contours act as border lines showing a continuous change of vg with respect to flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' ture of the dispersive band is also possible with the help of external perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' LIEB LADDER WITH QUASIPERIODIC NEXT NEAREST NEIGHBOR INTERACTION In the previous case the amplitude for E = 0 will be pinned at the top and down vertices of the diamond em- bedded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' From this standpoint we now decorate each arm of the rhombic plaquette by a finite generation quasiperi- oidic fibonacci kind of geometry with two different hop- pings tx and ty respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The generation sequence for this quasiperiodic structure follows the standard inflation rule X → XY and Y → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Based on this prescription regarding the anisotropy in off-diagonal term, there ex- ists three different types of atomic sites α (flanked by two X-bonds), β (in between X − Y pair) and γ (in be- tween Y − X pair).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Here we should mention that we consider the generations with X type of bond at their extremities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=', G2n+1, (n being integer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' This is only for convenience and does not alter the result Physics-wise as we go for thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 8: (Color online) Distribution of self-localized modes showing a typical three-subband pattern for large enough gen- eration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Hence if we start with a odd generation Fibonacci seg- ment that decorates each arm of the diamond, then one can decimate the chain n-times by employing the RSRG method to get back the original diamond structure with renormalized parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The recursive flows of the pa- rameters are governed by the following equations, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='ǫα(n + 1) = ǫα(n) + t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='x(n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='∆(n)[2E − (ǫβ(n) + ǫγ(n))] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='ǫβ(n + 1) = ǫα(n) + (E − ǫβ(n))t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='x(n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='∆(n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='x(n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='(E − ǫβ(n)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='ǫγ(n + 1) = ǫγ(n) + (E − ǫγ(n))t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='x(n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='∆(n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='y(n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='(E − ǫβ(n)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='ǫC(n + 1) = ǫα(n) + 2t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='x(n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='∆(n) [2E − (ǫβ(n) + ǫγ(n))] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='tx(n + 1) = t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='x(n)ty(n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='∆(n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='ty(n + 1) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='tx(n)ty(n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='(E − ǫβ(n)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='(8) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='where ∆(n) = [(E − ǫβ(n))(E − ǫγ(n))] − t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='y(n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='Obviously after decimation if we want to explore the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='same compact localized state (at E = ǫ) in this renor- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='malized lattice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' then due to the iterative procedure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' on- site potential is now a complicated function of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 4 *** *** 3 + + n 2 1 0 1 1 1 1 1 1 3 2 1 0 2 3 1 E7 And if we now extract roots from the eigenvalue equa- tion (E−ǫα) = 0, all the roots will produce a multifractal distribution of the set of compact localized states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Obvi- ously as we increase the generation of the fibonacci struc- ture, in the thermodynamic limit, all the self-localized modes exhibit a global three subband structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The pat- tern is already prominent in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Each subband can be fine scanned in the energy scale to bring out the inherent self-similarity and multifractality, the hallmark of the Fi- bonacci quasicrystals [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The self-similarity of the spec- trum have been checked by going over to higher enough generations, though we refrain from showing these data to save space here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' LIEB LADDER WITH FRACTAL TYPE OF LONG RANGE CONNECTION FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 9: (Color online) An infinite array of Lieb strip with long range connectivity decorated by fractal object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' We start this demonstration from the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 9 where a fi- nite generation of self-similar Vicsek geometry [57, 58] is grafted inside the basic Lieb motif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The longer range con- nection is here established through the aperiodic object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Also a uniform magnetic flux Φ may be applied in each small plaquette of the fractal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' It should be ap- preciated that while a Lieb geometry in its basic skeleton is known to support a robust type of central self-localized state, the inclusion of fractal structure of a finite genera- tion in each unit cell disturbs the translational ordering locally (though it is maintained on a global scale in the horizontal direction) in the transverse direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' This non-trivial competitive scenario makes the conventional methods of obtaining the self-localized states impossi- ble to be implemented, especially in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' We take the help of RSRG technique to bypass this issue and present an analytical formalism from which one can exactly determine the localized modes as a function of external flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Starting from a finite generation of scale invariant fractal network, after suitable steps of decima- tion [57, 58] one can produce a Lieb ladder geometry with a diamond plaquette embedded into it (as discussed in the previous discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The renormalized potential of the top vertex of the diamond is now a complicated function of energy and flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Therefore straightforward solving of the equation [E − ǫA(E, Φ)] = 0 gives us a in- teresting distribution of compact localized states in the E − Φ space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' This non-trivial distribution of eigenvalues as a func- tion of flux may be considered an equivalent dispersion relation since for an electron moving round a closed path, FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 10: (Color online) Distribution of self-localized states with applied flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' the magnetic flux behaves the similar physical role as that of the wave vector [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The distribution of eigen- modes compose an interesting miniband-like structure as a function of external perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The competition be- tween the global periodicity and the local fractal entity has a crucial impact on this spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' We can continu- ously engineer the magnetic flux to engineer the impris- onment of wave train with high selectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Moreover, there are a number of inter-twined band overlap, and a quite densely packed distribution of allowed modes, form- ing quasi-continuous E − Φ band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Close obser- vation of this eigenspectrum reveals the formation of in- teresting variants of the Hofstadter butterflies [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The spectral landscape is a quantum fractal, and encoding the gaps with appropriate topological quantum numbers remains an open problem for such deterministic fractals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Before ending this section we should put emphasis on a very pertinent point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' An aperiodic fractal object is inserted in the unit cell of the periodic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The self-similar pattern of the fractal entity will have the in- fluence on the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' All such self-localized modes are the consequences of destructive quantum interfer- ence and the geometrical configuration of the underly- ing system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' For this class of energy eigenvalue, the spa- tial span of the cluster of atomic sites containing non- vanishing amplitudes increases with the generation of the fractal geometry incorporated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Hence with an appropri- ate choice of the RSRG index n, the onset of localization and hence the spread of trapping island can be staggered, in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' This tunable delay of the extent of localization has already been studied for a wide varieties of fractal geometries [57, 58, 60, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' This comprehensive discus- sion regarding the manipulation of the geometry-induced localization makes the phenomenon of Aharonov-Bohm caging more interesting as well as challenging from the experimental point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='00 Φ/Φ 08 (a) (b) 1 2 N 1 2 N Φ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 11: (Color online) (a) Schematic diagram of elementary diamond-Lieb interferometer and (b) demonstrates the deco- ration of basic unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' DIAMOND-LIEB INTERFEROMETER In this section we investigate the spectral character- istics of a quantum network in which each arm of the Lieb-ladder geometry is ‘decorated’ by diamond-shaped Aharonov-Bohm (AB) interferometer [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Each elemen- tary interferometer is pierced by a invariable magnetic perturbation applied perpendicular to the plane of the in- terferometer, and traps a flux Φ (in unit of Φ0 = hc/e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' This type of diamond based interferometers have been formerly studied as the minimal prototypes of bipartite structures having nodes with different coordination num- bers, and representing a family of itinerant geometri- cally frustrated electronic systems [52]-[54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' We refer to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 11(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' A standard diamond-Lieb AB interferome- ter is shown pictorially there whereas Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 11(b) demon- strates that each diamond loop can take a shape of a quantum ring consisting of multiple lattice points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Each arm of the diamond may be decorated by N number of atomic scatterers between the vertices, such that the to- tal number of single level quantum dots contained in a single interferometer is 4(N + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' An uniform magnetic flux Φ may be allocated within each loop, and the elec- tron hopping is restricted to take the non-vanishing value for the nearest neighboring nodes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' To study the systematic spectral analysis we take the help of RSRG approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Each elementary loop of the interferometer is properly renormalized to transform it into a simple diamond having just four sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Due to this decimation process we will get three types of sites A, B and C (respectively marked by black, red and blue colored atomic sites in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 11(a)) with corresponding parameters given by ˜ǫA = ǫ + 6tUN−1(x) UN(x) ˜ǫB = ǫ + 4tUN−1(x) UN(x) ˜ǫC = ǫ + 2tUN−1(x) UN(x) tF (B) = te±i(N+1)θ/UN(x) (9) Here, UN(x) is the N-th order Chebyshev polynomial of second kind, and x = (E −ǫ)/2t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The ‘effective’ diamond ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='loops are then renormalized in a proper way (C types of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='sites are being decimated out) such that we will get back ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='the Lieb ladder with renormalized on-site potential and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='overlap integral respectively given by ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='˜ǫ4 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='˜ǫB + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='4tFtB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='(E − ǫC) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='˜ǫ6 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='˜ǫA + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='6tFtB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='(E − ǫC) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='˜t = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='2tF tB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='(E − ǫC) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='(10) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='We will now exploit all the above equations to extract the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='detailed information about the electronic spectrum and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='the nature of the eigenstates provided by such a model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Spectral landscape and inverse participation ratio To analyze we first put N = 0 here so that the quan- tum ring of elementary interferometer takes the form of a diamond (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 11(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The density of states with energy for different values of magnetic flux enclosed within each elementary interferometer is shown in the upper panel of the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' From the plot, we see that in absence of magnetic flux the density of states reflects the periodic nature of the geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' It consists of absolutely contin- uous zones populated by resonant eigenstates with sharp spikes at E = 0 and ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' But here it is to be noted that the localized character of those modes cannot be distinctly revealed because of its position within the con- tinuum of extended modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' But when we apply quarter flux quantum the central localized mode becomes iso- lated and prominent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' It is also seen from the plots that with the gradual increment of flux value the window of resonant modes in the DOS profile shrinks along the en- ergy scale and ultimately leads to extreme localization of eigenstates for half flux quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Actually, the effec- tive overlap parameter between the two axial extremities of the interferometer vanishes for this special flux value and this makes the complete absence of resonant modes to be possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' This is the basic physical background of extreme localization of excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' We should appreciate that this typical flux induced localization of wave train inside a charateristic trapping island is a subset of the usual Aharonov-Bohm caging [27] For the sake of completeness of the discussion related to the spectral property of such quantum interferometer model, we have also calculated the inverse participation ratio (IPR) to certify the above density of states plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' To formulate the localization of a normalized eigenstate the inverse participation ratio is defined as I = L � i=1 |ψi|4 (11) 9 (a) �4 �2 0 2 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='8 1 E Ρ (b) �4 �2 0 2 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='8 1 E Ρ (c) �4 �2 0 2 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='8 1 E Ρ (d) �4 �2 0 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='040 E IPR (e) �4 �2 0 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='06 E IPR (f) �4 �2 0 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='5 E IPR FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 12: (Color online) (Upper panel) Variation of density of states ρ(E) as a function of energy E of the excitation and (lower panel) indicates the variation of inversion participation ratio (IPR) wth energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The external magnetic flux values are respectively (a) Φ = 0, (b) Φ = Φ0/4 and (c) Φ = Φ0/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' It is known that for an extended mode IPR goes as 1/L, but it approaches to unity for a localized state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The lower panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 12 describes the variation of IPR with the energy of the injected projectile for different flux values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' It is evident from the plots that the IPR supports the spectral profile cited in the upper panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' As we see that with nominal strength of perturbation the central gap opens up around E = 0, clearly indicating the central localized mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The shrinking of resonant window with the gradual increment of flux is also ap- parent from the IPR plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' It is also interesting to ap- preciate that for half flux quantum IPR plot (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 12f) also demonstrates the AB-caging leading to the extreme localization of eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Flux dependent eigenspectrum �2 �1 0 1 2 �3 �2 �1 0 1 2 3 ���0 E FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 13: (Color online) Flux dependent allowed eigenspec- trum for the diamond-Lieb AB-interferometer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The pattern is flux periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 13 represents the essential graphical variation of allowed eigenspectrum for a diamond-Lieb AB interfer- ometer with N = 0 with respect to the external magnetic flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' With the increment of N, the number of scatterers in each elementary interferometer, the spectrum will be densely packed with several band crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The present variation is seen to be flux periodic of periodicity equal to one flux quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' It is needless to say that the eigen- spectrum is inevitably sensitive to the numerical values of the parameters of the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' However, the pe- riodicity retains for such spectrum after every single flux quantum change of the external perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' It is observed that there is a tendency of clustering of the allowed eigenvalues towards the edges of the eigen- spectrum as is clear from the above-mentioned diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' A number of band crossings are noticed and the spec- trum cites kind of a zero band gap semiconductor like behavior, mimicking Dirac point as observed in case of graphene, at Φ/Φ0 = ±i, i being an integer including zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' As we increase the complexity in each interferom- eter by increasing N, the central gap gets consequently filled up by more eigenstates, and the E −Φ contours get more flattened up forming a quasi-continuous spectrum, exotic in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The central eigenstate corresponding to eigenvalue E = 0 is a robust kind of mode irrespective of the application of perturbation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=', the existence of that state is insensitive to the value of the external flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' More- over, when the magnetic flux is set as Φ = (i+1/2)Φ0, we observe a spectral collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' In that case one can easily identify the localization character of the central state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Most importantly, it is evident from the spectral land- scape that the it consists of a set of discrete points (eigen- values) for half flux quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' This is the canonical case of extreme localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' For such special flux value the vanishing overlap parameter makes the geometry equiv- alent to discrete set of lattice points with zero connectiv- ity between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' This makes the excitation to be caged 10 within the trapping island.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Further it is to be noted that this AB-caging [27] may happen for any value of N, the number of eigenvalues in the discrete set depends on the choice of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' CLOSING REMARKS A methodical analysis of the flux induced tunable caging of excitation in a quasi-one dimensional Lieb net- work with long range connectivity is reported in this manuscript within the tight-binding framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' With the inclusion of second neighbor overlap integral in a dec- orated way, external source of perturbation can act as an important role for the selective caging of wave packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Flux dependent band engineering and hence the com- prehensive control over the group velocity of the wave train as well as the band curvature are studied in de- tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Decoration of the next nearest neighbor hopping in certain quasiperiodic fashion or by some determinis- tic fractal object is also demonstrated analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Real space renormalization group approach provides us a suit- able platform to obtain an exact prescription for the de- termination of self-localized modes induced by destruc- tive quantum interference effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' As we have seen that in the quasiperiodic Fibonacci variation the distribution of eigenstates shows a standard three-subband pattern while in case of fractal entity countably infinite number of localized modes cite an interesting quasi-continuous distribution against flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' We have also critically studied the spectral properties of a diamond Lieb interferome- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The energy spectrum shows an exotic feature com- prising extended, staggered and edge-localized eigenfunc- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The number of such states depend on the number of quantum dots present in each arm of the elementary diamond interferometer, and can populate the spectral landscape as densely as desired by the experimentalists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' A constant magnetic perturbation can be utilized to con- trol the positions of all such states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Moreover at special flux value the spectrum describes the Aharonov-Bohm caging of eigenstates leading to an interesting spectral collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Acknowledgments The author is thankful for the stimulating discussions regarding the results with Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Amrita Mukherjee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' The author also gratefully acknowledges the fruitful discus- sion made with Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Chakrabarti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Mukherjee, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Spracklen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Choudhury, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Goldman, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' ¨Ohberg, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Andersson, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Thomson, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 114, 245504 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Mukherjee and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} 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Takada, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Entin-Wohlman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Ya- mamoto, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Tarucha, New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' 16, 083015 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQfqfj3/content/2301.00546v1.pdf'} +page_content=' [38] J.' metadata={'source': 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b/-tFLT4oBgHgl3EQfDC7x/content/tmp_files/2301.11978v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..061c6c4131f0b6e139f23c73d1e235e23d9816f5 --- /dev/null +++ b/-tFLT4oBgHgl3EQfDC7x/content/tmp_files/2301.11978v1.pdf.txt @@ -0,0 +1,1301 @@ +MNRAS 000, 1–11 (2022) +Preprint 31 January 2023 +Compiled using MNRAS LATEX style file v3.0 +Analytical marginalisation over photometric redshift uncertainties in +cosmic shear analyses +Jaime Ruiz-Zapatero1 ★, Boryana Hadzhiyska2,3, David Alonso1, Pedro G. Ferreira1, Carlos García-García1 +and Arrykrishna Mootoovaloo1 +1Department of Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK +2Miller Institute for Basic Research in Science, University of California, Berkeley, CA, 94720, USA. +3Physics Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720. +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +As the statistical power of imaging surveys grows, it is crucial to account for all systematic uncertainties. This is normally +done by constructing a model of these uncertainties and then marginalizing over the additional model parameters. The resulting +high dimensionality of the total parameter spaces makes inferring the cosmological parameters significantly more costly using +traditional Monte-Carlo sampling methods. A particularly relevant example is the redshift distribution, 𝑝(𝑧), of the source +samples, which may require tens of parameters to describe fully. However, relatively tight priors can be usually placed on these +parameters through calibration of the associated systematics. In this paper we show, quantitatively, that a linearisation of the +theoretical prediction with respect to these calibratable systematic parameters allows us to analytically marginalise over these +extra parameters, leading to a factor ∼ 30 reduction in the time needed for parameter inference, while accurately recovering the +same posterior distributions for the cosmological parameters that would be obtained through a full numerical marginalisation +over 160 𝑝(𝑧) parameters. We demonstrate that this is feasible not only with current data and current achievable calibration +priors but also for future Stage-IV datasets. +Key words: cosmology: large-scale structure of Universe – gravitational lensing: weak – methods: data analysis +1 INTRODUCTION +In recent years unprecedentedly precise observations in cosmology +have uncovered a number of tensions between datasets that may +constitute both tantalising hints of new physics or a manifestation of +a lack of control over theoretical systematics (Heymans et al. 2021; +Riess et al. 2022). +At its simplest, the current cosmological paradigm, the Λ (denoting +the cosmological constant) cold dark matter model (ΛCDM), can be +described by only five parameters: Ω𝑚, Ω𝑏, 𝐴𝑠, 𝑛𝑠 and ℎ (see e.g. +Scott (2018) for a detailed review). However, in order to relate the +theoretical predictions of this model to actual physical observables, +it is necessary to extend it. Phenomenological models that describe +the astrophysical systems that form the basis of our observations, +as well as observational sources of systematic uncertainty, are then +appended to the core ΛCDM model. In the presence of large statistical +uncertainties, these models may consist of simple relationships in +terms of a handful of parameters. However, more precise data requires +an equally precise characterisation of these relationships, which leads +to an increase in the complexity of the model. Thus, the number +of parameters associated with these bridging models, colloquially +referred to as “nuisance” parameters, has steadily grown over the +years. +The term “nuisance” is accurate when describing these parameters. +★ E-mail: jaime.ruiz-zapatero@physics.ox.ac.uk +Not only are they generally uninteresting by comparison with the +fundamental cosmological parameters we aim to constraint, but the +increase in parameter dimensionality of the model makes exploring +their posterior distribution significantly more computationally costly. +Standard Markov Chain Monte-Carlo (MCMC), and other rejection- +based sampling methods (Metropolis et al. 1953; Foreman-Mackey +et al. 2013; Alsing & Handley 2021, among others) suffer from the +so-called “curse of dimensionality”, whereby the acceptance rate +of new samples decreases sharply with the number of parameters +(exponentially in the worst cases). +Nuisance parameters can be divided into two groups based on their +prior distributions: calibratable and non-calibratable parameters. The +non-calibratable parameters can only be constrained by the data and, +as such, typically have largely non-constraining priors. On the other +hand, we can place tighter priors on the calibratable parameters, +either by accurately characterising the instrument measurements or +by using independent external observations. In the case of cosmic +shear analyses, the impact of galaxy intrinsic alignments (Hirata & +Seljak 2004) is a standard example of a non-calibratable systematic. +On the calibratable side, the two best examples are multiplicative +shape measurement systematics, and the uncertainties in the redshift +distribution of the target source galaxies (Hoyle et al. 2018; Sánchez +& Bernstein 2019; Hildebrandt et al. 2020a; Stölzner et al. 2021; +Zhang et al. 2023). +Of these calibratable systematics the dominant source of uncer- +tainty in photometric surveys is the accuracy of redshift distributions, +© 2022 The Authors +arXiv:2301.11978v1 [astro-ph.CO] 27 Jan 2023 + +2 +Ruiz-Zapatero et al. +which are known to strongly affect the accuracy of cosmological con- +straints. The vital quantity to determine is the redshift distribution +of each tomographic sample of galaxies, 𝑝(𝑧). The fact that the un- +certainties in 𝑝(𝑧) can be calibrated with external spectroscopic data +(e.g. via direct calibration, (Lima et al. 2008; Wright et al. 2020), +clustering redshifts (Schneider et al. 2006; Newman 2008; Matthews +& Newman 2010; Schmidt et al. 2013), and shear ratios (Prat et al. +2018; Sánchez et al. 2022)), enables us to place relatively strong +priors on the redshift distribution, which in turn makes it possible to +use approximate methods to efficiently marginalise over these uncer- +tainties. +Analytical marginalisation schemes for photometric redshift un- +certainties have already been proposed in the literature. In Stölzner +et al. (2021) an analytic marginalisation scheme for photometric red- +shift uncertainties was proposed based on Gaussian mixture mod- +els and applied to the analysis of KV450 data (Hildebrandt et al. +2020b). Alternatively, in Zhang et al. (2023) a resampling approach +to marginalize over these uncertainties was proposed and applied to +the analysis HSC data. Here, we will explore the method initially pro- +posed in Hadzhiyska et al. (2020), further exploited in García-García +et al. (2023), and recently characterised in the context of the Laplace +approximation in Hadzhiyska et al. (2023). The method is based on +linearising the dependence of the theoretical prediction with respect +to the parameters defining the redshift distribution around their cali- +bration priors. This then allows one to analytically marginalise over +these parameters by modifying the covariance matrix of the data, +effectively assigning higher variance (as allowed by the calibration +prior) to the data modes most sensitive to variations in the 𝑝(𝑧). +The goal of this paper is to exhaustively validate this approximate +marginalisation scheme in the context of cosmic shear analyses. We +will do so by proving that we are able to obtain the same constraints +on cosmological parameters using this scheme, as well as employing +brute-force methods that sample the full parameter space exactly. We +will show this for both simple parametrisations of the 𝑝(𝑧) uncertain- +ties, in terms of shifts to the mean of the distribution, as well as using +completely general “non-parametric” models that treat the amplitude +of the 𝑝(𝑧) in narrowly-spaced intervals of 𝑧 as calibratable variables, +leading to a model with more than ∼ 100 nuisance parameters. In or- +der to numerically marginalize over such large parameter spaces we +develop an auto-differentiable code to obtain theoretical predictions +for the cosmic shear observables. This allows us to employ gradi- +ent based sampling algorithms, such as Hamiltonian Monte Carlo, +to beat the aforementioned curse of dimensionality. Finally, we will +show that the method is valid not only for current data, but also for +futuristic Stage-IV surveys, where photometric redshift uncertainties +will likely make up a large fraction of the total error budget. Inter- +estingly, our analysis will show that, in the context of cosmic shear +data, relatively inexpensive parametrisations of photometric redshift +uncertainties based on one free parameter per redshift bin (e.g. mean +shifts, or ranked discrete realisations (Cordero et al. 2022)), return ef- +fectively the same posterior distribution on cosmological parameters +as the most general non-parametric models. +This paper is structured as follows. In Section 2 we describe the +methods used in this work including the theory behind weak lens- +ing observables, the calibration of redshift distributions, and the +mathematics of analytical marginalisation via first-order expansion. +Section 3 presents the Dark Energy Survey data used to produce real- +istic source redshift distributions and their associated uncertainties, +as well as the models used to simulate future datasets. In Section +4 we describe the likelihood used to analyse these data, as well as +the different parametrisations used to describe 𝑝(𝑧) uncertainties. +Section 5 presents our results, quantifying the performance of ana- +lytical marginalisation methods. Finally, we present our conclusions +in Section 6. +2 METHODS +2.1 Cosmic shear power spectra +It is now commonplace to carry out the analysis of galaxy weak +lensing data tomographically. The full sample is split into redshift +bins and the two-point correlation functions of all pairs of bins are +measured and compared with their theoretical prediction. Let 𝛾𝛼( ˆn) +be a map of the spin-2 lensing shear field inferred from the sources in +the 𝛼-th redshift bin. Its relation with the three-dimensional matter +overdensity 𝛿𝑚(x) is (Bartelmann & Schneider 2001; Krause et al. +2017) +𝛾𝛼( ˆn) = +∫ 𝜒𝐻 +0 +𝑑𝜒 𝑞𝛼(𝜒) +� +−𝜒−2ðð∇−2𝛿𝑚(𝜒ˆn, 𝑧) +� +, +(1) +where ˆn is the sky direction, 𝜒 is the comoving radial distance at +redshift 𝑧, 𝜒𝐻 is the distance to the horizon, 𝑞𝛼(𝜒) is the weak +lensing radial kernel, and ð is the spin-raising differential operator, +acting on a spin-𝑠 quantity as (Newman & Penrose 1966): +ð 𝑠 𝑓 (𝜃, 𝜑) = −(sin 𝜃)𝑠 +� 𝜕 +𝜕𝜃 + +𝑖 +sin 𝜃 +𝜕 +𝜕𝜑 +� +(sin 𝜃)−𝑠 𝑠 𝑓 +(2) +and turning it into a spin-(𝑠 + 1) quantity. The weak lensing kernel +is1 +𝑞𝛼(𝜒) ≡ 3 +2 𝐻2 +0Ω𝑚 +𝜒 +𝑎(𝜒) +∫ ∞ +𝑧(𝜒) +𝑑𝑧′𝑝𝛼(𝑧′) 𝜒(𝑧′) − 𝜒 +𝜒(𝑧′) +, +(3) +where 𝐻0 ≡ 𝐻(𝑧 = 0) is the Hubble expansion rate today, Ω𝑚 is +the current matter density parameter and, most importantly for our +discussion, 𝑝𝛼(𝑧) is the redshift distribution in bin 𝛼, +The angular power spectrum of the 𝐸-mode components of two +maps 𝛼 and 𝛽, 𝐶 𝛼𝛽 +ℓ +can be related to the three-dimensional matter +power spectrum 𝑃(𝑘, 𝑧) via: +𝐶 𝛼𝛽 +ℓ += 𝐺2 +ℓ +∫ +𝑑𝜒 +𝜒2 𝑞𝛼(𝜒) 𝑞𝛽(𝜒) 𝑃 +� +𝑘 = ℓ + 1/2 +𝜒 +, 𝑧(𝜒) +� +, +(4) +where we have assumed the Limber approximation (Limber 1953; +Afshordi et al. 2004), which is valid for the broad weak lensing ker- +nels considered in this work. The scale-dependent lensing prefactor, +𝐺ℓ ≡ +√︄ +(ℓ + 2)! +(ℓ − 2)! +1 +(ℓ + 1/2)2 , +(5) +accounts for the difference between angular and three-dimensional +derivatives in Eq. 1 (i.e. 𝜒2ð2∇−2 � 1). This prefactor leads to +sub-percent differences for ℓ > 11 and can therefore be neglected +on small scales (Kilbinger et al. 2017). In this work we will use +the Halofit fitting function of Smith et al. (2003); Takahashi et al. +(2012) to describe the matter power spectrum. +The intrinsic alignment (IA) of galaxies due to local interactions +(gravitational or otherwise), is an important contaminant for cosmic +shear data that must be taken into account (Brown et al. 2002). For +simplicity, however, and since the focus of this work is the impact of +the marginalisation over redshift distribution uncertainties, we will +ignore the contribution from intrinsic alignments in this analysis. +1 Note that this is only strictly valid in ΛCDM (Ferreira 2019). +MNRAS 000, 1–11 (2022) + +Analytical marginalisation over photo-𝑧 uncertainties +3 +2.2 Redshift distribution uncertainties +The sub-samples that make up the redshift bins used in the tomo- +graphic cosmic shear analysis of an imaging survey are selected +based on the source photometry, either by simple cuts in the in- +ferred photometric redshifts (photo-𝑧), or by selecting directly in the +magnitude-color space of the sample, bypassing photo-𝑧 estimation +altogether. Regardless of the method used to select the sub-samples, +their true redshift distributions are inevitably subject to some level +of uncertainty, due to the lack of precise redshift measurements. +The 𝑝(𝑧) can however be calibrated through various methods, e.g.: +weighted direct calibration with a sufficiently complete spectroscopic +sample (Lima et al. 2008; Wright et al. 2020), clustering redshifts +(Schneider et al. 2006; Newman 2008; Matthews & Newman 2010; +Schmidt et al. 2013), and shear ratios (Prat et al. 2018; Sánchez et al. +2022). This typically leads to relatively tight priors on the 𝑝(𝑧), but +the residual uncertainties in this prior must be propagated into the +final parameter constraints. +To characterise these uncertainties, we will make use of two dif- +ferent methods, which encompass the range of model complexity we +may reasonably expect from current and future data. +• Method 1: 𝑧 shifts. Most cosmic shear analyses to date +(Miyazaki et al. 2012; Hildebrandt et al. 2020b; Heymans et al. +2021; Abbott et al. 2018a, 2022, among others) have summarised +the uncertainty in the calibrated 𝑝𝛼(𝑧) into a single parameter Δ𝑧𝛼 +that shifts the mean of the redshift distribution. I.e. let ˆ𝑝𝛼(𝑧) be the +best-guess redshift distribution. The true redshift distribution is then +𝑝𝛼(𝑧) = ˆ𝑝𝛼(𝑧 + Δ𝑧𝛼). +(6) +A prior on Δ𝑧𝛼 can be derived using the calibration methods listed +above. We will refer to this method as parametric. +This simple model turns out to be relatively well suited to describe +the impact of 𝑝(𝑧) uncertainties in the case of cosmic shear data. +Since weak lensing is a radially cumulative effect, the amplitude +of the weak lensing kernel (Eq. 3) is mostly sensitive to the mean +redshift of the sample, and thus much of the effect on cosmic shear +observables is well described by this parameter (Bonnett et al. 2016). +Other modes of 𝑝(𝑧) uncertainty, such as the distribution width, +may be more relevant for galaxy clustering observables, or for the +intrinsic alignment contribution to cosmic shear. Near-future cosmic +shear samples may indeed require a more sophisticated description +of the 𝑝(𝑧) uncertainty, and thus we turn to a more general method. +• Method 2: 𝑝(𝑧) bin heights. Most 𝑝(𝑧) calibration methods +(e.g. direct calibration or clustering redshifts) will produce a binned +measurement of the 𝑝(𝑧) with deterministic redshift bin ranges, and +uncertain bin heights. The most general method to propagate these +uncertainties is therefore to treat each bin height 𝑝𝑖 ≡ 𝑝(𝑧𝑖) as +a free parameter in the model, with a prior given by the calibration +uncertainties. The latter may be in the form of individual 1𝜎 errors for +each bin height, if the uncertainties are approximately uncorrelated, +or a full covariance matrix covering all bin heights. +The resulting parametrisation thus sidesteps any attempt at sum- +marising the uncertainty into effective parameters, and thus we will +refer to this method as non-parametric. The method therefore fully +propagates all calibration uncertainties into the final constraints with +minimal approximations. +The key practical difference between both methods, in the context +of error propagation, is the additional complexity they incur. The +parametric approach (Method 1) introduces one free parameter per +redshift bin. For 𝑂(5) bins, this is already enough to significantly +impact the performance of standard MCMC algorithms. In turn, the +non-parametric approach (Method 2) introduces tens or hundreds of +parameters per redshift bin, and one must resort to advanced sam- +pling methods in order to fully explore the resulting model without +assumptions. +2.3 Linearisation and analytical marginalisation +Let 𝛀 be the set of non-calibratable parameters of a model (in our +case this is the set of cosmological and non-calibratable nuisance +parameters) and 𝝂 the set of calibratable parameter such that the total +set of parameters is given by 𝜽 = 𝛀 ∪ 𝝂. Now consider the general +case of a Gaussian posterior distribution of the form +−2 log 𝑃(𝛀, 𝝂|d) =(d − t)𝑇 C−1(d − t) + (𝝂 − ¯𝝂)𝑇 P−1(𝝂 − ¯𝝂) +− 2 log 𝑃(𝛀) + const., +(7) +where d is the data. We assume a Gaussian calibration prior with +mean ¯𝝂 and covariance P, while 𝑃(𝛀) is the prior on 𝛀 (which is, as +per our assumption, broad). t(𝛀, 𝝂) is the theoretical prediction for +the data d which implicitly depends on both calibratable and non- +calibratable parameters. C is the covariance matrix of d, which is +parameter-independent. +Assuming a tight prior on 𝝂, we start by expanding the theory +prediction around ¯𝝂 +t ≃ ¯t + T(𝝂 − ¯𝝂), +where ¯t ≡ t(𝛀, ¯𝝂), +T ≡ 𝑑t +𝑑𝝂 +����𝝂=¯𝝂 +. +(8) +Substituting this approximation in Eq. 7, the posterior becomes Gaus- +sian in 𝝂, and thus the calibratable parameters can be marginalised +analytically. As shown in Hadzhiyska et al. (2020), the resulting +marginalised posterior is +−2 log 𝑃(𝛀|d) ≃(d − ¯t)𝑇 ˜C−1(d − ¯t) − 2 log 𝑃(𝛀) ++ log +� +det +� +T𝑇 C−1T + P−1�� ++ const., +(9) +where the modified covariance is +˜C ≡ C + TPT𝑇 . +(10) +Note that, strictly speaking, both the modified covariance and the +term in the second line of Eq. 9 depend on𝛀, which would in principle +complicate the evaluation of the likelihood. In practice, thisparameter +dependence can be neglected such that the value of 𝛀 at which these +terms are evaluated can be fixed during exploration of the posterior. +However, fixing 𝛀 at values with a bad fit to the data will result +in a mischaracterisation of the response of the theory vector to the +nuisance parameters leading to inaccurate marginalised posteriors. +Ideally, 𝛀 is fixed to its maximum a posteriori (MAP) value. However, +as shown in Hadzhiyska et al. (2020) and in preliminary results, no +appreciable differences are found in the marginalised posteriors for +𝛀 within 2𝜎 of the MAP. Note that the size of the 2𝜎 region will +depend on how constraining the data is. +This result is intuitively simple to understand if we think of T as +the response of the data to variations in the nuisance parameters. +After marginalising over the calibratable parameters, the resulting +distribution is a multi-variate Gaussian where the data covariance +has been updated in Eq. 10 by increasing the uncertainty in the data +modes that most prominently respond to variations in the nuisance +parameters. +In this work, 𝝂 corresponds to the parameters describing the red- +shift distribution uncertainties, i.e. one shift parameter per redshift +bin when using the parametric approach, or a set of 𝑝(𝑧) bin heights in +the non-parametric scheme. The method described above, however, +MNRAS 000, 1–11 (2022) + +4 +Ruiz-Zapatero et al. +0.00 +0.05 +0.10 +0.15 +0.20 +p(z)0 +p(z)1 +p(z)2 +p(z)3 +0.5 +1.0 +1.5 +z +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +z +0.5 +1.0 +1.5 +z +0.5 +1.0 +1.5 +z +0.5 +1.0 +1.5 +z +10 +3 +10 +2 +10 +1 +100 +Figure 1. Top row: normalized galaxies’s redshift distributions for each of the 4 redshift bins. Bottom row: correlation matrix obtained using the DIR algorithm +for each of the 4 galaxies’ redshift distributions. Note that for visualization purposes we display the absolute values of the each correlation matrix in logarithmic +scale. In this plot we can see that the covariance matrices obtained through the DIR algorithm are mostly diagonal. +is fully general and has in the past been applied to marginalise over +other types of nuisance parameters, including multiplicative shape +measurement biases (Hildebrandt et al. 2020b), as well as truly linear +parameters such as shot-noise (García-García et al. 2021) or system- +atic template amplitudes (Koukoufilippas et al. 2020). The aim of +this paper is thus to determine the applicability of this method to the +case of redshift distribution uncertainties. +3 DATA +In order to evaluate the performance of the analytical marginalisation +approach described in the previous section in the context of current +and future surveys, we make use of data from the first-year cosmic +shear analysis of the Dark Energy Survey (DES-Y1, Abbott et al. +(2018b)). The aim of this is twofold: first, to demonstrate that the +method can be successfully implemented in real data, with real-life +complications (e.g. noisy 𝑝(𝑧)s, numerical covariances, astrophysi- +cal and observational systematics) and, second, to demonstrate this +validity for future Stage-IV datasets in the presence of 𝑝(𝑧) cali- +bration uncertainties already achieved on current data. This section +describes the DES-Y1 data used, and the models used to generate +simulated future Stage-IV data. +3.1 DES-Y1 data and redshift distributions +The Dark Energy Survey is a photometric, 5-year survey, that has +observed 5000 deg2 of the sky using five different filter bands (grizY). +The observations were made with the 4m Blanco Telescope, provided +with the 570-Mpix Dark Energy Camera (DECam), from the Cerro +Tololo Inter-American Observatory (CTIO), in Chile. In this paper +we use cosmic shear data from the first data release (Abbott et al. +2018b), which covers 1786 deg2 before masking. In particular, we +use the public Metacalibration source catalog2, which is divided +in four redshift bins covering the range 𝑧 ≲ 1.6 (Hoyle et al. 2018). +We use the calibrated redshift distributions of the Metacalibra- +tion sample provided by García-García et al. (2023). The 𝑝(𝑧)s were +estimated via direct calibration (DIR Lima et al. (2008)), using the +COSMOS 30-band catalog (Laigle et al. 2016) as a calibrating sam- +ple. The uncertainties of the measured redshift distributions were +estimated analytically, as described in García-García et al. (2023), +accounting for both shot noise and sample variance, and represent +a realistic level of 𝑝(𝑧) uncertainty achieved by current existing +datasets. The redshift distributions were sampled on 40 bins of width +𝛿𝑧 = 0.04 covering the range 0 ≤ 𝑧 ≤ 1.6. Fig. 1 shows, in the first +row, the redshift distributions of the four Metacalibration samples +and their statistical uncertainties. Note that we estimated the full co- +variance matrix of the 𝑝(𝑧) bin heights. The covariance is dominated +by the diagonal, as can be seen in the bottom panels of Fig. 1. +We will also use the cosmic shear angular power spectra provided +by Nicola et al. (2021). A full description of the methods used to +estimate these power spectra, and their associated covariance matrix, +from the DES-Y1 data is provided by the authors. +3.2 Future Stage-IV data +We generate a simulated data vector corresponding to a Stage-IV +cosmic shear survey, such as the Legacy Survey of Space Time, at +the Rubin Observatory (LSST Dark Energy Science Collaboration +2012), or the Euclid survey (Spergel et al. 2015). Our aim is to ef- +fectively test the analytical marginalisation method in the low-noise +regime, where the inferred posterior is likely more sensitive to resid- +ual 𝑝(𝑧) uncertainties, and the error budget may become dominated +by these, rather than the statistical errors in the data themselves. +2 https://desdr-server.ncsa.illinois.edu/despublic/y1a1_ +files/ +MNRAS 000, 1–11 (2022) + +Analytical marginalisation over photo-𝑧 uncertainties +5 +For simplicity, we simulate the Stage-IV survey as having the +same redshift distributions as the DES-Y1 sample. This includes +both the 𝑝(𝑧)s themselves, and their calibration uncertainties. While +it is possible that techniques for inferring redshifts from photometry, +or the size and quality of calibrating spectroscopic samples, will +improve substantially by the time Stage-IV data are available, we +prefer to err on the side of caution and assume the same performance +as currently achieved. For instance it is possible that redshift estimates +will suffer commensurately with the increase in survey depth. The +results presented here are therefore conservative, and their validity +will only be reinforced if better 𝑝(𝑧) calibration samples are used in +the future. +We generate cosmic shear power spectra using CCL (Chisari et al. +2019) for the best-fit Planck 2018 cosmological parameters (Planck +Collaboration et al. 2020): Ω𝑏ℎ2 = 0.02237, Ω𝑐ℎ2 = 0.12, ℎ = +0.6736, 109𝐴𝑠 = 2.0830, 𝑛𝑠 = 0.9649, 𝑤0 = −1, 𝑤𝑎 = 0. We use +the same sampling in ℓ used for the DES-Y1 power spectra, and use +only scales in the range ℓ ∈ [30, 2000]. +We compute the covariance matrix of these power spectra an- +alytically, including a disconnected “Gaussian” component, and a +connected super-sample covariance contribution (SSC). +Cov +� +𝐶 𝛼 +ℓ , 𝐶𝜌𝜎 +ℓ′ +� += Cov𝐺 +� +𝐶 𝛼𝛽 +ℓ +, 𝐶𝜌𝜎 +ℓ′ +� ++ CovSSC +� +𝐶 𝛼𝛽 +ℓ +, 𝐶 𝜎𝜌 +ℓ′ +� +. +(11) +We estimate the Gaussian covariance using a simple mode-counting +approximation (Efstathiou 2004) as +Cov𝐺 +� +𝐶 𝛼𝛽 +ℓ +, 𝐶𝜌𝜎 +ℓ′ +� += 𝛿ℓℓ′ +𝐶 𝛼𝜌 +ℓ +𝐶𝛽𝜎 +ℓ ++ 𝐶 𝛼𝜎 +ℓ +𝐶𝛽𝜌 +ℓ +(2ℓ + 1) Δℓ 𝑓sky +, +(12) +where 𝑓sky is the fraction of the sky covered by the experiment. We +assume 𝑓sky = 0.4, as in the case of LSST (LSST Dark Energy Sci- +ence Collaboration 2012). The angular power spectra above contain +the contribution from shape noise in the auto-correlation, of the form +𝑁 𝛼𝛼 +ℓ += +𝜎2𝛾 +¯𝑛𝛼 +. +(13) +Here 𝜎𝛾 = 0.28 is the per-component ellipticity dispersion in each +source, and ¯𝑛𝛼 is the angular number density of sources in the 𝛼-th +redshift bin. We assume 𝑛𝛼 = 4 arcmin−2 in each redshift bin. +We compute the super-sample covariance contribution following: +CovSSC(𝐶 𝛼𝛽 +ℓ +, 𝐶𝜌𝜎 +ℓ′ ) = +∫ +d𝜒 𝑞𝛼(𝜒)𝑞𝛽(𝜒)𝑞𝜌(𝜒)𝑞𝜎(𝜒) +𝜒4 +× +(14) +𝜕𝑃(ℓ/𝜒, 𝑧) +𝜕𝛿LS +𝜕𝑃(ℓ′/𝜒, 𝑧) +𝜕𝛿LS +𝜎2 +LS(𝑧), +(15) +as in Nicola et al. (2021). 𝜕𝑃(𝑘, 𝑧)/𝜕𝛿LS is the response of the matter +power spectrum to a large-scale density fluctuation 𝛿LS, and the +quantity 𝜎2 +𝑏(𝑧) is the variance of the long wavelength mode over the +survey footprint. We estimate the latter as in Krause & Eifler (2017), +modelling the footprint simply as a circular cap of area 4𝜋 𝑓sky. We +estimate the response function using perturbation theory and the halo +model, as described in Krause & Eifler (2017), and as implemented +in CCL. +4 LIKELIHOOD +We extract cosmological parameter constraints using a Gaussian like- +lihood as described in Section 2.3. In order to validate the analytical +Parameter priors +Parameter +Prior +Parameter +Prior +Cosmology +Redshift calibration +Ωm +𝑈 (0.1, 0.9) +Δ𝑧1 +N0.0, 0.016) +Ωb +𝑈 (0.03, 0.07) +Δ𝑧2 +N(0.0, 0.017) +ℎ +𝑈 (0.55, 0.91) +Δ𝑧3 +N(0.0, 0.013) +𝑛s +𝑈 (0.87, 1.07) +Δ𝑧4 +N(0.0, 0.015) +𝜎8 +𝑈 (0.6, 0.9) +𝑝i +N( ¯𝑝𝑖, C) +Shear multiplicative bias +𝑚𝑖 +0.012 +Table 1. Prior distributions for the parameters considered in this work. Note +that the redshift calibration section contains the priors for both the Δ𝑧 and +𝑝𝛼 (𝑧) models which are not sampled simultaneously. +marginalisation approach, we will either use the full posterior dis- +tribution in Eq. 7, or the analytically marginalised version in Eq. +93. In the first case, 𝝂 includes all nuisance parameters describing +the redshift distribution uncertainties, and in both cases 𝛀 includes +all other model parameters. Specifically, 𝛀 contains the five ΛCDM +cosmological parameters (Ωm, Ωb, 𝜎8, 𝑛𝑠, ℎ). +When marginalising over redshift distribution uncertainties, 𝝂 will +contain either one redshift shift parameter Δ𝑧𝛼 for each redshift bin, +when employing the parametric description of 𝑝(𝑧) uncertainties +(Method 1), or a set of bin heights for each redshift bin determining +𝑝𝛼(𝑧), when using the non-parametric approach (Method 2). The +first case will introduce 4 new parameters to the model, while the +latter will introduce 4 × 40 = 160 new amplitude parameters, as +described in Section 3.1. +Table 1 shows the parameter priors used in this work. All cosmo- +logical parameters take uniform, largely uninformative priors. For +simplicity, the multiplicative bias parameters were fixed at the center +of the Gaussian priors from the official analysis of DES-Y1 (Ab- +bott et al. 2018a). When using Method 1 to numerically marginalise +over the 𝑝(𝑧) uncertainties, we used Gaussian priors on each of the +shift parameters Δ𝑧𝛼, following those used by DES-Y1 (Abbott et al. +2018a). When using Method 2 (marginalisation over 𝑝(𝑧) bin am- +plitudes), we assume a multi-variate Gaussian prior, with the 𝑝(𝑧) +covariance described in Sect. 3.1 and shown in Fig. 1. +For both 𝑝(𝑧) uncertainty models, when using analytical marginal- +isation, we use Eq. 9 and modify the covariance as in Eq. 10, with P +given by the priors described above. When using numerical marginal- +isation, we simply explore the posterior distribution of the full model, +including all the 𝑝(𝑧), 𝑝𝑖, parameters. In the case of Method 2, this +involves sampling a distribution with 165 parameters, of which the +bulk (160 parameters) describe the 𝑝(𝑧) uncertainty. This is not fea- +sible for standard Metropolis-Hastings MCMC methods Metropolis +et al. (1953); Hastings (1970) due to the curse of dimensionality, and +therefore we resort to a Hamiltonian Monte Carlo (HMC) approach. +HMC (MacKay 2002; Betancourt 2017) uses notions of Hamil- +tonian dynamics to draw trajectories on the parameter space along +which the sampler moves. This results in a much greater accep- +tance rate, and allows HMC to beat the dimensionality curse. HMC +can thus efficiently explore parameter spaces with large numbers of +dimensions in far less time than Metropolis-Hastings or nested sam- +pling techniques (Alsing & Handley 2021). The main difficulty of +3 Recall that we treat the term in the second line of Eq. 9 as a constant. +MNRAS 000, 1–11 (2022) + +6 +Ruiz-Zapatero et al. +0.2 +0.3 +0.4 +m +0.70 +0.75 +0.80 +S8 +0.6 +0.7 +0.8 +0.9 +8 +0.6 0.7 0.8 0.9 +8 +0.71 0.75 0.79 +S8 +z - Analytical marg. - DESY1 +z - Numerical marg. - DESY1 +z - No marg. - DESY1 +Figure 2. Marginalised posterior distributions for the combination of parame- +ters Ωm, 𝜎8 and 𝑆8 obtained when considering the Δ𝑧 model for photometric +uncertainties for DES-Y1 data. The blue contours correspond to the case +where the Δ𝑧 parameter are fixed. The magenta contours are obtained when +numerically marginalizing over the Δ𝑧 parameters. Finally, the black dashed +contours are obtained when analytically marginalizing over the Δ𝑧 parame- +ters. We can observe that the analytical and numerical marginalisation return +nearly identical posteriors. +using HMC is the need to calculate gradients of the log-posterior +to calculate the Hamiltonian equations of motion. The additional +computational cost of obtaining these derivatives numerically (e.g. +via adaptive finite differences) may outweigh the gains caused by the +higher acceptance rates of HMC. To overcome this problem we make +use of automatic differentiation (AD). To take advantage of AD, we +have developed a cosmological theoretical prediction code natively +written in the Julia programming language (Ruiz-Zapatero et al. +2023). Julia is a just-in-time (JIT) compiled language with C-like +performance and seamless AD integration, which can thus be used +to efficiently sample complex cosmological posteriors using HMC. +To sample the posterior distribution we use the No-U-Turns Sampler +(NUTS Hoffman & Gelman (2011)) implementation of HMC within +the Turing.jl package (Ge et al. 2018). +5 RESULTS +5.1 Linearising Δ𝑧 +Let us begin the discussion of our results by considering the simplest +of the two models of the photometric uncertainties studied in this +work, the Δ𝑧 model (called Method 1 above). As discussed in Section +4, this model introduces 4 new shift parameters Δ𝑧 (one per redshift +bin) in addition to the 5 ΛCDM parameters. All other nuisance +parameters are kept fixed. For the DES-Y1 and LSST-like datasets, +we will compare the result of analytically marginalizing over the Δ𝑧 +parameters against performing the full numerical marginalisation on +the corresponding cosmological constraints. In order to quantify the +contribution of redshift uncertainties to the total error budget, we will +also present results for the case when the Δ𝑧 parameters are fixed (i.e. +assuming perfect knowledge of the redshift distributions). +0.30 0.35 +m +0.82 +0.83 +0.84 +0.85 +S8 +0.75 +0.80 +0.85 +8 +0.75 +0.85 +8 +0.83 +0.85 +S8 +z - Analytical marg. - LSST +z - Numerical marg. - LSST +z - No marg. - LSST +Figure 3. Marginalised posterior distributions for the combination of parame- +ters Ωm, 𝜎8 and 𝑆8 obtained when considering the Δ𝑧 model for photometric +uncertainties for futuristic LSST-like data. The green contours correspond to +the case where the Δ𝑧 parameter are fixed. The orange contours are obtained +when numerically marginalizing over the Δ𝑧 parameters. Finally, the black +dashed contours are obtained when analytically marginalizing over the Δ𝑧 pa- +rameters. We can observe that the analytical and numerical marginalisations +return nearly identical posteriors. +Our results for DES-Y1 data are shown in Fig. 2, with the er- +rors on all parameters listed in Table 2. On the one hand, we find +that marginalizing analytically or numerically over the Δ𝑧 parame- +ters leads to the same marginalised posterior for the cosmological +parameters. On the other hand, fixing the Δ𝑧 parameters returns a +posterior distribution that is only mildly narrower than the marginal +distribution. For the DES-Y1 data, the impact of redshift uncertain- +ties in the final cosmological errors is relatively small (although not +negligible). Thus, if we truly wish to study the effect of marginal- +izing analytically as opposed to numerically over the Δ𝑧 parameters +we will have to consider futuristic LSST-like data, where the impact +of these uncertainties will likely be higher. +We show results for futuristic LSST-like data on Fig. 3, with the +parameter constraints listed in Table 2. First of all, in the case LSST- +like data we observe that not marginalising over the Δ𝑧 parameters +in the model results in significantly narrower posteriors, with the +final uncertainties shrinking by a factor ∼ 2. The impact of redshift +distribution uncertainties in this case is thus much more relevant, +and the accuracy of the analytical marginalisation scheme becomes +paramount. However, comparing the contours obtained by numerical +and analytical marginalisation, we observe that both methods return +largely equivalent posterior distributions, with the final uncertainties +changing by much less than 10%. This holds even in the case the +Δ𝑧 prior worsen by a factor 4 as seen in Figure A1, in Appendix A. +Therefore, linearizing the likelihood around the Δ𝑧 parameters will be +a good enough approximation for LSST-data, at least for relatively +simple parametrisations of the 𝑝(𝑧) uncertainty, which will allow +us to reduce the dimensionality of the model and make parameter +inference more efficient. +It is worth emphasizing that the results in this section are not +meant to be interpreted as forecasts on the constraining power of +MNRAS 000, 1–11 (2022) + +Analytical marginalisation over photo-𝑧 uncertainties +7 +Δ𝑧 model +Fixed +Numerical +Analytical +Ωm +DES-Y1 +0.333 ± 0.055 +0.3 ± 0.056 +0.306 ± 0.055 +LSST +0.311 ± 0.011 +0.317 ± 0.02 +0.317 ± 0.02 +𝜎8 +DES-Y1 +0.724 ± 0.072 +0.765 ± 0.077 +0.758 ± 0.076 +LSST +0.82 ± 0.015 +0.821 ± 0.027 +0.823 ± 0.027 +𝑆8 +DES-Y1 +0.753 ± 0.015 +0.756 ± 0.015 +0.756 ± 0.015 +LSST +0.833 ± 0.002 +0.833 ± 0.005 +0.833 ± 0.006 +Table 2. Numerical values for the mean and 1𝜎 confidence intervals for +the 1D marginalised posterior distributions of the cosmological parameters +Ωm, 𝜎8 and 𝑆8 obtained when considering the first method (𝑧 shifts) to +characterise the photometric redshift uncertainties. The first column shows +the values obtained when the Δ𝑧 parameters were kept fixed, the second +column when they were marginalised numerically and the third column when +they were marginalised analytically. In each row we display the constraints +obtained when using DES-Y1 or LSST-like data to constrain the models. +LSST on cosmological parameters, but only on our ability to ana- +lytically marginalize over photometric uncertainties in inferring the +underlying cosmology. The recovered constraints depend strongly on +assumptions such as the redshift calibration that LSST will be able to +achieve for the different samples involved. As such, the results pre- +sented here are only a conservative estimate of the effect of analytic +marginalisation on cosmological constraints. +5.2 Linearising 𝑝𝛼(𝑧) +In the previous section we have shown that, even for futuristic LSST- +like data, it is possible to marginalize over redshift uncertainties +analytically, assuming a relatively simple parametrisation of these +uncertainties. We now turn to more complex models to characterise +these uncertainties. +In order to do so we consider the previously discussed 𝑝𝛼(𝑧) +model (called Method 2 above), which turns the height of each bin +in the redshift distribution histograms into a free parameter. This +results in 40 new free parameters per redshift bin with a total of 160 +parameters for the data considered in this work. +We start by revisiting the DES-Y1 data analysis, presenting our +results in Fig. 4. As we observed in the previous section, we find that +even for the far more general 𝑝𝛼(𝑧) model there is no significant +difference between numerically marginalizing over the 𝑝𝛼(𝑧), or +doing so through our approximate analytical approach. Furthermore, +as before, fixing the shape of the redshift distribution leads to only +mildly tighter constraints. On the one hand, this means that the result +found for the Δ𝑧 model is not reliant on the simplicity of the model, +but instead inherent to the sensitivity of DES-Y1 data. On the other +hand, this also means that we must turn once again to futuristic LSST- +like data to study the impact of a more general parametrisation of +photometric uncertainties. +The results for futuristic LSST-like data are shown in Fig. 5. As in +the case of the Δ𝑧 parametrisation, we find that, in the case LSST- +like data, not including the 𝑝𝛼(𝑧) parameters in the model results +in significantly narrower posteriors. By looking at the corresponding +numerical values in Tab. 3, we see that the 𝑆8 constraints become +twice as tight when the 𝑝𝛼(𝑧) parameters are fixed. Most importantly, +we find that marginalizing over the 𝑝𝛼(𝑧) parameters analytically +or numerically yields almost indistinguishable posteriors. Thus, the +results found in Sect. 5.1 for the simple Δ𝑧 parametrisation, in fact +hold for significantly more general models of the uncertainty in the +galaxy redshift distributions. +Finally, in Fig. 6 we present the constraints obtained for the 160 +0.2 +0.3 +0.4 +m +0.70 +0.74 +0.78 +S8 +0.6 +0.7 +0.8 +0.9 +8 +0.6 0.7 0.8 0.9 +8 +0.71 0.75 0.79 +S8 +p (z) - Analytical marg. - DESY1 +p (z) - Numerical marg. - DESY1 +p (z) - No marg. - DESY1 +Figure 4. Marginalised posterior distributions for the combination of pa- +rameters Ωm, 𝜎8 and 𝑆8 obtained when considering the 𝑝𝛼 (𝑧) model for +photometric uncertainties for DES-Y1 data. The blue contours correspond to +the case where the 𝑝𝛼 (𝑧) parameter are fixed. The magenta contours are ob- +tained when numerically marginalising over the 𝑝𝛼 (𝑧) parameters. Finally, +the black dashed contours are obtained when analytically marginalizing over +the 𝑝𝛼 (𝑧) parameters. We can observe that the analytical and numerical +marginalisation return nearly identical posteriors. +0.30 0.35 +m +0.82 +0.83 +0.84 +0.85 +S8 +0.75 +0.80 +0.85 +0.90 +8 +0.75 +0.85 +8 +0.83 +0.85 +S8 +p (z) - Analytical marg. - LSST +p (z) - Numerical marg. - LSST +p (z) - No marg. - LSST +Figure 5. Marginalised posterior distributions for the combination of pa- +rameters Ωm, 𝜎8 and 𝑆8 obtained when considering the 𝑝𝛼 (𝑧) model for +photometric uncertainties for LSST-like futuristic data. The green contours +correspond to the case where the 𝑝𝛼 (𝑧) parameter are fixed. The orange +contours were obtained when numerically marginalizing over the 𝑝𝛼 (𝑧) pa- +rameters. Finally, the black dashed contours were obtained when analytically +marginalizing over the 𝑝𝛼 (𝑧) parameters. We can observe that the analytical +and numerical marginalization return nearly identical posteriors. +MNRAS 000, 1–11 (2022) + +8 +Ruiz-Zapatero et al. +𝑝𝛼 (𝑧) model +Fixed +Numerical +Analytical +Ωm +DES-Y1 +0.333 ± 0.056 +0.308 ± 0.055 +0.312 ± 0.057 +LSST +0.311 ± 0.011 +0.317 ± 0.02 +0.317 ± 0.021 +𝜎8 +DES-Y1 +0.723 ± 0.073 +0.755 ± 0.075 +0.75 ± 0.077 +LSST +0.824 ± 0.015 +0.816 ± 0.026 +0.815 ± 0.027 +𝑆8 +DES-Y1 +0.753 ± 0.015 +0.755 ± 0.015 +0.755 ± 0.015 +LSST +0.838 ± 0.002 +0.837 ± 0.006 +0.837 ± 0.006 +Table 3. Numerical values for the mean and 1𝜎 confidence intervals for the +1D marginalised posterior distributions of the cosmological parameters Ωm, +𝜎8 and 𝑆8 obtained when considering the second method (𝑝(𝑧) bin heights) +to characterise the photometric redshift uncertainties. The first column shows +the values obtained when the 𝑝𝛼 (𝑧) parameters are kept fixed, the second +column when they are marginalised numerically, and the third column when +they are marginalised analytically. In each row we display the constraints +obtained when using DES-Y1 or LSST-like data to constrain the models. +𝑝𝛼(𝑧) parameters for both the DES-Y1 (top panel) and LSST-like +data (bottom panel) in color bands. We observe that the posterior +distributions are largely dominated by the prior (shown in dashed +black line with error bars) and, thus, the redshift distribution is not +significantly self-calibrated by the data in either case. +Before moving to the next Section, it is worth stressing that con- +straining such a large parameter space has only been possible thanks +to the auto-differentiable nature of the code used to obtain theoret- +ical predictions, allowing us to use gradient-based samplers, much +more efficient that standard samplers. The development of such auto- +differentiable codes will therefore become imperative in the near fu- +ture given the increasing complexity of models used in cosmological +analyses. +5.3 Δ𝑧 vs 𝑝𝛼(𝑧) +In the previous sections we have focused in the impact of how we +marginalize over the different parametrisations of photometric red- +shift uncertainties. In this section we will focus instead on what we +marginalize over, i.e. the impact of the choice of parametrisation. +The question is then: Can a one-parameter-per-bin model (Δ𝑧 model) +capture all the meaningful modifications to photometric redshift dis- +tributions? +In order to answer this question, we constrain the cosmological +parameters for the Δ𝑧 and 𝑝𝛼(𝑧) models in the case with futuristic +LSST-like data. In both cases, we marginalize numerically over their +respective nuisance parameters. As shown in Fig. 7 and Tables 2 +and 3, both methods recover the same posterior distributions with +small differences. Thus, it is in principle possible that even Stage-IV +surveys will be able to use relatively simple models to describe the +redshift distribution of cosmic shear samples4. +6 CONCLUSIONS +One of the most significant obstacles to overcome in photometric +weak lensing surveys is the accurate modeling of redshift distribu- +tions, 𝑝(𝑧). Not only are our measurements prone to error, which can +bias the inferred cosmological parameters, but accounting for these +4 Note, however, this is likely not the case for photometric galaxy clustering +studies where other properties of the redshift distribution (e.g. its width) have +a stronger impact on the theoretical prediction (Nicola et al. 2020). +0.00 +0.05 +0.10 +0.15 +0.20 +p (z) +DESY1 +0.0 +0.5 +1.0 +1.5 +z +0.00 +0.05 +0.10 +0.15 +0.20 +p (z) +LSST +Figure 6. Posterior distributions for the 𝑝𝛼 (𝑧) parameters when considering +DES-Y1 data (top row) and futuristic LSST-like data (bottom row). The black +dashed line shows the mean of the Gaussian prior of the 𝑝𝛼 (𝑧) parameters. +The error bars show their corresponding error. +uncertainties is also a major inhibitor of efficient parameter inference. +In this paper, we investigate the impact of analytically marginalizing +over the uncertainties in the redshift distribution of galaxies in weak +lensing surveys, as initially proposed in Hadzhiyska et al. (2020). +In particular, we thoroughly quantify the validity of this approach +for a current weak lensing survey, DES, as well as for a futuristic +LSST-like survey, testing whether a fast analytic method proposed +in this work is capable of reproducing the posterior distributions and +constraints one arrives at when adopting the traditional method of +diligently varying tens or hundreds of nuisance parameters. +Our results show that, for present surveys, marginalizing over the +uncertainty in the redshift distribution of galaxies has only a mild +impact on the constraints on cosmological parameters, although one +that our analytical approximation is able to reproduce accurately. This +is true for the two parametrisations of the uncertainties considered +in this work, in terms of mean redshift shifts or redshift distribution +histogram heights. However, the impact of redshift distribution un- +certainties changes dramatically for future LSST-like surveys. In this +case, redshift uncertainties commensurate with current calibration +samples lead to an degradation in the final constraints on cosmo- +logical parameters of up to a factor ∼ 2. Capturing this effect for +an arbitrarily complex parametrisation of the redshift distribution +uncertainties is an a priori difficult task without resorting to a full +exploration of the parameter space. Nevertheless, we find that the +analytical approximate scheme explored here is still able to recover +the marginalised constraints on cosmological parameters to high +fidelity, even after marginalising over more than 100 nuisance pa- +MNRAS 000, 1–11 (2022) + +Analytical marginalisation over photo-𝑧 uncertainties +9 +0.30 0.35 +m +0.82 +0.83 +0.84 +0.85 +S8 +0.75 +0.80 +0.85 +0.90 +8 +0.6 +0.7 +0.8 +0.9 +h +0.95 +1.00 +ns +0.025 +0.050 +0.075 +b +0.04 0.06 +b +0.95 +1.00 +ns +0.6 0.7 0.8 0.9 +h +0.74 0.80 0.86 +8 +0.82 +0.84 +S8 +z - Numerical marg. - LSST +p (z) - Numerical marg. - LSST +Figure 7. Comparison between the obtained marginalised posterior distributions of the cosmological parameters when numerically marginalizing over the Δ𝑧 +(black dash-dotted) and 𝑝𝛼 (𝑧) (orange) photometric uncertainties models when applied to LSST-like futuristic data. We can observe that both prametrizations +of the photometric redshift uncertainties return identical posteriors for the cosmological parameters. +rameters. This means that, while future surveys will certainly have +to account for these uncertainties, they will be able to do so using +fast marginalisation methods without increasing the dimensionality +of their astrophysical and cosmological models. +Our results have also shown that simple parametrisations of the +redshift distribution for cosmic shear samples, in terms of shifts in +the mean redshift, are, surprisingly, able to reproduce the impact of +the full uncertainty on 𝑝(𝑧) on the final constraints to high precision. +Although this result will likely not hold for other probes (e.g. tomo- +graphic galaxy clustering), it should certainly simplify the analysis +of future cosmic shear data. +It is worth emphasizing that our work has focused exclusively on +the case of cosmic shear data, and that our conclusions only apply in +this context. The validity of the analytical approximation employed +here for general tomographic tracers of structure with uncertain ra- +dial kernels is not guaranteed, and future work should quantify its +performance on photometric clustering data – the other key probe +of the flagship “3×2pt” analysis of imaging surveys – and its cross +correlation with cosmic shear and CMB lensing data (Heymans et al. +2021; Abbott et al. 2022; García-García et al. 2021; White et al. +2022). +MNRAS 000, 1–11 (2022) + +10 +Ruiz-Zapatero et al. +ACKNOWLEDGEMENTS +We would like to thank Aně Slosar and Marius Millea for useful dis- +cussions. DA is supported by the Science and Technology Facilities +Council through an Ernest Rutherford Fellowship, grant reference +ST/P004474. PGF, CGG and AM are supported by European Re- +search Council Grant No: 693024 and the Beecroft Trust. JRZ is +supported by an STFC doctoral studentship. We made extensive use +of computational resources at the University of Oxford Department +of Physics, funded by the John Fell Oxford University Press Research +Fund. +We made extensive use of the numpy (Oliphant 2006; Van Der Walt +et al. 2011), scipy (Virtanen et al. 2020), astropy (Astropy Col- +laboration et al. 2013, 2018), healpy (Zonca et al. 2019), GetDist +Lewis (2019), and matplotlib (Hunter 2007) python packages. We +also make use of the Julia packages ForwardDiff.jl (Revels et al. +2016) and Turing.jl (Ge et al. 2018). +DATA AVAILABILITY +The code developed for this work as well as the derived datasets +produced (power spectra and covariances) are available upon request. +The catalogues and maps used were made publicly available by the +authors of the relevant papers, as described in the text. +REFERENCES +Abbott T. M. C., et al., 2018a, Phys. Rev. D, 98, 043526 +Abbott T. M. C., et al., 2018b, ApJS, 239, 18 +Abbott T. M. C., et al., 2022, Phys. Rev. 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Testing whether this +assumption might break in a realistic scenario, is therefore essential. +This is important in the context of Stage-IV since, even though it is +expected that spectroscopic samples and the associated calibration +MNRAS 000, 1–11 (2022) + +Analytical marginalisation over photo-𝑧 uncertainties +11 +0.3 +0.4 +m +0.80 +0.85 +S8 +0.8 +0.9 +8 +0.6 +0.7 +0.8 +0.9 +h +0.95 +1.00 +ns +0.04 +0.06 +b +0.04 +0.06 +b +0.95 1.00 +ns +0.6 0.7 0.8 0.9 +h +0.8 +0.9 +8 +0.80 +0.85 +S8 +z - Analytical marg. - LSST - 4 +z - Numerical marg. - LSST- 4 +Figure A1. Shows a comparison between the obtained marginalised posterior distributions of the cosmological parameters when analytically marginalizing over +the Δ𝑧 (black dashed) and when performing the full numerical marginalisation (orange) when analyzing LSST-like data. In both cases the Δ𝑧 prior distributions +where made 4 times wider. We can observe that despite significantly broadening the prior distributions the analytical marginalisation returns virtually identical +posteriors for the cosmological parameters. +techniques will improve over time, the increase in depth that LSST- +like surveys will represent may make the calibration of the faintest +samples in the survey particularly challenging. +To further stress-test our approximate method, we repeat our anal- +ysis of the LSST-like futuristic data using the Δ𝑧 model for redshift +uncertainties with priors 4 times larger than used in our fiducial +analysis (which themselves were based on existing calibration sam- +ples). The result of this test is shown in Fig. A1. Reassuringly, the +results show that, despite quadrupling the uncertainty in the redshift +nuisance parameters, the analytic marginalisation method yields vir- +tually the same constraints on the cosmological parameters as the +brute-force marginalisation, in spite of the significantly broader pos- +terior contours. This implicit validates the approximation that a first- +order expansion of the theory data vector with respect to a change +in redshift distribution is sufficient over a conservative range of cali- +bration priors. +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–11 (2022) + diff --git a/-tFLT4oBgHgl3EQfDC7x/content/tmp_files/load_file.txt b/-tFLT4oBgHgl3EQfDC7x/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3eaaf78c108ea9e31b0cfefed2758b1849a670ef --- /dev/null +++ b/-tFLT4oBgHgl3EQfDC7x/content/tmp_files/load_file.txt @@ -0,0 +1,1059 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf,len=1058 +page_content='MNRAS 000, 1–11 (2022) Preprint 31 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='0 Analytical marginalisation over photometric redshift uncertainties in cosmic shear analyses Jaime Ruiz-Zapatero1 ★, Boryana Hadzhiyska2,3, David Alonso1, Pedro G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Ferreira1, Carlos García-García1 and Arrykrishna Mootoovaloo1 1Department of Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK 2Miller Institute for Basic Research in Science, University of California, Berkeley, CA, 94720, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 3Physics Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' in original form ZZZ ABSTRACT As the statistical power of imaging surveys grows, it is crucial to account for all systematic uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' This is normally done by constructing a model of these uncertainties and then marginalizing over the additional model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The resulting high dimensionality of the total parameter spaces makes inferring the cosmological parameters significantly more costly using traditional Monte-Carlo sampling methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' A particularly relevant example is the redshift distribution, 𝑝(𝑧), of the source samples, which may require tens of parameters to describe fully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' However, relatively tight priors can be usually placed on these parameters through calibration of the associated systematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' In this paper we show, quantitatively, that a linearisation of the theoretical prediction with respect to these calibratable systematic parameters allows us to analytically marginalise over these extra parameters, leading to a factor ∼ 30 reduction in the time needed for parameter inference, while accurately recovering the same posterior distributions for the cosmological parameters that would be obtained through a full numerical marginalisation over 160 𝑝(𝑧) parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' We demonstrate that this is feasible not only with current data and current achievable calibration priors but also for future Stage-IV datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Key words: cosmology: large-scale structure of Universe – gravitational lensing: weak – methods: data analysis 1 INTRODUCTION In recent years unprecedentedly precise observations in cosmology have uncovered a number of tensions between datasets that may constitute both tantalising hints of new physics or a manifestation of a lack of control over theoretical systematics (Heymans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Riess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' At its simplest, the current cosmological paradigm, the Λ (denoting the cosmological constant) cold dark matter model (ΛCDM), can be described by only five parameters: Ω𝑚, Ω𝑏, 𝐴𝑠, 𝑛𝑠 and ℎ (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Scott (2018) for a detailed review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' However, in order to relate the theoretical predictions of this model to actual physical observables, it is necessary to extend it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Phenomenological models that describe the astrophysical systems that form the basis of our observations, as well as observational sources of systematic uncertainty, are then appended to the core ΛCDM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' In the presence of large statistical uncertainties, these models may consist of simple relationships in terms of a handful of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' However, more precise data requires an equally precise characterisation of these relationships, which leads to an increase in the complexity of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Thus, the number of parameters associated with these bridging models, colloquially referred to as “nuisance” parameters, has steadily grown over the years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The term “nuisance” is accurate when describing these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' ★ E-mail: jaime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='ruiz-zapatero@physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='uk Not only are they generally uninteresting by comparison with the fundamental cosmological parameters we aim to constraint, but the increase in parameter dimensionality of the model makes exploring their posterior distribution significantly more computationally costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Standard Markov Chain Monte-Carlo (MCMC), and other rejection- based sampling methods (Metropolis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 1953;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Alsing & Handley 2021, among others) suffer from the so-called “curse of dimensionality”, whereby the acceptance rate of new samples decreases sharply with the number of parameters (exponentially in the worst cases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Nuisance parameters can be divided into two groups based on their prior distributions: calibratable and non-calibratable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The non-calibratable parameters can only be constrained by the data and, as such, typically have largely non-constraining priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' On the other hand, we can place tighter priors on the calibratable parameters, either by accurately characterising the instrument measurements or by using independent external observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' In the case of cosmic shear analyses, the impact of galaxy intrinsic alignments (Hirata & Seljak 2004) is a standard example of a non-calibratable systematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' On the calibratable side, the two best examples are multiplicative shape measurement systematics, and the uncertainties in the redshift distribution of the target source galaxies (Hoyle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Sánchez & Bernstein 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Hildebrandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Stölzner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Of these calibratable systematics the dominant source of uncer- tainty in photometric surveys is the accuracy of redshift distributions, © 2022 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='11978v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='CO] 27 Jan 2023 2 Ruiz-Zapatero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' which are known to strongly affect the accuracy of cosmological con- straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The vital quantity to determine is the redshift distribution of each tomographic sample of galaxies, 𝑝(𝑧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The fact that the un- certainties in 𝑝(𝑧) can be calibrated with external spectroscopic data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' via direct calibration, (Lima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2020), clustering redshifts (Schneider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Newman 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Matthews & Newman 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Schmidt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2013), and shear ratios (Prat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Sánchez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2022)), enables us to place relatively strong priors on the redshift distribution, which in turn makes it possible to use approximate methods to efficiently marginalise over these uncer- tainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Analytical marginalisation schemes for photometric redshift un- certainties have already been proposed in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' In Stölzner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' (2021) an analytic marginalisation scheme for photometric red- shift uncertainties was proposed based on Gaussian mixture mod- els and applied to the analysis of KV450 data (Hildebrandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Alternatively, in Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' (2023) a resampling approach to marginalize over these uncertainties was proposed and applied to the analysis HSC data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Here, we will explore the method initially pro- posed in Hadzhiyska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' (2020), further exploited in García-García et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' (2023), and recently characterised in the context of the Laplace approximation in Hadzhiyska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The method is based on linearising the dependence of the theoretical prediction with respect to the parameters defining the redshift distribution around their cali- bration priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' This then allows one to analytically marginalise over these parameters by modifying the covariance matrix of the data, effectively assigning higher variance (as allowed by the calibration prior) to the data modes most sensitive to variations in the 𝑝(𝑧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The goal of this paper is to exhaustively validate this approximate marginalisation scheme in the context of cosmic shear analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' We will do so by proving that we are able to obtain the same constraints on cosmological parameters using this scheme, as well as employing brute-force methods that sample the full parameter space exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' We will show this for both simple parametrisations of the 𝑝(𝑧) uncertain- ties, in terms of shifts to the mean of the distribution, as well as using completely general “non-parametric” models that treat the amplitude of the 𝑝(𝑧) in narrowly-spaced intervals of 𝑧 as calibratable variables, leading to a model with more than ∼ 100 nuisance parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' In or- der to numerically marginalize over such large parameter spaces we develop an auto-differentiable code to obtain theoretical predictions for the cosmic shear observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' This allows us to employ gradi- ent based sampling algorithms, such as Hamiltonian Monte Carlo, to beat the aforementioned curse of dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Finally, we will show that the method is valid not only for current data, but also for futuristic Stage-IV surveys, where photometric redshift uncertainties will likely make up a large fraction of the total error budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Inter- estingly, our analysis will show that, in the context of cosmic shear data, relatively inexpensive parametrisations of photometric redshift uncertainties based on one free parameter per redshift bin (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' mean shifts, or ranked discrete realisations (Cordero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2022)), return ef- fectively the same posterior distribution on cosmological parameters as the most general non-parametric models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' This paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' In Section 2 we describe the methods used in this work including the theory behind weak lens- ing observables, the calibration of redshift distributions, and the mathematics of analytical marginalisation via first-order expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Section 3 presents the Dark Energy Survey data used to produce real- istic source redshift distributions and their associated uncertainties, as well as the models used to simulate future datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' In Section 4 we describe the likelihood used to analyse these data, as well as the different parametrisations used to describe 𝑝(𝑧) uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Section 5 presents our results, quantifying the performance of ana- lytical marginalisation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Finally, we present our conclusions in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2 METHODS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='1 Cosmic shear power spectra It is now commonplace to carry out the analysis of galaxy weak lensing data tomographically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The full sample is split into redshift bins and the two-point correlation functions of all pairs of bins are measured and compared with their theoretical prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Let 𝛾𝛼( ˆn) be a map of the spin-2 lensing shear field inferred from the sources in the 𝛼-th redshift bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Its relation with the three-dimensional matter overdensity 𝛿𝑚(x) is (Bartelmann & Schneider 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Krause et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2017) 𝛾𝛼( ˆn) = ∫ 𝜒𝐻 0 𝑑𝜒 𝑞𝛼(𝜒) � −𝜒−2ðð∇−2𝛿𝑚(𝜒ˆn, 𝑧) � , (1) where ˆn is the sky direction, 𝜒 is the comoving radial distance at redshift 𝑧, 𝜒𝐻 is the distance to the horizon, 𝑞𝛼(𝜒) is the weak lensing radial kernel, and ð is the spin-raising differential operator, acting on a spin-𝑠 quantity as (Newman & Penrose 1966): ð 𝑠 𝑓 (𝜃, 𝜑) = −(sin 𝜃)𝑠 � 𝜕 𝜕𝜃 + 𝑖 sin 𝜃 𝜕 𝜕𝜑 � (sin 𝜃)−𝑠 𝑠 𝑓 (2) and turning it into a spin-(𝑠 + 1) quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The weak lensing kernel is1 𝑞𝛼(𝜒) ≡ 3 2 𝐻2 0Ω𝑚 𝜒 𝑎(𝜒) ∫ ∞ 𝑧(𝜒) 𝑑𝑧′𝑝𝛼(𝑧′) 𝜒(𝑧′) − 𝜒 𝜒(𝑧′) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' (3) where 𝐻0 ≡ 𝐻(𝑧 = 0) is the Hubble expansion rate today,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Ω𝑚 is the current matter density parameter and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' most importantly for our discussion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 𝑝𝛼(𝑧) is the redshift distribution in bin 𝛼,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The angular power spectrum of the 𝐸-mode components of two maps 𝛼 and 𝛽,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 𝐶 𝛼𝛽 ℓ can be related to the three-dimensional matter power spectrum 𝑃(𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 𝑧) via: 𝐶 𝛼𝛽 ℓ = 𝐺2 ℓ ∫ 𝑑𝜒 𝜒2 𝑞𝛼(𝜒) 𝑞𝛽(𝜒) 𝑃 � 𝑘 = ℓ + 1/2 𝜒 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 𝑧(𝜒) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' (4) where we have assumed the Limber approximation (Limber 1953;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Afshordi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2004), which is valid for the broad weak lensing ker- nels considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The scale-dependent lensing prefactor, 𝐺ℓ ≡ √︄ (ℓ + 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' (ℓ − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 1 (ℓ + 1/2)2 , (5) accounts for the difference between angular and three-dimensional derivatives in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 𝜒2ð2∇−2 � 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' This prefactor leads to sub-percent differences for ℓ > 11 and can therefore be neglected on small scales (Kilbinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' In this work we will use the Halofit fitting function of Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' (2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Takahashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' (2012) to describe the matter power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The intrinsic alignment (IA) of galaxies due to local interactions (gravitational or otherwise), is an important contaminant for cosmic shear data that must be taken into account (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' For simplicity, however, and since the focus of this work is the impact of the marginalisation over redshift distribution uncertainties, we will ignore the contribution from intrinsic alignments in this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 1 Note that this is only strictly valid in ΛCDM (Ferreira 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' MNRAS 000, 1–11 (2022) Analytical marginalisation over photo-𝑧 uncertainties 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='2 Redshift distribution uncertainties The sub-samples that make up the redshift bins used in the tomo- graphic cosmic shear analysis of an imaging survey are selected based on the source photometry, either by simple cuts in the in- ferred photometric redshifts (photo-𝑧), or by selecting directly in the magnitude-color space of the sample, bypassing photo-𝑧 estimation altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Regardless of the method used to select the sub-samples, their true redshift distributions are inevitably subject to some level of uncertainty, due to the lack of precise redshift measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The 𝑝(𝑧) can however be calibrated through various methods, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' : weighted direct calibration with a sufficiently complete spectroscopic sample (Lima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2020), clustering redshifts (Schneider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Newman 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Matthews & Newman 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Schmidt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2013), and shear ratios (Prat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Sánchez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' This typically leads to relatively tight priors on the 𝑝(𝑧), but the residual uncertainties in this prior must be propagated into the final parameter constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' To characterise these uncertainties, we will make use of two dif- ferent methods, which encompass the range of model complexity we may reasonably expect from current and future data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Method 1: 𝑧 shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Most cosmic shear analyses to date (Miyazaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Hildebrandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Heymans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2018a, 2022, among others) have summarised the uncertainty in the calibrated 𝑝𝛼(𝑧) into a single parameter Δ𝑧𝛼 that shifts the mean of the redshift distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' let ˆ𝑝𝛼(𝑧) be the best-guess redshift distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The true redshift distribution is then 𝑝𝛼(𝑧) = ˆ𝑝𝛼(𝑧 + Δ𝑧𝛼).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' (6) A prior on Δ𝑧𝛼 can be derived using the calibration methods listed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' We will refer to this method as parametric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' This simple model turns out to be relatively well suited to describe the impact of 𝑝(𝑧) uncertainties in the case of cosmic shear data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Since weak lensing is a radially cumulative effect, the amplitude of the weak lensing kernel (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 3) is mostly sensitive to the mean redshift of the sample, and thus much of the effect on cosmic shear observables is well described by this parameter (Bonnett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Other modes of 𝑝(𝑧) uncertainty, such as the distribution width, may be more relevant for galaxy clustering observables, or for the intrinsic alignment contribution to cosmic shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Near-future cosmic shear samples may indeed require a more sophisticated description of the 𝑝(𝑧) uncertainty, and thus we turn to a more general method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Method 2: 𝑝(𝑧) bin heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Most 𝑝(𝑧) calibration methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' direct calibration or clustering redshifts) will produce a binned measurement of the 𝑝(𝑧) with deterministic redshift bin ranges, and uncertain bin heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The most general method to propagate these uncertainties is therefore to treat each bin height 𝑝𝑖 ≡ 𝑝(𝑧𝑖) as a free parameter in the model, with a prior given by the calibration uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The latter may be in the form of individual 1𝜎 errors for each bin height, if the uncertainties are approximately uncorrelated, or a full covariance matrix covering all bin heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The resulting parametrisation thus sidesteps any attempt at sum- marising the uncertainty into effective parameters, and thus we will refer to this method as non-parametric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The method therefore fully propagates all calibration uncertainties into the final constraints with minimal approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The key practical difference between both methods, in the context of error propagation, is the additional complexity they incur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The parametric approach (Method 1) introduces one free parameter per redshift bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' For 𝑂(5) bins, this is already enough to significantly impact the performance of standard MCMC algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' In turn, the non-parametric approach (Method 2) introduces tens or hundreds of parameters per redshift bin, and one must resort to advanced sam- pling methods in order to fully explore the resulting model without assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='3 Linearisation and analytical marginalisation Let 𝛀 be the set of non-calibratable parameters of a model (in our case this is the set of cosmological and non-calibratable nuisance parameters) and 𝝂 the set of calibratable parameter such that the total set of parameters is given by 𝜽 = 𝛀 ∪ 𝝂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Now consider the general case of a Gaussian posterior distribution of the form −2 log 𝑃(𝛀, 𝝂|d) =(d − t)𝑇 C−1(d − t) + (𝝂 − ¯𝝂)𝑇 P−1(𝝂 − ¯𝝂) − 2 log 𝑃(𝛀) + const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=', (7) where d is the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' We assume a Gaussian calibration prior with mean ¯𝝂 and covariance P, while 𝑃(𝛀) is the prior on 𝛀 (which is, as per our assumption, broad).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' t(𝛀, 𝝂) is the theoretical prediction for the data d which implicitly depends on both calibratable and non- calibratable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' C is the covariance matrix of d, which is parameter-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Assuming a tight prior on 𝝂, we start by expanding the theory prediction around ¯𝝂 t ≃ ¯t + T(𝝂 − ¯𝝂), where ¯t ≡ t(𝛀, ¯𝝂), T ≡ 𝑑t 𝑑𝝂 ����𝝂=¯𝝂 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' (8) Substituting this approximation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 7, the posterior becomes Gaus- sian in 𝝂, and thus the calibratable parameters can be marginalised analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' As shown in Hadzhiyska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' (2020), the resulting marginalised posterior is −2 log 𝑃(𝛀|d) ≃(d − ¯t)𝑇 ˜C−1(d − ¯t) − 2 log 𝑃(𝛀) + log � det � T𝑇 C−1T + P−1�� + const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=', (9) where the modified covariance is ˜C ≡ C + TPT𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' (10) Note that, strictly speaking, both the modified covariance and the term in the second line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 9 depend on𝛀, which would in principle complicate the evaluation of the likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' In practice, thisparameter dependence can be neglected such that the value of 𝛀 at which these terms are evaluated can be fixed during exploration of the posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' However, fixing 𝛀 at values with a bad fit to the data will result in a mischaracterisation of the response of the theory vector to the nuisance parameters leading to inaccurate marginalised posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Ideally, 𝛀 is fixed to its maximum a posteriori (MAP) value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' However, as shown in Hadzhiyska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' (2020) and in preliminary results, no appreciable differences are found in the marginalised posteriors for 𝛀 within 2𝜎 of the MAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Note that the size of the 2𝜎 region will depend on how constraining the data is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' This result is intuitively simple to understand if we think of T as the response of the data to variations in the nuisance parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' After marginalising over the calibratable parameters, the resulting distribution is a multi-variate Gaussian where the data covariance has been updated in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 10 by increasing the uncertainty in the data modes that most prominently respond to variations in the nuisance parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' In this work, 𝝂 corresponds to the parameters describing the red- shift distribution uncertainties, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' one shift parameter per redshift bin when using the parametric approach, or a set of 𝑝(𝑧) bin heights in the non-parametric scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The method described above, however, MNRAS 000, 1–11 (2022) 4 Ruiz-Zapatero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='20 p(z)0 p(z)1 p(z)2 p(z)3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='5 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='50 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='5 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='5 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='5 z 10 3 10 2 10 1 100 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Top row: normalized galaxies’s redshift distributions for each of the 4 redshift bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Bottom row: correlation matrix obtained using the DIR algorithm for each of the 4 galaxies’ redshift distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Note that for visualization purposes we display the absolute values of the each correlation matrix in logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' In this plot we can see that the covariance matrices obtained through the DIR algorithm are mostly diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' is fully general and has in the past been applied to marginalise over other types of nuisance parameters, including multiplicative shape measurement biases (Hildebrandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2020b), as well as truly linear parameters such as shot-noise (García-García et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2021) or system- atic template amplitudes (Koukoufilippas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The aim of this paper is thus to determine the applicability of this method to the case of redshift distribution uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 3 DATA In order to evaluate the performance of the analytical marginalisation approach described in the previous section in the context of current and future surveys, we make use of data from the first-year cosmic shear analysis of the Dark Energy Survey (DES-Y1, Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' (2018b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The aim of this is twofold: first, to demonstrate that the method can be successfully implemented in real data, with real-life complications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' noisy 𝑝(𝑧)s, numerical covariances, astrophysi- cal and observational systematics) and, second, to demonstrate this validity for future Stage-IV datasets in the presence of 𝑝(𝑧) cali- bration uncertainties already achieved on current data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' This section describes the DES-Y1 data used, and the models used to generate simulated future Stage-IV data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='1 DES-Y1 data and redshift distributions The Dark Energy Survey is a photometric, 5-year survey, that has observed 5000 deg2 of the sky using five different filter bands (grizY).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The observations were made with the 4m Blanco Telescope, provided with the 570-Mpix Dark Energy Camera (DECam), from the Cerro Tololo Inter-American Observatory (CTIO), in Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' In this paper we use cosmic shear data from the first data release (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2018b), which covers 1786 deg2 before masking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' In particular, we use the public Metacalibration source catalog2, which is divided in four redshift bins covering the range 𝑧 ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='6 (Hoyle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' We use the calibrated redshift distributions of the Metacalibra- tion sample provided by García-García et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The 𝑝(𝑧)s were estimated via direct calibration (DIR Lima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' (2008)), using the COSMOS 30-band catalog (Laigle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2016) as a calibrating sam- ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The uncertainties of the measured redshift distributions were estimated analytically, as described in García-García et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' (2023), accounting for both shot noise and sample variance, and represent a realistic level of 𝑝(𝑧) uncertainty achieved by current existing datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The redshift distributions were sampled on 40 bins of width 𝛿𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='04 covering the range 0 ≤ 𝑧 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 1 shows, in the first row, the redshift distributions of the four Metacalibration samples and their statistical uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Note that we estimated the full co- variance matrix of the 𝑝(𝑧) bin heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The covariance is dominated by the diagonal, as can be seen in the bottom panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' We will also use the cosmic shear angular power spectra provided by Nicola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' A full description of the methods used to estimate these power spectra, and their associated covariance matrix, from the DES-Y1 data is provided by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='2 Future Stage-IV data We generate a simulated data vector corresponding to a Stage-IV cosmic shear survey, such as the Legacy Survey of Space Time, at the Rubin Observatory (LSST Dark Energy Science Collaboration 2012), or the Euclid survey (Spergel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Our aim is to ef- fectively test the analytical marginalisation method in the low-noise regime, where the inferred posterior is likely more sensitive to resid- ual 𝑝(𝑧) uncertainties, and the error budget may become dominated by these, rather than the statistical errors in the data themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2 https://desdr-server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='ncsa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='edu/despublic/y1a1_ files/ MNRAS 000, 1–11 (2022) Analytical marginalisation over photo-𝑧 uncertainties 5 For simplicity, we simulate the Stage-IV survey as having the same redshift distributions as the DES-Y1 sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' This includes both the 𝑝(𝑧)s themselves, and their calibration uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' While it is possible that techniques for inferring redshifts from photometry, or the size and quality of calibrating spectroscopic samples, will improve substantially by the time Stage-IV data are available, we prefer to err on the side of caution and assume the same performance as currently achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' For instance it is possible that redshift estimates will suffer commensurately with the increase in survey depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The results presented here are therefore conservative, and their validity will only be reinforced if better 𝑝(𝑧) calibration samples are used in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' We generate cosmic shear power spectra using CCL (Chisari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2019) for the best-fit Planck 2018 cosmological parameters (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2020): Ω𝑏ℎ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='02237, Ω𝑐ℎ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='12, ℎ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='6736, 109𝐴𝑠 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='0830, 𝑛𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='9649, 𝑤0 = −1, 𝑤𝑎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' We use the same sampling in ℓ used for the DES-Y1 power spectra, and use only scales in the range ℓ ∈ [30, 2000].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' We compute the covariance matrix of these power spectra an- alytically, including a disconnected “Gaussian” component, and a connected super-sample covariance contribution (SSC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Cov � 𝐶 𝛼 ℓ , 𝐶𝜌𝜎 ℓ′ � = Cov𝐺 � 𝐶 𝛼𝛽 ℓ , 𝐶𝜌𝜎 ℓ′ � + CovSSC � 𝐶 𝛼𝛽 ℓ , 𝐶 𝜎𝜌 ℓ′ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' (11) We estimate the Gaussian covariance using a simple mode-counting approximation (Efstathiou 2004) as Cov𝐺 � 𝐶 𝛼𝛽 ℓ , 𝐶𝜌𝜎 ℓ′ � = 𝛿ℓℓ′ 𝐶 𝛼𝜌 ℓ 𝐶𝛽𝜎 ℓ + 𝐶 𝛼𝜎 ℓ 𝐶𝛽𝜌 ℓ (2ℓ + 1) Δℓ 𝑓sky , (12) where 𝑓sky is the fraction of the sky covered by the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' We assume 𝑓sky = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='4, as in the case of LSST (LSST Dark Energy Sci- ence Collaboration 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The angular power spectra above contain the contribution from shape noise in the auto-correlation, of the form 𝑁 𝛼𝛼 ℓ = 𝜎2𝛾 ¯𝑛𝛼 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' (13) Here 𝜎𝛾 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='28 is the per-component ellipticity dispersion in each source, and ¯𝑛𝛼 is the angular number density of sources in the 𝛼-th redshift bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' We assume 𝑛𝛼 = 4 arcmin−2 in each redshift bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' We compute the super-sample covariance contribution following: CovSSC(𝐶 𝛼𝛽 ℓ , 𝐶𝜌𝜎 ℓ′ ) = ∫ d𝜒 𝑞𝛼(𝜒)𝑞𝛽(𝜒)𝑞𝜌(𝜒)𝑞𝜎(𝜒) 𝜒4 × (14) 𝜕𝑃(ℓ/𝜒, 𝑧) 𝜕𝛿LS 𝜕𝑃(ℓ′/𝜒, 𝑧) 𝜕𝛿LS 𝜎2 LS(𝑧), (15) as in Nicola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 𝜕𝑃(𝑘, 𝑧)/𝜕𝛿LS is the response of the matter power spectrum to a large-scale density fluctuation 𝛿LS, and the quantity 𝜎2 𝑏(𝑧) is the variance of the long wavelength mode over the survey footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' We estimate the latter as in Krause & Eifler (2017), modelling the footprint simply as a circular cap of area 4𝜋 𝑓sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' We estimate the response function using perturbation theory and the halo model, as described in Krause & Eifler (2017), and as implemented in CCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 4 LIKELIHOOD We extract cosmological parameter constraints using a Gaussian like- lihood as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' In order to validate the analytical Parameter priors Parameter Prior Parameter Prior Cosmology Redshift calibration Ωm 𝑈 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='9) Δ𝑧1 N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='016) Ωb 𝑈 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='03, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='07) Δ𝑧2 N(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='017) ℎ 𝑈 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='55, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='91) Δ𝑧3 N(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='013) 𝑛s 𝑈 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='87, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='07) Δ𝑧4 N(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='015) 𝜎8 𝑈 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='9) 𝑝i N( ¯𝑝𝑖, C) Shear multiplicative bias 𝑚𝑖 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='012 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Prior distributions for the parameters considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Note that the redshift calibration section contains the priors for both the Δ𝑧 and 𝑝𝛼 (𝑧) models which are not sampled simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' marginalisation approach, we will either use the full posterior dis- tribution in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 7, or the analytically marginalised version in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' In the first case, 𝝂 includes all nuisance parameters describing the redshift distribution uncertainties, and in both cases 𝛀 includes all other model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Specifically, 𝛀 contains the five ΛCDM cosmological parameters (Ωm, Ωb, 𝜎8, 𝑛𝑠, ℎ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' When marginalising over redshift distribution uncertainties, 𝝂 will contain either one redshift shift parameter Δ𝑧𝛼 for each redshift bin, when employing the parametric description of 𝑝(𝑧) uncertainties (Method 1), or a set of bin heights for each redshift bin determining 𝑝𝛼(𝑧), when using the non-parametric approach (Method 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The first case will introduce 4 new parameters to the model, while the latter will introduce 4 × 40 = 160 new amplitude parameters, as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Table 1 shows the parameter priors used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' All cosmo- logical parameters take uniform, largely uninformative priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' For simplicity, the multiplicative bias parameters were fixed at the center of the Gaussian priors from the official analysis of DES-Y1 (Ab- bott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' When using Method 1 to numerically marginalise over the 𝑝(𝑧) uncertainties, we used Gaussian priors on each of the shift parameters Δ𝑧𝛼, following those used by DES-Y1 (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' When using Method 2 (marginalisation over 𝑝(𝑧) bin am- plitudes), we assume a multi-variate Gaussian prior, with the 𝑝(𝑧) covariance described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='1 and shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' For both 𝑝(𝑧) uncertainty models, when using analytical marginal- isation, we use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 9 and modify the covariance as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 10, with P given by the priors described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' When using numerical marginal- isation, we simply explore the posterior distribution of the full model, including all the 𝑝(𝑧), 𝑝𝑖, parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' In the case of Method 2, this involves sampling a distribution with 165 parameters, of which the bulk (160 parameters) describe the 𝑝(𝑧) uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' This is not fea- sible for standard Metropolis-Hastings MCMC methods Metropolis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' (1953);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Hastings (1970) due to the curse of dimensionality, and therefore we resort to a Hamiltonian Monte Carlo (HMC) approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' HMC (MacKay 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Betancourt 2017) uses notions of Hamil- tonian dynamics to draw trajectories on the parameter space along which the sampler moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' This results in a much greater accep- tance rate, and allows HMC to beat the dimensionality curse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' HMC can thus efficiently explore parameter spaces with large numbers of dimensions in far less time than Metropolis-Hastings or nested sam- pling techniques (Alsing & Handley 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The main difficulty of 3 Recall that we treat the term in the second line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 9 as a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' MNRAS 000, 1–11 (2022) 6 Ruiz-Zapatero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='4 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='80 S8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='9 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='9 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='79 S8 z - Analytical marg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' - DESY1 z - Numerical marg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' - DESY1 z - No marg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' - DESY1 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Marginalised posterior distributions for the combination of parame- ters Ωm, 𝜎8 and 𝑆8 obtained when considering the Δ𝑧 model for photometric uncertainties for DES-Y1 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The blue contours correspond to the case where the Δ𝑧 parameter are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The magenta contours are obtained when numerically marginalizing over the Δ𝑧 parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Finally, the black dashed contours are obtained when analytically marginalizing over the Δ𝑧 parame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' We can observe that the analytical and numerical marginalisation return nearly identical posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' using HMC is the need to calculate gradients of the log-posterior to calculate the Hamiltonian equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The additional computational cost of obtaining these derivatives numerically (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' via adaptive finite differences) may outweigh the gains caused by the higher acceptance rates of HMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' To overcome this problem we make use of automatic differentiation (AD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' To take advantage of AD, we have developed a cosmological theoretical prediction code natively written in the Julia programming language (Ruiz-Zapatero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Julia is a just-in-time (JIT) compiled language with C-like performance and seamless AD integration, which can thus be used to efficiently sample complex cosmological posteriors using HMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' To sample the posterior distribution we use the No-U-Turns Sampler (NUTS Hoffman & Gelman (2011)) implementation of HMC within the Turing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='jl package (Ge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 5 RESULTS 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='1 Linearising Δ𝑧 Let us begin the discussion of our results by considering the simplest of the two models of the photometric uncertainties studied in this work, the Δ𝑧 model (called Method 1 above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' As discussed in Section 4, this model introduces 4 new shift parameters Δ𝑧 (one per redshift bin) in addition to the 5 ΛCDM parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' All other nuisance parameters are kept fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' For the DES-Y1 and LSST-like datasets, we will compare the result of analytically marginalizing over the Δ𝑧 parameters against performing the full numerical marginalisation on the corresponding cosmological constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' In order to quantify the contribution of redshift uncertainties to the total error budget, we will also present results for the case when the Δ𝑧 parameters are fixed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' assuming perfect knowledge of the redshift distributions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='35 m 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='85 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='85 S8 z - Analytical marg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' - LSST z - Numerical marg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' - LSST z - No marg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' - LSST Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Marginalised posterior distributions for the combination of parame- ters Ωm, 𝜎8 and 𝑆8 obtained when considering the Δ𝑧 model for photometric uncertainties for futuristic LSST-like data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The green contours correspond to the case where the Δ𝑧 parameter are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The orange contours are obtained when numerically marginalizing over the Δ𝑧 parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Finally, the black dashed contours are obtained when analytically marginalizing over the Δ𝑧 pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' We can observe that the analytical and numerical marginalisations return nearly identical posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Our results for DES-Y1 data are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2, with the er- rors on all parameters listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' On the one hand, we find that marginalizing analytically or numerically over the Δ𝑧 parame- ters leads to the same marginalised posterior for the cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' On the other hand, fixing the Δ𝑧 parameters returns a posterior distribution that is only mildly narrower than the marginal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' For the DES-Y1 data, the impact of redshift uncertain- ties in the final cosmological errors is relatively small (although not negligible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Thus, if we truly wish to study the effect of marginal- izing analytically as opposed to numerically over the Δ𝑧 parameters we will have to consider futuristic LSST-like data, where the impact of these uncertainties will likely be higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' We show results for futuristic LSST-like data on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 3, with the parameter constraints listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' First of all, in the case LSST- like data we observe that not marginalising over the Δ𝑧 parameters in the model results in significantly narrower posteriors, with the final uncertainties shrinking by a factor ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The impact of redshift distribution uncertainties in this case is thus much more relevant, and the accuracy of the analytical marginalisation scheme becomes paramount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' However, comparing the contours obtained by numerical and analytical marginalisation, we observe that both methods return largely equivalent posterior distributions, with the final uncertainties changing by much less than 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' This holds even in the case the Δ𝑧 prior worsen by a factor 4 as seen in Figure A1, in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Therefore, linearizing the likelihood around the Δ𝑧 parameters will be a good enough approximation for LSST-data, at least for relatively simple parametrisations of the 𝑝(𝑧) uncertainty, which will allow us to reduce the dimensionality of the model and make parameter inference more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' It is worth emphasizing that the results in this section are not meant to be interpreted as forecasts on the constraining power of MNRAS 000, 1–11 (2022) Analytical marginalisation over photo-𝑧 uncertainties 7 Δ𝑧 model Fixed Numerical Analytical Ωm DES-Y1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='333 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='306 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='055 LSST 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='311 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='317 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='317 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='02 𝜎8 DES-Y1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='724 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='072 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='765 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='077 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='758 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='076 LSST 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='821 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='823 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='027 𝑆8 DES-Y1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='753 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='756 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='756 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='015 LSST 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='833 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='833 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='833 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='006 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Numerical values for the mean and 1𝜎 confidence intervals for the 1D marginalised posterior distributions of the cosmological parameters Ωm, 𝜎8 and 𝑆8 obtained when considering the first method (𝑧 shifts) to characterise the photometric redshift uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The first column shows the values obtained when the Δ𝑧 parameters were kept fixed, the second column when they were marginalised numerically and the third column when they were marginalised analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' In each row we display the constraints obtained when using DES-Y1 or LSST-like data to constrain the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' LSST on cosmological parameters, but only on our ability to ana- lytically marginalize over photometric uncertainties in inferring the underlying cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The recovered constraints depend strongly on assumptions such as the redshift calibration that LSST will be able to achieve for the different samples involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' As such, the results pre- sented here are only a conservative estimate of the effect of analytic marginalisation on cosmological constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='2 Linearising 𝑝𝛼(𝑧) In the previous section we have shown that, even for futuristic LSST- like data, it is possible to marginalize over redshift uncertainties analytically, assuming a relatively simple parametrisation of these uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' We now turn to more complex models to characterise these uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' In order to do so we consider the previously discussed 𝑝𝛼(𝑧) model (called Method 2 above), which turns the height of each bin in the redshift distribution histograms into a free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' This results in 40 new free parameters per redshift bin with a total of 160 parameters for the data considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' We start by revisiting the DES-Y1 data analysis, presenting our results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' As we observed in the previous section, we find that even for the far more general 𝑝𝛼(𝑧) model there is no significant difference between numerically marginalizing over the 𝑝𝛼(𝑧), or doing so through our approximate analytical approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Furthermore, as before, fixing the shape of the redshift distribution leads to only mildly tighter constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' On the one hand, this means that the result found for the Δ𝑧 model is not reliant on the simplicity of the model, but instead inherent to the sensitivity of DES-Y1 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' On the other hand, this also means that we must turn once again to futuristic LSST- like data to study the impact of a more general parametrisation of photometric uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The results for futuristic LSST-like data are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' As in the case of the Δ𝑧 parametrisation, we find that, in the case LSST- like data, not including the 𝑝𝛼(𝑧) parameters in the model results in significantly narrower posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' By looking at the corresponding numerical values in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 3, we see that the 𝑆8 constraints become twice as tight when the 𝑝𝛼(𝑧) parameters are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Most importantly, we find that marginalizing over the 𝑝𝛼(𝑧) parameters analytically or numerically yields almost indistinguishable posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Thus, the results found in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='1 for the simple Δ𝑧 parametrisation, in fact hold for significantly more general models of the uncertainty in the galaxy redshift distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Finally, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 6 we present the constraints obtained for the 160 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='3 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='79 S8 p (z) - Analytical marg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' - DESY1 p (z) - Numerical marg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' - DESY1 p (z) - No marg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' - DESY1 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Marginalised posterior distributions for the combination of pa- rameters Ωm, 𝜎8 and 𝑆8 obtained when considering the 𝑝𝛼 (𝑧) model for photometric uncertainties for DES-Y1 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The blue contours correspond to the case where the 𝑝𝛼 (𝑧) parameter are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The magenta contours are ob- tained when numerically marginalising over the 𝑝𝛼 (𝑧) parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Finally, the black dashed contours are obtained when analytically marginalizing over the 𝑝𝛼 (𝑧) parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' We can observe that the analytical and numerical marginalisation return nearly identical posteriors.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='85 S8 p (z) - Analytical marg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' - LSST p (z) - Numerical marg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' - LSST p (z) - No marg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' - LSST Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Marginalised posterior distributions for the combination of pa- rameters Ωm, 𝜎8 and 𝑆8 obtained when considering the 𝑝𝛼 (𝑧) model for photometric uncertainties for LSST-like futuristic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The green contours correspond to the case where the 𝑝𝛼 (𝑧) parameter are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The orange contours were obtained when numerically marginalizing over the 𝑝𝛼 (𝑧) pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Finally, the black dashed contours were obtained when analytically marginalizing over the 𝑝𝛼 (𝑧) parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' We can observe that the analytical and numerical marginalization return nearly identical posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' MNRAS 000, 1–11 (2022) 8 Ruiz-Zapatero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 𝑝𝛼 (𝑧) model Fixed Numerical Analytical Ωm DES-Y1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='333 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='308 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='312 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='057 LSST 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='311 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='317 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='317 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='021 𝜎8 DES-Y1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='723 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='073 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='755 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='077 LSST 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='824 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='816 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='815 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='027 𝑆8 DES-Y1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='753 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='755 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='755 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='015 LSST 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='838 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='837 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='837 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='006 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Numerical values for the mean and 1𝜎 confidence intervals for the 1D marginalised posterior distributions of the cosmological parameters Ωm, 𝜎8 and 𝑆8 obtained when considering the second method (𝑝(𝑧) bin heights) to characterise the photometric redshift uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The first column shows the values obtained when the 𝑝𝛼 (𝑧) parameters are kept fixed, the second column when they are marginalised numerically, and the third column when they are marginalised analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' In each row we display the constraints obtained when using DES-Y1 or LSST-like data to constrain the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 𝑝𝛼(𝑧) parameters for both the DES-Y1 (top panel) and LSST-like data (bottom panel) in color bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' We observe that the posterior distributions are largely dominated by the prior (shown in dashed black line with error bars) and, thus, the redshift distribution is not significantly self-calibrated by the data in either case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Before moving to the next Section, it is worth stressing that con- straining such a large parameter space has only been possible thanks to the auto-differentiable nature of the code used to obtain theoret- ical predictions, allowing us to use gradient-based samplers, much more efficient that standard samplers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The development of such auto- differentiable codes will therefore become imperative in the near fu- ture given the increasing complexity of models used in cosmological analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='3 Δ𝑧 vs 𝑝𝛼(𝑧) In the previous sections we have focused in the impact of how we marginalize over the different parametrisations of photometric red- shift uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' In this section we will focus instead on what we marginalize over, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' the impact of the choice of parametrisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The question is then: Can a one-parameter-per-bin model (Δ𝑧 model) capture all the meaningful modifications to photometric redshift dis- tributions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' In order to answer this question, we constrain the cosmological parameters for the Δ𝑧 and 𝑝𝛼(𝑧) models in the case with futuristic LSST-like data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' In both cases, we marginalize numerically over their respective nuisance parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 7 and Tables 2 and 3, both methods recover the same posterior distributions with small differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Thus, it is in principle possible that even Stage-IV surveys will be able to use relatively simple models to describe the redshift distribution of cosmic shear samples4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 6 CONCLUSIONS One of the most significant obstacles to overcome in photometric weak lensing surveys is the accurate modeling of redshift distribu- tions, 𝑝(𝑧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Not only are our measurements prone to error, which can bias the inferred cosmological parameters, but accounting for these 4 Note, however, this is likely not the case for photometric galaxy clustering studies where other properties of the redshift distribution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' its width) have a stronger impact on the theoretical prediction (Nicola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='20 p (z) DESY1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='5 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='20 p (z) LSST Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Posterior distributions for the 𝑝𝛼 (𝑧) parameters when considering DES-Y1 data (top row) and futuristic LSST-like data (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The black dashed line shows the mean of the Gaussian prior of the 𝑝𝛼 (𝑧) parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The error bars show their corresponding error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' uncertainties is also a major inhibitor of efficient parameter inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' In this paper, we investigate the impact of analytically marginalizing over the uncertainties in the redshift distribution of galaxies in weak lensing surveys, as initially proposed in Hadzhiyska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' In particular, we thoroughly quantify the validity of this approach for a current weak lensing survey, DES, as well as for a futuristic LSST-like survey, testing whether a fast analytic method proposed in this work is capable of reproducing the posterior distributions and constraints one arrives at when adopting the traditional method of diligently varying tens or hundreds of nuisance parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Our results show that, for present surveys, marginalizing over the uncertainty in the redshift distribution of galaxies has only a mild impact on the constraints on cosmological parameters, although one that our analytical approximation is able to reproduce accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' This is true for the two parametrisations of the uncertainties considered in this work, in terms of mean redshift shifts or redshift distribution histogram heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' However, the impact of redshift distribution un- certainties changes dramatically for future LSST-like surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' In this case, redshift uncertainties commensurate with current calibration samples lead to an degradation in the final constraints on cosmo- logical parameters of up to a factor ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Capturing this effect for an arbitrarily complex parametrisation of the redshift distribution uncertainties is an a priori difficult task without resorting to a full exploration of the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Nevertheless, we find that the analytical approximate scheme explored here is still able to recover the marginalised constraints on cosmological parameters to high fidelity, even after marginalising over more than 100 nuisance pa- MNRAS 000, 1–11 (2022) Analytical marginalisation over photo-𝑧 uncertainties 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='30 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='86 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='84 S8 z - Numerical marg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' - LSST p (z) - Numerical marg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' - LSST Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Comparison between the obtained marginalised posterior distributions of the cosmological parameters when numerically marginalizing over the Δ𝑧 (black dash-dotted) and 𝑝𝛼 (𝑧) (orange) photometric uncertainties models when applied to LSST-like futuristic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' We can observe that both prametrizations of the photometric redshift uncertainties return identical posteriors for the cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' This means that, while future surveys will certainly have to account for these uncertainties, they will be able to do so using fast marginalisation methods without increasing the dimensionality of their astrophysical and cosmological models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Our results have also shown that simple parametrisations of the redshift distribution for cosmic shear samples, in terms of shifts in the mean redshift, are, surprisingly, able to reproduce the impact of the full uncertainty on 𝑝(𝑧) on the final constraints to high precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Although this result will likely not hold for other probes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' tomo- graphic galaxy clustering), it should certainly simplify the analysis of future cosmic shear data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' It is worth emphasizing that our work has focused exclusively on the case of cosmic shear data, and that our conclusions only apply in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The validity of the analytical approximation employed here for general tomographic tracers of structure with uncertain ra- dial kernels is not guaranteed, and future work should quantify its performance on photometric clustering data – the other key probe of the flagship “3×2pt” analysis of imaging surveys – and its cross correlation with cosmic shear and CMB lensing data (Heymans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' García-García et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' White et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' MNRAS 000, 1–11 (2022) 10 Ruiz-Zapatero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We would like to thank Aně Slosar and Marius Millea for useful dis- cussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' DA is supported by the Science and Technology Facilities Council through an Ernest Rutherford Fellowship, grant reference ST/P004474.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' PGF, CGG and AM are supported by European Re- search Council Grant No: 693024 and the Beecroft Trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' JRZ is supported by an STFC doctoral studentship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' We made extensive use of computational resources at the University of Oxford Department of Physics, funded by the John Fell Oxford University Press Research Fund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' We made extensive use of the numpy (Oliphant 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Van Der Walt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2011), scipy (Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2020), astropy (Astropy Col- laboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2013, 2018), healpy (Zonca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2019), GetDist Lewis (2019), and matplotlib (Hunter 2007) python packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' We also make use of the Julia packages ForwardDiff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='jl (Revels et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2016) and Turing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='jl (Ge et al.' 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H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=', van den Busch J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=', Heymans C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=', 2020, A&A, 637, A100 Zhang T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=', Rau M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=', Mandelbaum R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=', Li X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=', Moews B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=', 2023, MNRAS, 518, 709 Zonca A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=', Singer L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=', Lenz D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=', Reinecke M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=', Rosset C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=', Hivon E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=', Gorski K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=', 2019, Journal of Open Source Software, 4, 1298 APPENDIX A: STRESS-TESTING THE APPROXIMATION As described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' 2, the approximation used here to analytically marginalise over the redshift calibration parameters assumes a suffi- ciently tight prior on these parameters, such that the dependence of the theory prediction on them can be linearised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Testing whether this assumption might break in a realistic scenario, is therefore essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' This is important in the context of Stage-IV since, even though it is expected that spectroscopic samples and the associated calibration MNRAS 000, 1–11 (2022) Analytical marginalisation over photo-𝑧 uncertainties 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='4 m 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='9 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content='85 S8 z - Analytical marg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' - LSST - 4 z - Numerical marg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' - LSST- 4 Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Shows a comparison between the obtained marginalised posterior distributions of the cosmological parameters when analytically marginalizing over the Δ𝑧 (black dashed) and when performing the full numerical marginalisation (orange) when analyzing LSST-like data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' In both cases the Δ𝑧 prior distributions where made 4 times wider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' We can observe that despite significantly broadening the prior distributions the analytical marginalisation returns virtually identical posteriors for the cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' techniques will improve over time, the increase in depth that LSST- like surveys will represent may make the calibration of the faintest samples in the survey particularly challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' To further stress-test our approximate method, we repeat our anal- ysis of the LSST-like futuristic data using the Δ𝑧 model for redshift uncertainties with priors 4 times larger than used in our fiducial analysis (which themselves were based on existing calibration sam- ples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' The result of this test is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' Reassuringly, the results show that, despite quadrupling the uncertainty in the redshift nuisance parameters, the analytic marginalisation method yields vir- tually the same constraints on the cosmological parameters as the brute-force marginalisation, in spite of the significantly broader pos- terior contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' This implicit validates the approximation that a first- order expansion of the theory data vector with respect to a change in redshift distribution is sufficient over a conservative range of cali- bration priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} +page_content=' MNRAS 000, 1–11 (2022)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfDC7x/content/2301.11978v1.pdf'} diff --git 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b/09E2T4oBgHgl3EQf5AgM/content/tmp_files/2301.04185v1.pdf.txt @@ -0,0 +1,931 @@ +Synthesis and processing of lithium-loaded plastic +scintillators on the kilogram scale +Michael J. Forda*, Elisabeth Aigeldingera, Felicia Sutantoa, Natalia P. Zaitsevaa, +Viacheslav A. Lia, M. Leslie Carmana, Andrew Glenna, Cristian R. Catalaa, Steven A. +Dazeleya, Nathaniel Bowdena*1 + +aLawrence Livermore National Laboratory +7000 East Avenue, Livermore, CA 94550 + +Abstract +Plastic scintillators that can discriminate between gamma rays, fast neutrons, and +thermal neutrons were synthesized and characterized while considering the balance +between processing and performance at the kilogram scale. These trade-offs were +necessitated by the inclusion of 0.1 wt. % lithium-6 to enable detection of thermal +neutrons. The synthesis and processing of these plastic scintillators on the kilogram +scale required consideration of many factors. First, a comonomer (methacrylic acid) +was used to solubilize salts of lithium-6, which allow for a thermal-neutron capture +reaction that produces scintillation light following energy transfer. Second, scintillation +performance and processability were considered because the increasing content of the +comonomer resulted in a sharp decrease in the light output. The use of small amounts +of comonomer (≤3 wt. %) resulted in better performance but required high processing +temperatures. At large scales, these high temperatures could initiate an exothermic +polymerization that results in premature curing and/or defects. The deleterious effects +of the comonomer may be mitigated by using m-terphenyl as a primary dye rather than +2,5-diphenyloxazole (PPO), which has been traditionally used in organic scintillators. +Finally, the curing environment was controlled to avoid defects like cracking and +discoloration while maintaining solubility of dopants during curing. For scintillators +that were produced from kilogram-scale batches of precursors, the effective attenuation +of scintillation light was characterized. + +* This is to indicate the corresponding author. + Email address: bowden20@llnl.gov; ford40@llnl.gov + + +Keywords: pulse-shape discrimination, plastic scintillators, inverse beta decay, neutron +detection, large-scale detectors + +1. Introduction +The rising cost of fossil fuels and concerns about greenhouse gas emissions +motivate a revival of nuclear power generation.[1] However, the expansion of nuclear +power throughout the world and the construction of novel reactor types may challenge +the resources available for implementing conventional safeguards. Direct and +nonintrusive measurements of reactor operation using offer one approach to address +this concern. +These measurements can be facilitated by the development of novel detectors +to enable near-field (ca. 10-100 m) monitoring of antineutrinos produced by a +reactor.[2–10] Previous reports highlight the ability of organic scintillators to monitor +this antineutrino flux. For example, the Precision Reactor Oscillation and Spectrum +Experiment (PROSPECT)[11] recently demonstrated the measurement of the +antineutrino spectrum from 235U at the High Flux Isotope Reactor at Oak Ridge +National Laboratory. Like in many experiments that use organic scintillators to monitor +antineutrino flux, PROSPECT utilized about 4 tons of a liquid scintillator loaded with +a neutron capture agent (6Li in this case), which can detect signatures of inverse beta +decay. In this experiment, the scintillation light is associated with signatures of inverse +beta decay using a measurement scheme called pulse-shape discrimination (PSD).[12] +After monitoring a scintillation pulse over time, a prompt signal of scintillation light +can be associated with the antineutrino energy, and the delayed signal of scintillation +light can be associated with neutron capture by 6Li. Detectors like PROSPECT contain +scintillator with this capability of PSD, making them useful for uniquely identifying +capture reactions and rejecting fast-neutron background events. +For continued development of novel detectors, the phase of the detector +material as it relates to the mobility of the detector may be considered. Liquid +scintillators are relatively inexpensive, are easy to manufacture, and have good + +performance. However, liquid scintillators may require consideration of potential +hazards (e.g., flammability), handling, and storage when used in mobile detectors. +Conversely, plastic scintillators are less hazardous than liquid scintillators since they +are in the solid state, and plastic scintillators are self-supporting, which is useful for +mobility. However, plastic scintillators have not been widely available as materials +capable of PSD until recently[13,14]. Additionally, the PSD performance of plastic +scintillators worsens as the length of the scintillator increases due to light attenuation; +this attenuation is detrimental to the performance of large-volume (ton-scale) +detectors.[15] Thus, a mobile detector may require further consideration of these trade- +offs and further development of detector materials like plastic scintillators. +One development that is important for plastic scintillators is the capability of +these materials to discriminate between gamma rays, fast neutrons, and thermal +neutrons (Figure 1a). This capability necessitates the inclusion of a neutron capture +agent like 155Gd, 157Gd, 10B, or 6Li, and various reports describe attempts to incorporate +neutron capture agents into plastic scintillators. In particular, the doping of scintillators +with 6Li may be preferable since the 6Li(n,t) capture reaction produces a localized, +mono-energetic energy deposition that can be efficiently identified via PSD and energy +selections.[16,17] +Carboxylate salts of 6Li have been previously used to incorporate 6Li into +plastic scintillators while obtaining materials that are transparent. In one example, a 6Li +salt of methacrylic acid (MAA) was copolymerized with styrene. The 6Li salt of MAA +was not soluble in the plastic scintillator precursors at appreciable amounts; additional +MAA was needed to dissolve the 6Li salt of MAA and increase the 6Li content. This +necessity of additional MAA highlights how solubility of the polar 6Li compounds in +the nonpolar matrix must be considered to produce plastic scintillators. Despite the need +for additional MAA, these scintillators were promising for thermal-neutron detection. +At a small scale (≈ 1 cm), the plastic scintillator was responsive to an incident beam of +thermal neutrons from a research reactor.[18] However, the optical attenuation +properties were not assessed for this material but become important for large-volume +detectors, where the longest side of a single scintillator may be on the order of 10-100 +cm. + +Other developments of these scintillators that contain 6Li focused on +exploration of additional carboxylates.[19–22] One report described an investigation of +16 different 6Li salts. These 6Li salts were dissolved in a comonomer mixture of styrene +and methacrylic acid (90:10 styrene:MAA) to determine the maximum solubility of 6Li +at 60 oC. Additional 6Li could be dissolved as the MAA content increased, but the +scintillation light output and figure of merit (FoM) for PSD decreased as MAA and 6Li +content increased.[22] Thus, improvements in scintillation performance should +consider the balance between processing and performance. This consideration is +especially important in large plastic scintillators where thermal runaway[23] becomes +a concern for processing, and attenuation becomes a concern for performance. +In this report, we describe the synthesis and processing of lithium-loaded +plastic scintillators on the kilogram scale. We considered aspects related to the +composition of the plastic scintillator like the primary dye, secondary dye, monomers, +and lithium salts (Figure 1b) as well as aspects related to processing and curing like +dissolution and temperature of cure (Figure 1c). We first synthesized various 6Li salts +and characterized their solubility at different temperatures and with various +concentrations of comonomers. Then, we considered trade-offs in processing and +performance at the 10 g scale by evaluating the scintillation performance upon addition +of comonomer and 6Li salt in plastic scintillators that contained 2,5-diphenyloxazole +(PPO) as the primary dye. The light output of scintillation was reduced as the content +of the comonomer MAA increased in these scintillators. +We then compared the performance of plastic scintillators that contain PPO +vs. m-terphenyl (mTP) as the primary dye. The performance of plastic scintillators that +contain mTP as the primary dye is less sensitive to the addition of MAA. When scaling +our synthesis to 1 kg (Figure 1d), we targeted compositions that were available at scale +and allowed for lower temperatures of processing to avoid thermally initiated +polymerization and thermal runaway. All of these measures considered the potential for +production of large-scale plastic scintillators, which will be useful for applications like +mobile antineutrino detectors. + + +2. Materials and methods +Styrene (99%, Sigma), vinyl toluene (99%, Sigma), divinyl benzene (Sigma, +technical grade), and methyl methacrylate (99%, VWR) were passed through an +alumina column to remove inhibitor. Methacrylic acid (99%, Sigma) was dried over +sodium chloride and distilled under vacuum to remove inhibitor. All monomers were +sparged with nitrogen for > 30 min before being stored in a nitrogen-filled glovebox. +Figure 1. a) Plastic scintillators that are capable of PSD and contain lithium-6 distinguish between gamma rays, fast +neutrons, and thermal neutrons, as shown in this PSD plot. The gradient scale bar represents a relative population of data +points. b) The synthesis of plastic scintillators requires consideration of many components, as shown in this schematic. +The chemical structures, starting from the bottom left and going clockwise, are those of m-terphenyl, 2,5-diphenyloxazole, +Exalite 404, styrene, methacrylic acid, divinylbenzene, and three 6Li salts of carboxylic acids. c) The processing of the +precursors requires control of dissolution and curing conditions. d) Control of synthesis and processing has enabled the +production of large plastic scintillators, scaling to rectangular prisms like the ones in this photograph. The large +scintillators are about 0.41 m in length (total mass of about 1.5 kg). + + +0.8 +.0 +0.0 +0.5 +0.8 +Thermal neutrons +0.7 +Fas, neurone +0.3 +0.4 +Gamma raye +0.2 +0.2 +0.1 +100 200 +300 +4010 +Approt, enengy +e!All monomers except methacrylic acid were stored in an inert atmosphere at −20 oC. +Methacrylic acid was stored at room temperature. m-terphenyl (mTP, Smolecule) was +purified by recrystallization from toluene. L-231 (Luperox, 1,1-di(t-butylperoxy)-3,3,5- +trimethylcyclohexane) was used as a radical initiator after sparging for > 30 min with +dry nitrogen and kept at −20 °C until needed. 2,5-diphenyloxazole (PPO, scintillation +grade from Sigma), 1,4-bis(2-methylstyryl)benzene (bisMSB, Luxottica/Exciton), and +1,4-bis(9,9-diethyl-7-(tert-pentyl)-9H-fluoren-2-yl)benzene (E404, Luxottica/Exciton) +were used as received without further purification. +6Li carboxylate salts were synthesized by first suspending 6Li2CO3 (National +Isotope Development Center) in a 1:1 mixture of water (deionized) and methanol +(99.8%, Sigma). Excess carboxylic acid (1.02 equivalent excess) was mixed into a 1:1 +mixture of water and methanol. The basic suspension was slowly added to the acid +solution, and this mixture was heated to reflux for > 4 hours. The solution was filtered, +and the 6Li salt was precipitated by adding excess volume of cold acetone. The 6Li salt +was collected by vacuum filtration and washed with acetone, followed by drying under +vacuum at 80 oC. This procedure was used for carboxylate salts of pentanoic acid, +hexanoic acid, octanoic acid, 2-methylpropanoic acid, 2-methylbutanoic acid, 3- +methylbutanoic acid, and 2-ethylhexanoic acid. All acids were purchased from Sigma +or VWR and used as received. +Plastics were synthesized in a dry nitrogen environment. For initial evaluation, +plastics were synthesized using 10 g of precursor materials. For the production of large +plastic scintillators, the amount of precursors used was up to 2.7 kg. All materials that +were not stored in a dry nitrogen environment were dried under vacuum. 6Li salts were +easiest to process when they were first dissolved in a 1:1 mixture by weight of +styrene:MAA at elevated temperatures (about 60-80 oC) before adding the remainder +of the plastic composition. A typical synthesis would involve dissolving the primary +dye (e.g., PPO) and the secondary dye (e.g., bis-MSB or E404) in styrene or vinyl +toluene (VT). The monomer VT was used as the polymer matrix for plastics that used +mTP as the primary dye. VT was used for these plastics as we observed less consistency +in solubility for mTP-based plastics that used styrene. The performance of plastic +scintillators that used styrene and VT were compared, and we observed no meaningful + +difference in performance for these plastics. This precursor solution was heated to about +60-80 oC and mixed with a solution of the 6Li salt in 1:1 mixture by weight of +styrene:MAA. DVB (typically 5 wt. %) and initiator (0.08 wt. % for 10 g plastics; 0.01 +wt. % for plastics batches greater than 400 g) were added. This mixture that contains +the precursor solution mentioned earlier and the solution that contains 6Li was poured +into a mould and sealed. The mould was placed in a nitrogen-filled oven and cured at +elevated temperatures. In one experiment, the viscosity of the precursor solution was +monitored using a rotary viscometer (Brookfield DV2T). +A typical curing profile would consist of heating for 7 days at 60 oC, followed +by a temperature ramp to 75 oC over one day. The scintillators were cured in convection +ovens (Cascade TEK) that were fitted with gas lines. Dry nitrogen flowed into solvent- +resistance plastic bags that contained the mould inside the oven. The bags maintained +a positive pressure of nitrogen. The plastic would stay at 75 oC for four days and then +cool to room temperature over the course of one day. Following curing, the scintillators +were removed from the moulds, then machined and polished. All photographs of +samples were taken using a Nikon D750 and were globally edited in Adobe Lightroom +for colour and exposure corrections. +For initial scintillator characterization, samples of mass equal to 10 g were +measured. The outer edge and one face of the scintillators were wrapped and covered +with Teflon tape. The exposed face was coupled with optical grease to a Hamamatsu +R6231-100-SEL photomultiplier tube (PMT). Signals from the PMT were recorded at +a sampling rate of 200 MS/s using a 14-bit CompuScope 14200 waveform digitizer. A +relative quantification of light output (LO) was measured using ionizing radiation from +137Cs incident upon the plastic scintillator. The values of LO that we report in this +manuscript are specific to our measurement system and thus should only be used for +relative comparison. We normalized the value of LO to measurements of the +commercial scintillator EJ-200. The location of 500 keVee was defined by the value of +the pulse integral at 50% of the height of the 137Cs Compton edge. For many +measurements, duplicate samples were synthesized, and averages are reported. For one +condition, 9 samples were replicated in multiple batches and were measured. The +standard deviation of these measurements was within 7% of the average value, which + +could be representative of the standard deviation related to contributions from +measurement and synthesis. Where standard deviation is not reported, a conservative +value of 10% of the value given could be assumed. +The measurement of effective attenuation length was performed with a longer +scintillator bar (1″ x 1″ x 16″). This measurement employed a setup that was identical +to the setup used to characterize scintillator bars in an antineutrino detector called +SANDD (Segmented AntiNeutrino Directional Detector).[24] This bar was wrapped +with polytetrafluoroethylene tape (POLY-TEMP PN-16050), and a pair of 1" +Hamamatsu R1924A-100 PMTs were mounted at either end of the scintillator bar using +EJ-550 silicone optical grease. The PMT operating voltage was set at -1100 V, and +signals were digitized using a Struck SIS3316 digitizer module (250 MS/s, 14 bit, 5 V +dynamic range). The energy threshold was set at approximately 0.1 MeVee, and 1600 +ns-long waveforms were sent to disk and stored in ROOT data format. The charge +response difference between the two PMTs due to gain and optical coupling variation +was corrected using a collimated 137Cs gamma-ray source directed at the center of the +plastic bar. Here, lead bricks were used for shaping the gamma-ray source into a fan +beam of about 0.5 cm width. +To obtain the effective attenuation length of the scintillator bar, we performed +a series of collimated 22Na measurements at regular intervals along the length of the +bar. In each measurement, we identified the location of the Compton continuum +maximum of the 1.275 MeV gamma-ray by fitting the energy response with a Gaussian +profile while varying the range of the fit to find the minimum χ2 (best fit). The Compton +continuum maximum position was identified as the mean of the Gaussian profile that +yielded the minimum χ2. The associated uncertainty was estimated by varying the range +of the fit until the χ2 value exceeded the 68% confidence level (CL) of the minimum χ2; +the uncertainty was the corresponding range of the mean of the Gaussian profile. +To measure PSD in smaller plastics, plastic scintillators were exposed to a +252Cf source. The source was shielded behind 5.1 cm of lead to reduce the gamma-ray +flux. To obtain a flux of thermal neutrons, high density polyethylene was also used as +a moderator for 252Cf. The measurements of scintillation from plastic scintillators + +exposed to 252Cf were integrated over time to determine the total charge (Qtotal). The +charge of the delayed component of the signal (Qtail) was determined from a delayed +fraction of the scintillation pulse. Scintillation pulses due to interactions of the +scintillator with neutrons have a larger fraction of Qtail relative to Qtotal; therefore, a +comparison of Qtail relative to Qtotal can be used to distinguish between scintillation due +to neutrons vs. gamma rays. The PSD was quantified using a figure of merit (FoM) that +is determined from histograms of the ratio of the charge of the delayed component +relative to the total charge, as described in previous reports[15,25]. Briefly, the FoM is: +𝐹𝑜𝑀 = +〈𝑛,𝑡〉−〈𝛾〉 +𝐹𝑊𝐻𝑀𝑛,𝑡+𝐹𝑊𝐻𝑀𝛾 + + +(2) +In this equation, 〈n,t〉-〈γ〉 represents the difference between the average value of the +neutron and gamma-ray signals, and FWHMn,t +FWHMγ represents the sum of the full- +width at half of the maximum value of the distributions of the thermal-neutron and +gamma-ray signals at the electron-equivalent energy of the thermal-neutron spot. For +plastics that don’t contain 6Li, the same equation was used for FoM, but the position +and FWHM of the neutron peak is used at an electron-equivalent energy near the 137Cs +Compton edge. +For PSD of the larger scintillator bar (1″ × 1″ × 16″), an identical setup as the +measurement of effective attenuation length was used. The bar was irradiated with an +uncollimated 252Cf source, and lead bricks with a total thickness of 6″ were placed +between the detector and the 252Cf source to reduce the gamma-ray flux. The charge +integration limits were optimized, and the best parameters were found to be [tL-20 ns ≤ +Qtotal ≤ tL+1300 ns] and [tL+24 ns ≤ Qtail ≤ tL+1300 ns], where tL is the leading edge of +the waveform. Assuming light transport behaves exponentially along the length of the +scintillator bar, we can eliminate the dependence of energy on event position by +reconstructing the energy as 𝐸 = √𝐸𝐴𝐸𝐵 , where EA and EB are the charges collected +by the two PMTs. + +3. Results and discussion +3a. Selection of lithium-6 salt based on solubility at moderate +temperatures + +To make large-scale production of plastic scintillators easier, our selection of +materials focused on those materials that had simple processing requirements (e.g., +temperatures below 80 oC). This requirement is necessary since high processing +temperatures for large plastic scintillators could thermally initiate the polymerization. +After dissolution of all dopants, we monitored the viscosity over time of liquid +precursors with and without the crosslinker divinylbenzene (DVB) at a temperature of +50 oC (Figure 2). The liquid precursors did not contain a radical initiator that initiates +polymerization; thus, any increase in viscosity is due to thermally initiated +polymerization. For precursor liquids that contained DVB, the viscosity began to +increase to measurable values above 0.1 Pa s after about 15600 s (4.3 hours) at 50 oC. +The viscosity further increased, reaching values greater than 40 Pa s after about 37400 +s (10.4 hours) at 50 oC. While these timescales may be appropriate to process plastic +Figure 2. The viscosity of liquid precursors containing 30 wt. % PPO and 0.2 wt. % +bis-MSB can increase over time as polymerization occurs, and this increase in viscosity +prevents processing of the liquid into a mould. In this case, precursors with 8 wt. % +DVB and without DVB are compared while held at a temperature of 50 oC. + +scintillators with industrial equipment, we observed that complete dissolution of all +components often required > 12 hours of stirring at elevated temperatures. +The premature onset of polymerization caused this increase in viscosity, which +could prevent trapped air bubbles from escaping or even prohibit transfer of the liquid +precursor to a mould. For precursor liquids that did not contain DVB, the viscosity +began to increase to measurable values above 0.1 Pa s after about 175000 s (48.6 hours) +at 50 oC. As before, the viscosity further increased and reached values greater than 40 +Pa s after 275000 s (76.4 hours) 50 oC. Without DVB, the working time of the liquid +precursors can be increased. Still, the limited working time of these materials highlights +the need for simple processing requirements like low temperatures. +Table 1. Summary of solubility tests. Note that some salts that formed a gel phase +were initially soluble. For solubility tests at 23 oC, the solubility was observed after 20 +hours of mixing. For solubility tests at 65 oC, the solubility was observed after 30 +minutes at 65 oC following 2 hours at 50 oC. +Acid used for 6Li salt +Solubility, 23 oC, 85:15 styrene:MAA +Solubility, 65 oC, 85:15 styrene:MAA +Pentanoic acid +Insoluble +Soluble +Hexanoic acid +Insoluble +Soluble +Octanoic acid +Insoluble +Soluble +2-methylpropanoic acid +Formed gel +Soluble +2-methylbutanoic acid +Formed gel +Soluble +3-methylbutanoic acid +Soluble +Soluble +2-ethylhexanoic acid +Formed gel +Soluble + + Another requirement for simple processing relates to the relative solubility of +the 6Li salts. Thus, we evaluated the solubility of various carboxylate salts of 6Li while + +considering the need for simple processing requirements. We synthesized 6Li salts of +pentanoic acid, hexanoic acid, octanoic acid, 2-methylpropanoic acid, 2- +methylbutanoic acid, 3-methylbutanoic acid, and 2-ethylhexanoic acid (Figure 3a). +The synthesis and solubility of these salts have been described previously[22], but our +focus is on solubility for synthesis of large plastics, which requires further +considerations related to processing that have not been reported. We added the 6Li salts +to liquid precursors that contained all monomers and dopants. The composition of the +liquid precursor was as follows: 30 wt. % 2,5-diphenyloxazole (PPO); 0.2 wt. % 1,4- +bis(2-methylstyryl)benzene (bisMSB); 5 wt. % DVB; an equivalent amount of 6Li salt +to obtain 0.1 wt. % 6Li; and the remainder was a mixture of 85 wt. % styrene and 15 +wt. % methacrylic acid (MAA). The MAA is necessary to dissolve the 6Li salt; other +monomers like methyl methacrylate and methyl acrylate do not dissolve the 6Li salt. +All precursors were mixed into a single vial and allowed to equilibrate for 20 +hours at room temperature (23 oC). Lithium-6 2-methylbutanoate was readily soluble +in the mixture initially. However, an opaque gel formed within 2 hours and persisted +after 20 hours (Figure 3b). Lithium-6 2-ethylhexanoate also formed a gel after initial +dissolution. For lithium-6 2-methylpropanoate, an opaque gel was observed after 20 +hours, but this mixture never fully dissolved, suggesting low solubility of this 6Li salt. +Similarly, 6Li salts of the linear alkyl carboxylic acids (pentanoic acid, hexanoic acid, +and octanoic acid) never fully dissolved at 23 oC. For lithium-6 3-methylbutanoate, the +liquid precursor remained clear after 20 hours at 23 oC. +The opaque gels that we observed could be destabilized after heating to +elevated temperatures. All vials were heated to 50 oC for 2 hours, which improved +dissolution of all components that were insoluble at room temperature. For example, +the liquids that contained 6Li salts of 2-methylbutanoic acid and 2-ethylhexanoic acid +became transparent. Further heating to 65 oC for 30 minutes improved dissolution; all + +liquid precursors with different 6Li salts were transparent after this heating step except +for the precursor that contained lithium-6 2-methylpropanoate (Figure 3c). +Based on this analysis, we selected the 6Li salt of 3-methylbutanoic acid for +the synthesis of large plastics; however, other 6Li salts like lithium-6 pentanoate and +lithium-6 2-methylbutanoate may also be suitable given that they form transparent +precursors after dissolution at 65 oC. Furthermore, we sometimes observed the +formation of a gel phase for precursors that contained lithium-6 3-methylbutanoate +when using less MAA, which highlights how plastic scintillators that contain this 6Li +salt are not immune to this processing challenge. +To avoid the formation of this gel phase, the 6Li salt could be dissolved +separately from the rest of the dopants. A 1:1 mixture of styrene and MAA was +sufficient to avoid thermally initiated homopolymerization of MAA during dissolution; +homopolymerization of MAA results in an opaque material. The remainder of the +styrene needed to form the final plastic was used to dissolve the primary and secondary +Figure 3. a) Chemical structures of 6Li salts that were studied; from left to right the structures correspond to +6Li salts of pentanoic acid, hexanoic acid, octanoic acid, 2-methylpropanoic acid, 2-methylbutanoic acid, 3- +methylbutanoic acid, and 2-ethylhexanoic acid. b) Photographs of liquid precursors in vials following +equilibration for 20 hours at 23 oC. c) Photographs of liquid precursors in vials following equilibration at an +additional 2 hours at 50 oC plus 30 minutes at 65 oC. All liquid precursors contain all components of a plastic +scintillator except the radical initiator. The precursors differ in the 6Li salt that was added; from left to right, the +vials contain 6Li salts of pentanoic acid, hexanoic acid, octanoic acid, 2-methylpropanoic acid, 3-methylbutanoic +acid, 2-methylbutanoic acid, and 2-ethylhexanoic acid. The vials are 28 mm in diameter. + +b)23 °C; 85:15 styrene:MAA +c)65C:85:15styrene:MAAdye in a separate container. Then, the two separate mixtures could be heated to 60-80 +oC and mixed at elevated temperatures before adding DVB and the radical initiator and +casting in a mould. +3b. Effect of comonomer on scintillation performance +Importantly, the effect of the comonomer that solubilizes the 6Li salts on +scintillation performance should be evaluated. The addition of non-aromatic +comonomers like methyl methacrylate (MMA) or methacrylic acid (MAA) can reduce +scintillation performance.[22,26] When 26 wt. % of MMA was used in a plastic +scintillator, the light yield reduced by 5% when compared to a plastic scintillator that +contained only polystyrene as the matrix. When 58 wt. % of MMA was used in a plastic +scintillator, the light yield reduced by 18%.[26] The total amount of the comonomer +Figure 4. a) A detrimental effect of MAA and 6Li salts reduce light output (LO) of scintillators that contain PPO +as a primary dye. 6Li salt may also influence light output. b) The LO decreases as the content of MAA increases, +which can be observed in the histograms that show the 137Cs Compton edge. c) The photograph of these vials highlights +the solubility threshold of lithium-6 3-methylbutanoate at 65 oC. The vials are 28 mm in diameter. d) Less substantial +effect on LO by MAA and 6Li salt for scintillators that contain mTP instead of PPO. The 6Li salt used for all samples +referenced in this figure was lithium-6 3-methylbutanoate. For a) and b), the average (dashed line) and standard +deviation (grey shaded region) of scintillators that do not contain any comonomer are shown for reference. + +65°C:6LitO +MAA weight content: +6.3% +6.9% +7.6% +8.2% +9.5% +Precipitate still +presentMAA added for dissolution of 6Li salts is typically less than 20 wt. % of the total +material, so we instead focused on comonomer addition at these lower concentrations. +Table 2. Summary of effect of composition on light output. +Primary dye +Co-monomer +Co-monomer content +Lithium salt content +Light output +PPO +N/A +0 +0 +1.05 +mTP +N/A +0 +0 +1.12 + + + + + +PPO +MMA +0.6 +0 +1.03 +PPO +MMA +3 +0 +1.07 +PPO +MMA +6 +0 +1.05 +PPO +MMA +13 +0 +1.09 + + + + + +PPO +MAA +0.6 +0 +0.96 +PPO +MAA +2 +0 +0.81 +PPO +MAA +3 +0 +0.78 +PPO +MAA +5 +0 +0.77 +PPO +MAA +6 +0 +0.73 +PPO +MAA +13 +0 +0.64 + + + + + +PPO +MAA +3 +1.7 +0.61 +PPO +MAA +5 +1.7 +0.57 +PPO +MAA +6 +1.7 +0.56 +PPO +MAA +13 +1.7 +0.51 + + + + + +mTP +MAA +3 +1.7 +0.93 +mTP +MAA +5 +1.7 +0.95 +mTP +MAA +6 +1.7 +0.93 +mTP +MAA +13 +1.7 +0.92 + +Even though MMA does not solubilize 6Li salts that we studied, we used this +comonomer as a non-aromatic additive to compare its effect on performance to +scintillators that contain the solubilizing comonomer, MAA. We also compared the +performance of plastic scintillators that did not contain any comonomer. The average +light output (LO) of three separate samples that did not contain any comonomer was +1.05 with a standard deviation of 0.07 (Figure 4a, Table 2). With 0.6 wt. % MMA +added, the LO was 1.03. As the amount of MMA increased, there was no clear trend in + +the LO. At 13 wt. % MMA, the LO was 1.09, and all measured values of LO were +within a standard deviation of the average value of plastic scintillators that did not +contain a comonomer. Thus, at these concentrations of MMA comonomer, the energy +transfer and light emission do not appear to be affected. +The same trend did not persist when using MAA. At 0.6 wt. % MAA, the LO +was 0.96, which corresponds to a 9% reduction in LO when compared to plastic +scintillators without this comonomer. At 1.6% MAA, the LO was further reduced to +0.81, which is a 23% reduction. The LO continues to decrease as MAA content +increases (Figure 4b, Table 2), but the magnitude of reduction appears to taper as the +MAA content exceeds 3 wt. %. At 13 wt. % MAA, the LO was reduced to 0.64, which +corresponds to a 40% reduction. The discrepancy between the effects of MMA and +MAA on LO indicates that the decrease in LO upon addition of MAA does not result +from simple dilution by a non-aromatic material. Rather, this decrease in LO suggests +that MAA may be detrimental to processes that affect scintillation like energy transfer +or emission. Although the exact mechanism is not fully elucidated, it is possible that +the heteroatoms on PPO (N, O) may interact with the polar acid functional group on +MAA. +Notably, the LO further decreases upon addition of 6Li salts to plastics that +contain PPO as the primary dye along with MAA as a comonomer when compared to +Figure 5. PSD distributions used to calculate FoM for comparison of plastics that contain PPO (a) and mTP (b). +Note that the FoMs for 6Li-plastics are given for discrimination between thermal neutrons and gamma rays whereas +FoMs for plastics without 6Li correspond to discrimination between fast neutrons and gamma rays. + +the addition of MAA alone. At 13 wt. % MAA, the LO decreased from 0.64 to 0.52 +when adding the 6Li salt. The reduction in LO with increasing MAA content and +addition of 6Li salt poses a challenge related to processing: higher content of MAA +allows for processing of the plastic scintillator at lower temperatures but reduces LO. +Plastic scintillators with lower content of MAA are more difficult to process due to poor +solubility of the 6Li salt. For example, lithium-6 3-methylbutanoate is not fully soluble +at 65 oC when the MAA content is equal to or less than 8 wt. % (Figure 4c). The cause +of the reduction in performance upon addition of 6Li salt may be similar to the reduction +in performance upon addition of MAA; PPO may have unfavourable interactions and/or +reactivity with these polar molecules. +To test this idea, we compared the scintillation performance when using 30 wt. +% m-terphenyl (mTP) as a primary dye instead of PPO. The chemical structure of mTP +(Figure 1b) does not contain any heteroatoms (i.e., only contains C, H). Plastics that +used mTP as the primary dye but did not contain MAA or a 6Li salt had an average LO +of 1.12 (Figure 4d, Table 2). Plastics that contained MAA and 6Li had a slight +reduction in LO with values of 0.93, 0.95, 0.93, and 0.92 at 3 wt. %, 5 wt. %, 6 wt. %, +and 13 wt. % MAA. At 13 wt. % MAA, the LO of a plastic that contains 6Li was reduced +by 18% when compared to a plastic that contained no MAA or 6Li. This value of LO is +nearly double the LO of an equivalent plastic that contained PPO instead of mTP as the +primary dye. +We also compared the FoM for PSD. For plastics that contain PPO but no 6Li, +the FoM is 3.05, which provides a baseline for comparison after addition of MAA and +6Li. Note that this FoM compares discrimination between gamma rays and fast neutrons +whereas the FoM for plastics that contain 6Li compares discrimination between gamma +rays and thermal neutrons. The energy range used to determine FoM was 450-550 +keVee, but that range was adjusted to capture the thermal neutron peak. Upon addition +of 13 wt. % MAA and 6Li, the FoM decreases to 2.24 and decreases for all plastics that +were measured that contained MAA (Figure 5a). +These PSD results can also be compared to plastics that contain mTP instead +of PPO (Figure 5b). For plastics that contain mTP but no 6Li, the FoM is 2.73. Upon + +addition of MAA and 6Li, the FoM is between 2.95 and 3.24. This increase is mostly +attributed to the high value of Qtail/Qtotal for the thermal-neutron capture spot along with +the observation that plastics that contain mTP are less sensitive to the addition of the +polar compounds that enable thermal-neutron capture. The FoM for plastics that +contained PPO decreased when MAA and 6Li were added whereas the FoM for plastics +that contained mTP increased when MAA and 6Li were added. +3c. Production of large plastic scintillators +As shown above, mTP may be promising for high performance scintillators +that contain 6Li; however, the inconsistencies with solubility and the current availability +at large scales in sufficient purity from commercial suppliers make mTP less suitable +Figure 6. a,b) Conditions that are not optimized for the production of large plastics produce defects like cracks and +bubbles (a) and discoloration (b). The left-most scintillator in (a) does not contain defects and serves as a reference. +These scintillators in (a) have a diameter of about 5.5 cm. The plastic scintillator in (b) is 40 cm in length. c) When the +curing conditions and environment are controlled, plastics scintillators can be produced in large scale, as shown in this +photograph of a scintillator that is 5 cm in width atop a sheet of paper. d) The effective attenuation length was measured +by placing a collimated gamma-ray source at set distances away from two PMTs and measuring the PMT response +(black). The data were fitted with an exponential profile to estimate the effective attenuation length (19-21cm). e) Large +plastic scintillators are capable of PSD, as shown in this distribution that demonstrates an ability to separate signals from +thermal neutrons and gamma rays. + +b)for the production of large plastic scintillators. For these large plastic scintillators, we +selected PPO despite its lower LO and lower FoM at smaller scales. +Similarly, other secondary dyes like Exalite 404 (E404) may have the best +performance in the plastic scintillators that we evaluated[27], but the cost of E404 might +be prohibitive when compared to a secondary dye like 1,4-bis(2-methylstyryl)benzene +(bisMSB). +When processing the plastics, all precursors except DVB and the radical +initiator are slowly heated to temperatures between 60 and 80 oC until full dissolution. +The plastics are cured in glass or aluminium moulds. To control the rate of +polymerization, the radical initiator is added at concentrations of 0.01 wt. %, and +plastics are cured at an initial temperature of 60 oC. Various times of curing were used, +and a typical recipe would involve curing for 7 days at 60 oC followed by an additional +4 days of curing at 75 oC. Excessive radical concentration and/or heating during +processing and curing could lead to defects like cracks and bubbles (Figure 6a). +Precipitation of precursors that have lower solubility may occur if temperatures are too +low. When curing the plastics, oxygen is displaced by a steady flow of nitrogen; without +nitrogen flow, discoloration can occur (Figure 6b). These precautionary measures +allow us to produce plastic scintillators without defects and with minimal +discoloration (Figure 6c). The effective attenuation of a plastic scintillator that was 16” +long was measured by placing a collimated gamma-ray source near the scintillator and +measuring the total light detected by PMTs that are mounted on each end of the plastic +scintillator (Figure 6d). The effective attenuation was determined to be about 19-21 +cm, which is comparable to the value obtained in previous experiments.[24] This plastic +also had PSD capability; a thermal-neutron spot is clearly separated from the gamma +signal (Figure 6e). After we optimized our process for synthesis of these large plastics, +we outsourced production to Eljen Technologies who is currently producing 6Li-loaded +prototypes with dimensions exceeding 0.5 m; full characterization of scintillator +performance at these large scales will be the subject of a future publication. + +4. Conclusion +By careful control of composition and processing, plastic scintillators that can +discriminate between gamma rays, fast neutrons, and thermal neutrons can be produced +at a scale of 1 kg or greater. The solubility of dopants that enable scintillation +functionality and solubilizing additives like methacrylic acid (MAA) that may +negatively affect performance were considered. Synthesis and processing procedures +were developed for large plastic scintillators containing 0.1 wt. % 6Li and high +concentration (30 wt. %) of PPO used as a primary dye. These scintillators were capable +of PSD. In these studies, various 6Li salts of aliphatic carboxylic acids were evaluated, +and many were found to be suitable for the production of large plastic scintillators with +the addition of MAA. The amount of MAA that was added to solubilize 6Li salts +affected the scintillation performance but also determined the temperature that plastic +scintillators could be produced at. An alternative way to avoid the deleterious effects of +MAA was discovered; use of m-terphenyl instead of PPO improved plastic scintillators. +However, m-terphenyl may have limitations like availability at large volumes. +With these considerations in mind, methods for the preparation of plastic +scintillators loaded with 6Li were established and demonstrated. Large-volume pieces +that could be used for large detectors were produced.[24,28] Such detectors will be +important for future safeguards related to nuclear power production and for unravelling +unknown aspects of particle physics. +Acknowledgements +This work was performed under the auspices of the U.S. Department of Energy by +Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344 and +was supported by the LLNL-LDRD Program under Project No. 20-SI-003, release +number LLNL-JRNL-839909. We would like to thank Jacob Kim for careful reading +and discussion of this manuscript. + +The authors declare no competing interests. + +References +[1] +W.J. Nuttall, Nuclear Renaissance: Technologies and Policies for the Future of +Nuclear Power, CRC Press, 2022. +[2] +J.F. Cavaignac, A. Hoummada, D.H. Koang, B. Vignon, Y. Declais, H. de Kerret, +H. Pessard, J.M. Thenard, Indication for neutrino oscillation from a high statistics +experiment at the bugey reactor, Phys. Lett. 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Equip. 942 (2019) 162334. https://doi.org/10.1016/j.nima.2019.162334. + + + diff --git a/09E2T4oBgHgl3EQf5AgM/content/tmp_files/load_file.txt b/09E2T4oBgHgl3EQf5AgM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..92c8dd75b40267068154c42ba023d32fd1a2d42a --- /dev/null +++ b/09E2T4oBgHgl3EQf5AgM/content/tmp_files/load_file.txt @@ -0,0 +1,1383 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf,len=1382 +page_content='Synthesis and processing of lithium-loaded plastic scintillators on the kilogram scale Michael J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Forda*, Elisabeth Aigeldingera, Felicia Sutantoa, Natalia P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Zaitsevaa, Viacheslav A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Lia, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Leslie Carmana, Andrew Glenna, Cristian R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Catalaa, Steven A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Dazeleya, Nathaniel Bowdena*1 aLawrence Livermore National Laboratory 7000 East Avenue, Livermore, CA 94550 Abstract Plastic scintillators that can discriminate between gamma rays, fast neutrons, and thermal neutrons were synthesized and characterized while considering the balance between processing and performance at the kilogram scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' These trade-offs were necessitated by the inclusion of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='1 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' % lithium-6 to enable detection of thermal neutrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The synthesis and processing of these plastic scintillators on the kilogram scale required consideration of many factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' First, a comonomer (methacrylic acid) was used to solubilize salts of lithium-6, which allow for a thermal-neutron capture reaction that produces scintillation light following energy transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Second, scintillation performance and processability were considered because the increasing content of the comonomer resulted in a sharp decrease in the light output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The use of small amounts of comonomer (≤3 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' %) resulted in better performance but required high processing temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' At large scales, these high temperatures could initiate an exothermic polymerization that results in premature curing and/or defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The deleterious effects of the comonomer may be mitigated by using m-terphenyl as a primary dye rather than 2,5-diphenyloxazole (PPO), which has been traditionally used in organic scintillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Finally, the curing environment was controlled to avoid defects like cracking and discoloration while maintaining solubility of dopants during curing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' For scintillators that were produced from kilogram-scale batches of precursors, the effective attenuation of scintillation light was characterized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' This is to indicate the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Email address: bowden20@llnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='gov;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' ford40@llnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='gov Keywords: pulse-shape discrimination, plastic scintillators, inverse beta decay, neutron detection, large-scale detectors 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Introduction The rising cost of fossil fuels and concerns about greenhouse gas emissions motivate a revival of nuclear power generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' [1] However, the expansion of nuclear power throughout the world and the construction of novel reactor types may challenge the resources available for implementing conventional safeguards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Direct and nonintrusive measurements of reactor operation using offer one approach to address this concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' These measurements can be facilitated by the development of novel detectors to enable near-field (ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' 10-100 m) monitoring of antineutrinos produced by a reactor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' [2–10] Previous reports highlight the ability of organic scintillators to monitor this antineutrino flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' For example, the Precision Reactor Oscillation and Spectrum Experiment (PROSPECT)[11] recently demonstrated the measurement of the antineutrino spectrum from 235U at the High Flux Isotope Reactor at Oak Ridge National Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Like in many experiments that use organic scintillators to monitor antineutrino flux, PROSPECT utilized about 4 tons of a liquid scintillator loaded with a neutron capture agent (6Li in this case), which can detect signatures of inverse beta decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' In this experiment, the scintillation light is associated with signatures of inverse beta decay using a measurement scheme called pulse-shape discrimination (PSD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' [12] After monitoring a scintillation pulse over time, a prompt signal of scintillation light can be associated with the antineutrino energy, and the delayed signal of scintillation light can be associated with neutron capture by 6Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Detectors like PROSPECT contain scintillator with this capability of PSD, making them useful for uniquely identifying capture reactions and rejecting fast-neutron background events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' For continued development of novel detectors, the phase of the detector material as it relates to the mobility of the detector may be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Liquid scintillators are relatively inexpensive, are easy to manufacture, and have good performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' However, liquid scintillators may require consideration of potential hazards (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=', flammability), handling, and storage when used in mobile detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Conversely, plastic scintillators are less hazardous than liquid scintillators since they are in the solid state, and plastic scintillators are self-supporting, which is useful for mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' However, plastic scintillators have not been widely available as materials capable of PSD until recently[13,14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Additionally, the PSD performance of plastic scintillators worsens as the length of the scintillator increases due to light attenuation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' this attenuation is detrimental to the performance of large-volume (ton-scale) detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' [15] Thus, a mobile detector may require further consideration of these trade- offs and further development of detector materials like plastic scintillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' One development that is important for plastic scintillators is the capability of these materials to discriminate between gamma rays, fast neutrons, and thermal neutrons (Figure 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' This capability necessitates the inclusion of a neutron capture agent like 155Gd, 157Gd, 10B, or 6Li, and various reports describe attempts to incorporate neutron capture agents into plastic scintillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' In particular, the doping of scintillators with 6Li may be preferable since the 6Li(n,t)\uf061 capture reaction produces a localized, mono-energetic energy deposition that can be efficiently identified via PSD and energy selections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' [16,17] Carboxylate salts of 6Li have been previously used to incorporate 6Li into plastic scintillators while obtaining materials that are transparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' In one example, a 6Li salt of methacrylic acid (MAA) was copolymerized with styrene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The 6Li salt of MAA was not soluble in the plastic scintillator precursors at appreciable amounts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' additional MAA was needed to dissolve the 6Li salt of MAA and increase the 6Li content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' This necessity of additional MAA highlights how solubility of the polar 6Li compounds in the nonpolar matrix must be considered to produce plastic scintillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Despite the need for additional MAA, these scintillators were promising for thermal-neutron detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' At a small scale (≈ 1 cm), the plastic scintillator was responsive to an incident beam of thermal neutrons from a research reactor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' [18] However, the optical attenuation properties were not assessed for this material but become important for large-volume detectors, where the longest side of a single scintillator may be on the order of 10-100 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Other developments of these scintillators that contain 6Li focused on exploration of additional carboxylates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' [19–22] One report described an investigation of 16 different 6Li salts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' These 6Li salts were dissolved in a comonomer mixture of styrene and methacrylic acid (90:10 styrene:MAA) to determine the maximum solubility of 6Li at 60 oC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Additional 6Li could be dissolved as the MAA content increased, but the scintillation light output and figure of merit (FoM) for PSD decreased as MAA and 6Li content increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' [22] Thus, improvements in scintillation performance should consider the balance between processing and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' This consideration is especially important in large plastic scintillators where thermal runaway[23] becomes a concern for processing, and attenuation becomes a concern for performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' In this report, we describe the synthesis and processing of lithium-loaded plastic scintillators on the kilogram scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' We considered aspects related to the composition of the plastic scintillator like the primary dye, secondary dye, monomers, and lithium salts (Figure 1b) as well as aspects related to processing and curing like dissolution and temperature of cure (Figure 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' We first synthesized various 6Li salts and characterized their solubility at different temperatures and with various concentrations of comonomers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Then, we considered trade-offs in processing and performance at the 10 g scale by evaluating the scintillation performance upon addition of comonomer and 6Li salt in plastic scintillators that contained 2,5-diphenyloxazole (PPO) as the primary dye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The light output of scintillation was reduced as the content of the comonomer MAA increased in these scintillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' We then compared the performance of plastic scintillators that contain PPO vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' m-terphenyl (mTP) as the primary dye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The performance of plastic scintillators that contain mTP as the primary dye is less sensitive to the addition of MAA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' When scaling our synthesis to 1 kg (Figure 1d), we targeted compositions that were available at scale and allowed for lower temperatures of processing to avoid thermally initiated polymerization and thermal runaway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' All of these measures considered the potential for production of large-scale plastic scintillators, which will be useful for applications like mobile antineutrino detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Materials and methods Styrene (99%, Sigma), vinyl toluene (99%, Sigma), divinyl benzene (Sigma, technical grade), and methyl methacrylate (99%, VWR) were passed through an alumina column to remove inhibitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Methacrylic acid (99%, Sigma) was dried over sodium chloride and distilled under vacuum to remove inhibitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' All monomers were sparged with nitrogen for > 30 min before being stored in a nitrogen-filled glovebox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' a) Plastic scintillators that are capable of PSD and contain lithium-6 distinguish between gamma rays, fast neutrons, and thermal neutrons, as shown in this PSD plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The gradient scale bar represents a relative population of data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' b) The synthesis of plastic scintillators requires consideration of many components, as shown in this schematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The chemical structures, starting from the bottom left and going clockwise, are those of m-terphenyl, 2,5-diphenyloxazole, Exalite 404, styrene, methacrylic acid, divinylbenzene, and three 6Li salts of carboxylic acids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' c) The processing of the precursors requires control of dissolution and curing conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' d) Control of synthesis and processing has enabled the production of large plastic scintillators, scaling to rectangular prisms like the ones in this photograph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The large scintillators are about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='41 m in length (total mass of about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='5 kg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='8 Thermal neutrons 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='7 Fas, neurone 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='4 Gamma raye 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='1 100 200 300 4010 Approt, enengy e!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='All monomers except methacrylic acid were stored in an inert atmosphere at −20 oC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Methacrylic acid was stored at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' m-terphenyl (mTP, Smolecule) was purified by recrystallization from toluene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' L-231 (Luperox, 1,1-di(t-butylperoxy)-3,3,5- trimethylcyclohexane) was used as a radical initiator after sparging for > 30 min with dry nitrogen and kept at −20 °C until needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' 2,5-diphenyloxazole (PPO, scintillation grade from Sigma), 1,4-bis(2-methylstyryl)benzene (bisMSB, Luxottica/Exciton), and 1,4-bis(9,9-diethyl-7-(tert-pentyl)-9H-fluoren-2-yl)benzene (E404, Luxottica/Exciton) were used as received without further purification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' 6Li carboxylate salts were synthesized by first suspending 6Li2CO3 (National Isotope Development Center) in a 1:1 mixture of water (deionized) and methanol (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='8%, Sigma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Excess carboxylic acid (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='02 equivalent excess) was mixed into a 1:1 mixture of water and methanol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The basic suspension was slowly added to the acid solution, and this mixture was heated to reflux for > 4 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The solution was filtered, and the 6Li salt was precipitated by adding excess volume of cold acetone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The 6Li salt was collected by vacuum filtration and washed with acetone, followed by drying under vacuum at 80 oC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' This procedure was used for carboxylate salts of pentanoic acid, hexanoic acid, octanoic acid, 2-methylpropanoic acid, 2-methylbutanoic acid, 3- methylbutanoic acid, and 2-ethylhexanoic acid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' All acids were purchased from Sigma or VWR and used as received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Plastics were synthesized in a dry nitrogen environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' For initial evaluation, plastics were synthesized using 10 g of precursor materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' For the production of large plastic scintillators, the amount of precursors used was up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='7 kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' All materials that were not stored in a dry nitrogen environment were dried under vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' 6Li salts were easiest to process when they were first dissolved in a 1:1 mixture by weight of styrene:MAA at elevated temperatures (about 60-80 oC) before adding the remainder of the plastic composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' A typical synthesis would involve dissolving the primary dye (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=', PPO) and the secondary dye (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=', bis-MSB or E404) in styrene or vinyl toluene (VT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The monomer VT was used as the polymer matrix for plastics that used mTP as the primary dye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' VT was used for these plastics as we observed less consistency in solubility for mTP-based plastics that used styrene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The performance of plastic scintillators that used styrene and VT were compared, and we observed no meaningful difference in performance for these plastics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' This precursor solution was heated to about 60-80 oC and mixed with a solution of the 6Li salt in 1:1 mixture by weight of styrene:MAA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' DVB (typically 5 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' %) and initiator (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='08 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' % for 10 g plastics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='01 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' % for plastics batches greater than 400 g) were added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' This mixture that contains the precursor solution mentioned earlier and the solution that contains 6Li was poured into a mould and sealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The mould was placed in a nitrogen-filled oven and cured at elevated temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' In one experiment, the viscosity of the precursor solution was monitored using a rotary viscometer (Brookfield DV2T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' A typical curing profile would consist of heating for 7 days at 60 oC, followed by a temperature ramp to 75 oC over one day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The scintillators were cured in convection ovens (Cascade TEK) that were fitted with gas lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Dry nitrogen flowed into solvent- resistance plastic bags that contained the mould inside the oven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The bags maintained a positive pressure of nitrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The plastic would stay at 75 oC for four days and then cool to room temperature over the course of one day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Following curing, the scintillators were removed from the moulds, then machined and polished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' All photographs of samples were taken using a Nikon D750 and were globally edited in Adobe Lightroom for colour and exposure corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' For initial scintillator characterization, samples of mass equal to 10 g were measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The outer edge and one face of the scintillators were wrapped and covered with Teflon tape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The exposed face was coupled with optical grease to a Hamamatsu R6231-100-SEL photomultiplier tube (PMT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Signals from the PMT were recorded at a sampling rate of 200 MS/s using a 14-bit CompuScope 14200 waveform digitizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' A relative quantification of light output (LO) was measured using ionizing radiation from 137Cs incident upon the plastic scintillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The values of LO that we report in this manuscript are specific to our measurement system and thus should only be used for relative comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' We normalized the value of LO to measurements of the commercial scintillator EJ-200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The location of 500 keVee was defined by the value of the pulse integral at 50% of the height of the 137Cs Compton edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' For many measurements, duplicate samples were synthesized, and averages are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' For one condition, 9 samples were replicated in multiple batches and were measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The standard deviation of these measurements was within 7% of the average value, which could be representative of the standard deviation related to contributions from measurement and synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Where standard deviation is not reported, a conservative value of 10% of the value given could be assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The measurement of effective attenuation length was performed with a longer scintillator bar (1″ x 1″ x 16″).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' This measurement employed a setup that was identical to the setup used to characterize scintillator bars in an antineutrino detector called SANDD (Segmented AntiNeutrino Directional Detector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' [24] This bar was wrapped with polytetrafluoroethylene tape (POLY-TEMP PN-16050), and a pair of 1" Hamamatsu R1924A-100 PMTs were mounted at either end of the scintillator bar using EJ-550 silicone optical grease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The PMT operating voltage was set at -1100 V, and signals were digitized using a Struck SIS3316 digitizer module (250 MS/s, 14 bit, 5 V dynamic range).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The energy threshold was set at approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='1 MeVee, and 1600 ns-long waveforms were sent to disk and stored in ROOT data format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The charge response difference between the two PMTs due to gain and optical coupling variation was corrected using a collimated 137Cs gamma-ray source directed at the center of the plastic bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Here, lead bricks were used for shaping the gamma-ray source into a fan beam of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='5 cm width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' To obtain the effective attenuation length of the scintillator bar, we performed a series of collimated 22Na measurements at regular intervals along the length of the bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' In each measurement, we identified the location of the Compton continuum maximum of the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='275 MeV gamma-ray by fitting the energy response with a Gaussian profile while varying the range of the fit to find the minimum χ2 (best fit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The Compton continuum maximum position was identified as the mean of the Gaussian profile that yielded the minimum χ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The associated uncertainty was estimated by varying the range of the fit until the χ2 value exceeded the 68% confidence level (CL) of the minimum χ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' the uncertainty was the corresponding range of the mean of the Gaussian profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' To measure PSD in smaller plastics, plastic scintillators were exposed to a 252Cf source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The source was shielded behind 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='1 cm of lead to reduce the gamma-ray flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' To obtain a flux of thermal neutrons, high density polyethylene was also used as a moderator for 252Cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The measurements of scintillation from plastic scintillators exposed to 252Cf were integrated over time to determine the total charge (Qtotal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The charge of the delayed component of the signal (Qtail) was determined from a delayed fraction of the scintillation pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Scintillation pulses due to interactions of the scintillator with neutrons have a larger fraction of Qtail relative to Qtotal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' therefore, a comparison of Qtail relative to Qtotal can be used to distinguish between scintillation due to neutrons vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' gamma rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The PSD was quantified using a figure of merit (FoM) that is determined from histograms of the ratio of the charge of the delayed component relative to the total charge, as described in previous reports[15,25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Briefly, the FoM is: 𝐹𝑜𝑀 = 〈𝑛,𝑡〉−〈𝛾〉 𝐹𝑊𝐻𝑀𝑛,𝑡+𝐹𝑊𝐻𝑀𝛾 (2) In this equation, 〈n,t〉-〈γ〉 represents the difference between the average value of the neutron and gamma-ray signals, and FWHMn,t +FWHMγ represents the sum of the full- width at half of the maximum value of the distributions of the thermal-neutron and gamma-ray signals at the electron-equivalent energy of the thermal-neutron spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' For plastics that don’t contain 6Li, the same equation was used for FoM, but the position and FWHM of the neutron peak is used at an electron-equivalent energy near the 137Cs Compton edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' For PSD of the larger scintillator bar (1″ × 1″ × 16″), an identical setup as the measurement of effective attenuation length was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The bar was irradiated with an uncollimated 252Cf source, and lead bricks with a total thickness of 6″ were placed between the detector and the 252Cf source to reduce the gamma-ray flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The charge integration limits were optimized, and the best parameters were found to be [tL-20 ns ≤ Qtotal ≤ tL+1300 ns] and [tL+24 ns ≤ Qtail ≤ tL+1300 ns], where tL is the leading edge of the waveform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Assuming light transport behaves exponentially along the length of the scintillator bar, we can eliminate the dependence of energy on event position by reconstructing the energy as 𝐸 = √𝐸𝐴𝐸𝐵 , where EA and EB are the charges collected by the two PMTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Results and discussion 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Selection of lithium-6 salt based on solubility at moderate temperatures To make large-scale production of plastic scintillators easier, our selection of materials focused on those materials that had simple processing requirements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=', temperatures below 80 oC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' This requirement is necessary since high processing temperatures for large plastic scintillators could thermally initiate the polymerization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' After dissolution of all dopants, we monitored the viscosity over time of liquid precursors with and without the crosslinker divinylbenzene (DVB) at a temperature of 50 oC (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The liquid precursors did not contain a radical initiator that initiates polymerization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' thus, any increase in viscosity is due to thermally initiated polymerization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' For precursor liquids that contained DVB, the viscosity began to increase to measurable values above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='1 Pa s after about 15600 s (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='3 hours) at 50 oC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The viscosity further increased, reaching values greater than 40 Pa s after about 37400 s (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='4 hours) at 50 oC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' While these timescales may be appropriate to process plastic Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The viscosity of liquid precursors containing 30 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' % PPO and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='2 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' % bis-MSB can increase over time as polymerization occurs, and this increase in viscosity prevents processing of the liquid into a mould.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' In this case, precursors with 8 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' % DVB and without DVB are compared while held at a temperature of 50 oC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' scintillators with industrial equipment, we observed that complete dissolution of all components often required > 12 hours of stirring at elevated temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The premature onset of polymerization caused this increase in viscosity, which could prevent trapped air bubbles from escaping or even prohibit transfer of the liquid precursor to a mould.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' For precursor liquids that did not contain DVB, the viscosity began to increase to measurable values above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='1 Pa s after about 175000 s (48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='6 hours) at 50 oC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' As before, the viscosity further increased and reached values greater than 40 Pa s after 275000 s (76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='4 hours) 50 oC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Without DVB, the working time of the liquid precursors can be increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Still, the limited working time of these materials highlights the need for simple processing requirements like low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Summary of solubility tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Note that some salts that formed a gel phase were initially soluble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' For solubility tests at 23 oC, the solubility was observed after 20 hours of mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' For solubility tests at 65 oC, the solubility was observed after 30 minutes at 65 oC following 2 hours at 50 oC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Acid used for 6Li salt Solubility, 23 oC, 85:15 styrene:MAA Solubility, 65 oC, 85:15 styrene:MAA Pentanoic acid Insoluble Soluble Hexanoic acid Insoluble Soluble Octanoic acid Insoluble Soluble 2-methylpropanoic acid Formed gel Soluble 2-methylbutanoic acid Formed gel Soluble 3-methylbutanoic acid Soluble Soluble 2-ethylhexanoic acid Formed gel Soluble Another requirement for simple processing relates to the relative solubility of the 6Li salts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Thus, we evaluated the solubility of various carboxylate salts of 6Li while considering the need for simple processing requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' We synthesized 6Li salts of pentanoic acid, hexanoic acid, octanoic acid, 2-methylpropanoic acid, 2- methylbutanoic acid, 3-methylbutanoic acid, and 2-ethylhexanoic acid (Figure 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The synthesis and solubility of these salts have been described previously[22], but our focus is on solubility for synthesis of large plastics, which requires further considerations related to processing that have not been reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' We added the 6Li salts to liquid precursors that contained all monomers and dopants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The composition of the liquid precursor was as follows: 30 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' % 2,5-diphenyloxazole (PPO);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='2 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' % 1,4- bis(2-methylstyryl)benzene (bisMSB);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' 5 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' % DVB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' an equivalent amount of 6Li salt to obtain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='1 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' % 6Li;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' and the remainder was a mixture of 85 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' % styrene and 15 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' % methacrylic acid (MAA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The MAA is necessary to dissolve the 6Li salt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' other monomers like methyl methacrylate and methyl acrylate do not dissolve the 6Li salt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' All precursors were mixed into a single vial and allowed to equilibrate for 20 hours at room temperature (23 oC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Lithium-6 2-methylbutanoate was readily soluble in the mixture initially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' However, an opaque gel formed within 2 hours and persisted after 20 hours (Figure 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Lithium-6 2-ethylhexanoate also formed a gel after initial dissolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' For lithium-6 2-methylpropanoate, an opaque gel was observed after 20 hours, but this mixture never fully dissolved, suggesting low solubility of this 6Li salt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Similarly, 6Li salts of the linear alkyl carboxylic acids (pentanoic acid, hexanoic acid, and octanoic acid) never fully dissolved at 23 oC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' For lithium-6 3-methylbutanoate, the liquid precursor remained clear after 20 hours at 23 oC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The opaque gels that we observed could be destabilized after heating to elevated temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' All vials were heated to 50 oC for 2 hours, which improved dissolution of all components that were insoluble at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' For example, the liquids that contained 6Li salts of 2-methylbutanoic acid and 2-ethylhexanoic acid became transparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Further heating to 65 oC for 30 minutes improved dissolution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' all liquid precursors with different 6Li salts were transparent after this heating step except for the precursor that contained lithium-6 2-methylpropanoate (Figure 3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Based on this analysis, we selected the 6Li salt of 3-methylbutanoic acid for the synthesis of large plastics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' however, other 6Li salts like lithium-6 pentanoate and lithium-6 2-methylbutanoate may also be suitable given that they form transparent precursors after dissolution at 65 oC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Furthermore, we sometimes observed the formation of a gel phase for precursors that contained lithium-6 3-methylbutanoate when using less MAA, which highlights how plastic scintillators that contain this 6Li salt are not immune to this processing challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' To avoid the formation of this gel phase, the 6Li salt could be dissolved separately from the rest of the dopants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' A 1:1 mixture of styrene and MAA was sufficient to avoid thermally initiated homopolymerization of MAA during dissolution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' homopolymerization of MAA results in an opaque material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The remainder of the styrene needed to form the final plastic was used to dissolve the primary and secondary Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' a) Chemical structures of 6Li salts that were studied;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' from left to right the structures correspond to 6Li salts of pentanoic acid, hexanoic acid, octanoic acid, 2-methylpropanoic acid, 2-methylbutanoic acid, 3- methylbutanoic acid, and 2-ethylhexanoic acid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' b) Photographs of liquid precursors in vials following equilibration for 20 hours at 23 oC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' c) Photographs of liquid precursors in vials following equilibration at an additional 2 hours at 50 oC plus 30 minutes at 65 oC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' All liquid precursors contain all components of a plastic scintillator except the radical initiator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The precursors differ in the 6Li salt that was added;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' from left to right, the vials contain 6Li salts of pentanoic acid, hexanoic acid, octanoic acid, 2-methylpropanoic acid, 3-methylbutanoic acid, 2-methylbutanoic acid, and 2-ethylhexanoic acid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The vials are 28 mm in diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' b)23 °C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' 85:15 styrene:MAA c)65C:85:15styrene:MAAdye in a separate container.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Then, the two separate mixtures could be heated to 60-80 oC and mixed at elevated temperatures before adding DVB and the radical initiator and casting in a mould.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Effect of comonomer on scintillation performance Importantly, the effect of the comonomer that solubilizes the 6Li salts on scintillation performance should be evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The addition of non-aromatic comonomers like methyl methacrylate (MMA) or methacrylic acid (MAA) can reduce scintillation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' [22,26] When 26 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' % of MMA was used in a plastic scintillator, the light yield reduced by 5% when compared to a plastic scintillator that contained only polystyrene as the matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' When 58 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' % of MMA was used in a plastic scintillator, the light yield reduced by 18%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' [26] The total amount of the comonomer Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' a) A detrimental effect of MAA and 6Li salts reduce light output (LO) of scintillators that contain PPO as a primary dye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' 6Li salt may also influence light output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' b) The LO decreases as the content of MAA increases, which can be observed in the histograms that show the 137Cs Compton edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' c) The photograph of these vials highlights the solubility threshold of lithium-6 3-methylbutanoate at 65 oC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The vials are 28 mm in diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' d) Less substantial effect on LO by MAA and 6Li salt for scintillators that contain mTP instead of PPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The 6Li salt used for all samples referenced in this figure was lithium-6 3-methylbutanoate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' For a) and b), the average (dashed line) and standard deviation (grey shaded region) of scintillators that do not contain any comonomer are shown for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' 65°C:6LitO MAA weight content: 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='3% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='9% 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='6% 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='2% 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='5% Precipitate still presentMAA added for dissolution of 6Li salts is typically less than 20 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' % of the total material, so we instead focused on comonomer addition at these lower concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Summary of effect of composition on light output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Primary dye Co-monomer Co-monomer content Lithium salt content Light output PPO N/A 0 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='05 mTP N/A 0 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='12 PPO MMA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='6 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='03 PPO MMA 3 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='07 PPO MMA 6 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='05 PPO MMA 13 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='09 PPO MAA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='96 PPO MAA 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='81 PPO MAA 3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='78 PPO MAA 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='77 PPO MAA 6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='73 PPO MAA 13 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='64 PPO MAA 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='61 PPO MAA 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='57 PPO MAA 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='56 PPO MAA 13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='51 mTP MAA 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='93 mTP MAA 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='95 mTP MAA 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='93 mTP MAA 13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='92 Even though MMA does not solubilize 6Li salts that we studied, we used this comonomer as a non-aromatic additive to compare its effect on performance to scintillators that contain the solubilizing comonomer, MAA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' We also compared the performance of plastic scintillators that did not contain any comonomer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The average light output (LO) of three separate samples that did not contain any comonomer was 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='05 with a standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='07 (Figure 4a, Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' With 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='6 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' % MMA added, the LO was 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' As the amount of MMA increased, there was no clear trend in the LO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' At 13 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' % MMA, the LO was 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='09, and all measured values of LO were within a standard deviation of the average value of plastic scintillators that did not contain a comonomer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Thus, at these concentrations of MMA comonomer, the energy transfer and light emission do not appear to be affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The same trend did not persist when using MAA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' At 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='6 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' % MAA, the LO was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='96, which corresponds to a 9% reduction in LO when compared to plastic scintillators without this comonomer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' At 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='6% MAA, the LO was further reduced to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='81, which is a 23% reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The LO continues to decrease as MAA content increases (Figure 4b, Table 2), but the magnitude of reduction appears to taper as the MAA content exceeds 3 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' At 13 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' % MAA, the LO was reduced to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='64, which corresponds to a 40% reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The discrepancy between the effects of MMA and MAA on LO indicates that the decrease in LO upon addition of MAA does not result from simple dilution by a non-aromatic material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Rather, this decrease in LO suggests that MAA may be detrimental to processes that affect scintillation like energy transfer or emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Although the exact mechanism is not fully elucidated, it is possible that the heteroatoms on PPO (N, O) may interact with the polar acid functional group on MAA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Notably, the LO further decreases upon addition of 6Li salts to plastics that contain PPO as the primary dye along with MAA as a comonomer when compared to Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' PSD distributions used to calculate FoM for comparison of plastics that contain PPO (a) and mTP (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Note that the FoMs for 6Li-plastics are given for discrimination between thermal neutrons and gamma rays whereas FoMs for plastics without 6Li correspond to discrimination between fast neutrons and gamma rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' the addition of MAA alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' At 13 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' % MAA, the LO decreased from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='64 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='52 when adding the 6Li salt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The reduction in LO with increasing MAA content and addition of 6Li salt poses a challenge related to processing: higher content of MAA allows for processing of the plastic scintillator at lower temperatures but reduces LO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Plastic scintillators with lower content of MAA are more difficult to process due to poor solubility of the 6Li salt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' For example, lithium-6 3-methylbutanoate is not fully soluble at 65 oC when the MAA content is equal to or less than 8 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' % (Figure 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The cause of the reduction in performance upon addition of 6Li salt may be similar to the reduction in performance upon addition of MAA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' PPO may have unfavourable interactions and/or reactivity with these polar molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' To test this idea, we compared the scintillation performance when using 30 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' % m-terphenyl (mTP) as a primary dye instead of PPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The chemical structure of mTP (Figure 1b) does not contain any heteroatoms (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=', only contains C, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Plastics that used mTP as the primary dye but did not contain MAA or a 6Li salt had an average LO of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='12 (Figure 4d, Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Plastics that contained MAA and 6Li had a slight reduction in LO with values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='93, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='95, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='93, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='92 at 3 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' %, 5 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' %, 6 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' %, and 13 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' % MAA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' At 13 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' % MAA, the LO of a plastic that contains 6Li was reduced by 18% when compared to a plastic that contained no MAA or 6Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' This value of LO is nearly double the LO of an equivalent plastic that contained PPO instead of mTP as the primary dye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' We also compared the FoM for PSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' For plastics that contain PPO but no 6Li, the FoM is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='05, which provides a baseline for comparison after addition of MAA and 6Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Note that this FoM compares discrimination between gamma rays and fast neutrons whereas the FoM for plastics that contain 6Li compares discrimination between gamma rays and thermal neutrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The energy range used to determine FoM was 450-550 keVee, but that range was adjusted to capture the thermal neutron peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Upon addition of 13 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' % MAA and 6Li, the FoM decreases to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='24 and decreases for all plastics that were measured that contained MAA (Figure 5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' These PSD results can also be compared to plastics that contain mTP instead of PPO (Figure 5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' For plastics that contain mTP but no 6Li, the FoM is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Upon addition of MAA and 6Li, the FoM is between 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='95 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' This increase is mostly attributed to the high value of Qtail/Qtotal for the thermal-neutron capture spot along with the observation that plastics that contain mTP are less sensitive to the addition of the polar compounds that enable thermal-neutron capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The FoM for plastics that contained PPO decreased when MAA and 6Li were added whereas the FoM for plastics that contained mTP increased when MAA and 6Li were added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Production of large plastic scintillators As shown above, mTP may be promising for high performance scintillators that contain 6Li;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' however, the inconsistencies with solubility and the current availability at large scales in sufficient purity from commercial suppliers make mTP less suitable Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' a,b) Conditions that are not optimized for the production of large plastics produce defects like cracks and bubbles (a) and discoloration (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The left-most scintillator in (a) does not contain defects and serves as a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' These scintillators in (a) have a diameter of about 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='5 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The plastic scintillator in (b) is 40 cm in length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' c) When the curing conditions and environment are controlled, plastics scintillators can be produced in large scale, as shown in this photograph of a scintillator that is 5 cm in width atop a sheet of paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' d) The effective attenuation length was measured by placing a collimated gamma-ray source at set distances away from two PMTs and measuring the PMT response (black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The data were fitted with an exponential profile to estimate the effective attenuation length (19-21cm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' e) Large plastic scintillators are capable of PSD, as shown in this distribution that demonstrates an ability to separate signals from thermal neutrons and gamma rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' b)for the production of large plastic scintillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' For these large plastic scintillators, we selected PPO despite its lower LO and lower FoM at smaller scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Similarly, other secondary dyes like Exalite 404 (E404) may have the best performance in the plastic scintillators that we evaluated[27], but the cost of E404 might be prohibitive when compared to a secondary dye like 1,4-bis(2-methylstyryl)benzene (bisMSB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' When processing the plastics, all precursors except DVB and the radical initiator are slowly heated to temperatures between 60 and 80 oC until full dissolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The plastics are cured in glass or aluminium moulds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' To control the rate of polymerization, the radical initiator is added at concentrations of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='01 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' %, and plastics are cured at an initial temperature of 60 oC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Various times of curing were used, and a typical recipe would involve curing for 7 days at 60 oC followed by an additional 4 days of curing at 75 oC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Excessive radical concentration and/or heating during processing and curing could lead to defects like cracks and bubbles (Figure 6a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Precipitation of precursors that have lower solubility may occur if temperatures are too low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' When curing the plastics, oxygen is displaced by a steady flow of nitrogen;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' without nitrogen flow, discoloration can occur (Figure 6b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' These precautionary measures allow us to produce plastic scintillators without defects and with minimal discoloration (Figure 6c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The effective attenuation of a plastic scintillator that was 16” long was measured by placing a collimated gamma-ray source near the scintillator and measuring the total light detected by PMTs that are mounted on each end of the plastic scintillator (Figure 6d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The effective attenuation was determined to be about 19-21 cm, which is comparable to the value obtained in previous experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' [24] This plastic also had PSD capability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' a thermal-neutron spot is clearly separated from the gamma signal (Figure 6e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' After we optimized our process for synthesis of these large plastics, we outsourced production to Eljen Technologies who is currently producing 6Li-loaded prototypes with dimensions exceeding 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='5 m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' full characterization of scintillator performance at these large scales will be the subject of a future publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Conclusion By careful control of composition and processing, plastic scintillators that can discriminate between gamma rays, fast neutrons, and thermal neutrons can be produced at a scale of 1 kg or greater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The solubility of dopants that enable scintillation functionality and solubilizing additives like methacrylic acid (MAA) that may negatively affect performance were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Synthesis and processing procedures were developed for large plastic scintillators containing 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='1 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' % 6Li and high concentration (30 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' %) of PPO used as a primary dye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' These scintillators were capable of PSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' In these studies, various 6Li salts of aliphatic carboxylic acids were evaluated, and many were found to be suitable for the production of large plastic scintillators with the addition of MAA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The amount of MAA that was added to solubilize 6Li salts affected the scintillation performance but also determined the temperature that plastic scintillators could be produced at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' An alternative way to avoid the deleterious effects of MAA was discovered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' use of m-terphenyl instead of PPO improved plastic scintillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' However, m-terphenyl may have limitations like availability at large volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' With these considerations in mind, methods for the preparation of plastic scintillators loaded with 6Li were established and demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Large-volume pieces that could be used for large detectors were produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' [24,28] Such detectors will be important for future safeguards related to nuclear power production and for unravelling unknown aspects of particle physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Acknowledgements This work was performed under the auspices of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344 and was supported by the LLNL-LDRD Program under Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' 20-SI-003, release number LLNL-JRNL-839909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' We would like to thank Jacob Kim for careful reading and discussion of this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' The authors declare no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' References [1] W.' 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Chaley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Jobez, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Jourde, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Kajfasz, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' de Kerret, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Koang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Lefièvre, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Léon, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Lesquoy, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Mallet, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Menthe, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Metref, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Poinsignon, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Potheau, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Provasi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Stutz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Thion, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Thomas, J.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Accel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Spectrometers Detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Assoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Equip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' 374 (1996) 164–187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='org/10.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='1103/RevModPhys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='011003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' [7] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Bowden, A.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Greaves, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Hagmann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Lund, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Mengesha, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Weinbeck, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Winant, Experimental results from an antineutrino detector for cooperative monitoring of nuclear reactors, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Methods Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Accel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Spectrometers Detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Assoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Equip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' 572 (2007) 985–998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='nima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' [8] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Kuroda, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Oguri, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Kato, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Nakata, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Inoue, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Ito, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Minowa, A mobile antineutrino detector with plastic scintillators, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Methods Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Accel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Spectrometers Detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Assoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Equip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' 690 (2012) 41–47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='nima.' 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M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Lindner, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Bazoma, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Bouvier, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Bui, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Communeau, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Rigalleau, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Roy, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Vilajosana, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Yermia, Online monitoring of the Osiris reactor with the Nucifer neutrino detector, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' D.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Subedi, Observation of Reactor Antineutrinos with a Rapidly Deployable Surface-Level Detector, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' 13 (2020) 034028.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='org/10.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Wilhelmi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Wise, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Yeh, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Yen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Zhang, Measurement of the Antineutrino Spectrum from $^{235}$U Fission at HFIR with PROSPECT, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' 122 (2019) 251801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='251801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' [13] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Brooks, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Pringle, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Funt, Pulse Shape Discrimination in a Plastic Scintillator, IRE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' 7 (1960) 35–38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='1109/TNS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='1960.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='4315733.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' [14] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Zaitseva, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Rupert, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' PaweŁczak, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Glenn, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Martinez, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Carman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Faust, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Cherepy, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Payne, Plastic scintillators with efficient neutron/gamma pulse shape discrimination, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Methods Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Accel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Spectrometers Detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Assoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Equip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' 668 (2012) 88–93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='nima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='071.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' [15] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Zaitseva, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Glenn, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Mabe, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Carman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Hurlbut, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Inman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Payne, Recent developments in plastic scintillators with pulse shape discrimination, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Methods Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Accel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Spectrometers Detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Assoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Equip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' 889 (2018) 97–104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='nima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='093.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' [16] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Bertrand, M.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Accel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Spectrometers Detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Assoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Equip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' 776 (2015) 114–128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='nima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' [17] I.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Methods Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Accel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Spectrometers Detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Assoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Equip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' 751 (2014) 62–69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='nima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='027.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' [18] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Breukers, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Bartle, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Edgar, Transparent lithium loaded plastic scintillators for thermal neutron detection, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Methods Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Accel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Spectrometers Detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Assoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Equip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' 701 (2013) 58–61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='nima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='080.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' [19] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Zaitseva, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Glenn, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Paul Martinez, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Carman, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Pawełczak, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Faust, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Payne, Pulse shape discrimination with lithium-containing organic scintillators, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Methods Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Accel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Spectrometers Detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Assoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Equip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' 729 (2013) 747–754.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='nima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' [20] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Cherepy, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Sanner, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Beck, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Swanberg, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Tillotson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Payne, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Hurlbut, Bismuth- and lithium-loaded plastic scintillators for gamma and neutron detection, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Methods Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Accel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Spectrometers Detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Assoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Equip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' 778 (2015) 126–132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='nima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' [21] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Mabe, A.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Spectrometers Detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Assoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Equip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' 806 (2016) 80–86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='nima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' [22] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Frangville, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Hamel, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Bertrand, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Montbarbon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Grabowski, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Lynde, Large solubility of lithium carboxylates reaching high rates of 6 Li incorporation in polystyrene-based plastic scintillators for fast/thermal 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Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' 22 (1978) 1207–1223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='1002/app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='070220505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} 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+page_content=' Duvall, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Jovanovic, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Li, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Mabe, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Reedy, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Wu, SANDD: A directional antineutrino detector with segmented 6Li-doped pulse-shape-sensitive plastic scintillator, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Methods Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Accel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Spectrometers Detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Assoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Equip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' 1006 (2021) 165409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='nima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='165409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' [25] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Mabe, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Glenn, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Carman, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Zaitseva, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Payne, Transparent plastic scintillators for neutron detection based on lithium salicylate, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Methods Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Accel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Spectrometers Detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Assoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Equip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' 806 (2016) 80–86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='nima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' [26] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Paul Martinez, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Pawelczak, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Glenn, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Leslie Carman, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Zaitseva, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Payne, Pulse shape discrimination in non-aromatic plastics, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Methods Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Accel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Spectrometers Detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Assoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Equip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' 771 (2015) 28–31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='nima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' [27] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Zaitseva, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Glenn, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Carman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Mabe, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Payne, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Marom, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Wang, Multiple dye interactions in plastic scintillators: Effects on pulse shape discrimination, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Methods Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Accel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Spectrometers Detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Assoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Equip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' 978 (2020) 164455.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='nima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='164455.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' [28] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Li, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Classen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Dazeley, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Duvall, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Jovanovic, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Mabe, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Reedy, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Sutanto, A prototype for SANDD: A highly-segmented pulse- shape-sensitive plastic scintillator detector incorporating silicon photomultiplier arrays, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Methods Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Accel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Spectrometers Detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Assoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' Equip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' 942 (2019) 162334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='nima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} +page_content='162334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQf5AgM/content/2301.04185v1.pdf'} diff --git a/0NFQT4oBgHgl3EQfDTUM/vector_store/index.faiss b/0NFQT4oBgHgl3EQfDTUM/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..70b81b67476aeb88dd907249003bb9c81cc9dc1c --- /dev/null +++ b/0NFQT4oBgHgl3EQfDTUM/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:679baa2b66b51f430412223eec24c31ed36cd8834154f4696776efa2f3086a05 +size 6029357 diff --git a/1dE0T4oBgHgl3EQfuQGc/content/tmp_files/2301.02603v1.pdf.txt b/1dE0T4oBgHgl3EQfuQGc/content/tmp_files/2301.02603v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..36e07a1e3e918d68ce1c097594ddf77b2150cf7d --- /dev/null +++ b/1dE0T4oBgHgl3EQfuQGc/content/tmp_files/2301.02603v1.pdf.txt @@ -0,0 +1,790 @@ +Contextual Autonomy Evaluation of Unmanned +Aerial Vehicles in Subterranean Environments +Ryan Donald +Peter Gavriel +Adam Norton +S. Reza Ahmadzadeh +PeARL lab and NERVE Center +University of Massachusetts Lowell +Lowell, USA +Ryan Donald@student.uml.edu +Peter Gavriel@uml.edu +Adam Norton@uml.edu +Reza Ahmadzadeh@uml.edu +Abstract—In this paper we focus on the evaluation of con- +textual autonomy for robots. More specifically, we propose a +fuzzy framework for calculating the autonomy score for a +small Unmanned Aerial Systems (sUAS) for performing a task +while considering task complexity and environmental factors. +Our framework is a cascaded Fuzzy Inference System (cFIS) +composed of combination of three FIS which represent dif- +ferent contextual autonomy capabilities. We performed several +experiments to test our framework in various contexts, such as +endurance time, navigation, take off/land, and room clearing, +with seven different sUAS. We introduce a predictive measure +which improves upon previous predictive measures, allowing for +previous real-world task performance to be used in predicting +future mission performance. +Index Terms—Contextual Autonomy, Unmanned Aerial Vehi- +cles, Fuzzy Systems +I. INTRODUCTION +In today’s world, robots are expected to become increasingly +present by assisting humans in performing various tasks in +different environments. While some robots have been designed +for a single purpose, others can accomplish a variety of tasks +with different levels of autonomy. Measuring robot autonomy +is an important and ever evolving concept and existing meth- +ods for evaluating robot autonomy can be categorized into +two main families: contextual and non-contextual. While the +former methods consider mission and task-specific measures +(e.g., ALFUS [1], ACL [2]), the latter only rely on implicit +system capabilities and do not consider the mission and +environment features (e.g., NCAP [3], [4]). +Our study in this paper focuses on evaluating the contextual +autonomy for small Unmanned Aerial Systems (sUAS). Exist- +ing methods such as ALFUS [1] and MPP [5] share a similar +shortcoming in that neither provides a simple implementation +for use with real-world systems. Another drawback of existing +methods that our approach addresses is the lack of a consistent +process for breaking down tasks into sub-tasks and combining +scores calculated for sub-tasks into a unified score for the given +task. In this paper we propose a method for evaluating the +contextual autonomy of sUAS based on a fuzzy interface that +allows the operator to design and modify the evaluation system +using linguistic reasoning. We designed four indoor tasks +(endurance time, navigation, takeoff/land, and room clearing) +and tested our interface in various experiments with seven +different sUAS. Our results show that the proposed approach +calculates a contextual autonomy score that can be used to +rank the systems for each context. +II. RELATED WORK +Some of the first and more simplistic methods of categoriz- +ing autonomous systems are the Levels of Automation (LOA) +proposed by Sheridan [6] and its later expansion [7]. LOA +defines automation as “the full or partial replacement of a +function previously carried out by the human operator” in a 1 +to 10 range; 1 being full control by the human and 10 being +full control by the computer. LOA does not accurately describe +how outside factors can affect the autonomous capability of a +system. While it could theoretically be applied to a robot, it +would not be accurate as it fails to accommodate for differing +degrees of difficulty in tasks, and environmental factors. +Another evaluation method is known as the Autonomy +Control Levels (ACL) [2]. ACL is designed for Unmanned +Aerial Vehicles (UAV), and operates on a similar basis of uti- +lizing autonomy levels from 0-10, with 0 being fully remotely +controlled by a pilot, and 10 being a human-like system. +These levels closely resemble the 10 LOA, following the same +concept. The ACL characterizes each system according to +four metrics, which attempt to categorize different areas of +autonomous behaviors for the system. In each of these, an +autonomy level from 0-10 is given based upon these behaviors. +This system has a similar drawback, in that it does not account +for difficulties in the mission itself. +Another method is the Autonomy and Technological Readi- +ness Assessment (ATRA) [8]. ATRA attempts to combine both +the basic theory behind the Autonomy Level, and the Tech- +nology Readiness Level (TRL) metric into one framework [8]. +TRL utilizes these two metrics in an attempt to evaluate the +autonomy level provided by different technologies onboard the +UAS. This is emphasized as a solution for the gap between +existing theoretical work and technological advances in the +UAS autonomy space. +Autonomy Levels for Unmanned Systems (ALFUS) is a +method for defining the autonomy of a system in terms of +three different axes [9]. ALFUS has a strong theoretical basis, +but somewhat impractical in the real-world implementation +due to the lack of maturity in some of these systems, as well +arXiv:2301.02603v1 [cs.RO] 6 Jan 2023 + +as the inability of most, if not all, available systems to reach +the upper levels of the three axes. +The three axes mentioned are known as the Mission Com- +plexity (MC), Environmental Complexity (EC), and Human +Independence (HI) axes. Each one of these axes pertains itself +to a different aspect of the contextual autonomy of a system. +The MC axis pertains mostly towards the difficulty of the tasks +and movements required of the system to complete the task +(e.g. maneuvers, speed, searching). Alternatively, the EC axis +concerns itself with the difficulty in the performance of the +task caused by environmental factors (e.g. Lighting, Obstacles, +Enclosed Spaces). Lastly, the HI axis is representative of the +level of independence between the user and the system (e.g. +task planning, task execution). +Due to the ability to split the representation of a system’s +autonomy into these three axes, it allows for the character- +ization and evaluation of system’s autonomy in real world +tests, including the impact that both the environment and the +mission profile can have on the system’s autonomy. Our work +in this paper is based off many of the ideas put forward +through ALFUS, and we utilize it as a foundational part of +our contextual autonomy evaluation. +The Mission Performance Potential is proposed as a method +for the evaluation of a unmanned system’s autonomous per- +formance, as well as a predictor for future missions [5]. +This method provides a metric which represents the max- +imum performance of a system in a given mission at a +given autonomy level. Uniquely, this method includes both +non-contextual autonomy metrics, and contextual autonomy +metrics, and provides a single output prediction based on both +types of data. +One of the drawbacks of MPP is that it only provides +a prediction of the performance of a system at a specified +autonomy level for a specified mission. In other words, this +does not evaluate how a real system performs, but rather the +maximum potential for a system to perform. Our approach +instead calculated the actual autonomy of a system based on +actual data from real-world experiments. +Fig. 1: Our cascaded Fuzzy Inference System used for calcu- +lating a contextual autonomy score for a performed task. +III. FRAMEWORK +ALFUS’ summary model works with a set of metrics for +each of its three axes, as well as a system of levels from +0 to 10. These levels are based upon possible answers from +those metrics, to provide a level evaluation of a system. As +a generic framework, ALFUS tends to have a very broad, +and somewhat open to interpretation, definition of metrics. +For instance, in the case of the EC axis, it ranges from a +“simple environment,” to an “extreme environment.” However, +the summary model describes the system in terms of an +autonomy level for each axis, while the Contextual Autonomy +Capability within ALFUS provides an actual score for each +axis. Due to the autonomy level evaluation, there is some +ambiguity when characterizing systems. This is one of the +main concerns with ALFUS, in that while it does provide a +strong theoretical background, the actual implementation of +the ideas with real-world systems is not as clear. +We utilize Takagi-Sugeno Fuzzy Inference Systems (FIS) +as a means to combine different metrics in an evaluation of +an sUAS which is both easy to use, and allows us to use +some data which is either not easily defined numerically, or +inherently qualitative about the environment, combined with +standard quantitative metrics. Fuzzy inferences also allow for +slight deviations in a metric to not cause a drastic change +in the evaluation of that sUAS. We designed a set of tests +with various mission and environment complexity levels (see +Section V), and defined a fuzzy inference system for each test. +Unlike MPP [5], our fuzzy inference systems are based on the +three-axis model used in ALFUS, by creating an individual FIS +for metrics associated with each axis (i.e., MC, EC, HI), and +an additional FIS which combines these three outputs into a +single score. This structure representing a cascaded FIS (cFIS) +is illustrated in Fig. 1. For each test, the outcome of the FIS +for all three axis is fed into a combining FIS that produces a +final autonomy score. Each FIS in our cFIS is a Sugeno-type +FIS with multiple inputs and one output. For each input of an +FIS, we consider three membership functions (MFs) labeled +as low, medium, and high. Without loss of generality, we used +triangular MFs, however, other types of MF can be used. The +input variables used in different tests and their corresponding +MF parameters have been reported in Table II. The output of +each Sugeno-type FIS has five singleton MFs (i.e., constant): +very bad, bad, medium, good, very good. Our FIS’ use a +triangular fuzzifier and a Sugeno defuzzifier (i.e., weighted +average output). For each FIS, we defined a rule base (i.e., a +set of linguistic rules). +In the cFIS structure in Fig. 1, the defuzzified output of +each FIS is a value in the range of [0, 1]. For the initial three +FIS, 0 and 1 represent the lowest and highest complexity, +respectively. In the case of the final FIS, 0 and 1 represent the +lowest and the highest autonomy, respectively. If we define the +singleton value of each output function as zi, and the degree +to which each output is weighted based upon the ruleset as +wi, then the output final score can be calculated as follows: +s = +�N +i=1 wizi +�N +i=1 wi +(1) +where N represents the number of rules in the rule base. +Table I reports an example of the fuzzy ruleset we used. The +advantage of this system is that we can utilize many different + +Human +Independence +FIS +Environmental +Output +Input Data +Complexity +TestFIS +Score +FIS +Mission +Complexity +FISFig. 2: From left to right, top to bottom: Cleo Robotics Dronut, +Flyability Elios 2, Lumenier Nighthawk 3, Parrot ANAFI USA GOV, +Skydio X2D, Teal Drones Golden Eagle, Vantage Robotics Vesper +types of data, and clearly define the ranges for each value, +allowing the pilots performing the tests to provide feedback +on the membership functions and rulesets. +Mission Complexity Axis +Low +Medium +High +Environment +Complexity +Axis +Low +Very Bad +Bad +Medium +Medium +Bad +Medium +Good +High +Medium +Good +Very good +TABLE I: Fuzzy Ruleset utilized in our final combinational FIS +IV. UAS PLATFORMS +Fig. 2 illustrates seven sUAS platforms evaluated in our ex- +periments. The platforms include: the Cleo Robotics Dronut1, +Flyability Elios 22, Lumenier Nighthawk 33, Parrot ANAFI +USA GOV4, Skydio X2D5, Teal Drones Golden Eagle6, and +Vantage Robotics Vesper7. These platforms provide a wide +ranging set of capabilities and use cases. For instance, Parrot, +Skydio X2D, Golden Eagle, and Vesper were developed for +outdoor reconnaissance, whereas the Dronut and Elios 2 were +developed for indoor reconnaissance and inspection, specif- +ically in urban and industrial environments. Previously, we +have used the same set of sUAS for a non-contextual bench- +marking [4], [10]. In our evaluations, we have anonymized the +data by assigning the platforms labels A through G without +any specific ordering or correlation. +V. TEST DESIGN +To evaluate the contextual autonomy of our platforms, we +have designed several tests across a spectrum of areas. The +variables for which we collected data for each test is reported +in Table II. In this section, we describe each test briefly. +As shown in Fig. 3 all tests have been designed for indoor +environments. +1https://cleorobotics.com/ +2https://www.flyability.com/elios-2 +3https://www.lumenier.com/ +4https://www.parrot.com/us/drones/anafi +5https://www.skydio.com/skydio-x2 +6https://tealdrones.com/suas-golden-eagle/ +7https://vantagerobotics.com/vesper/ +A. Runtime Endurance +This family of tests focuses on the battery life of the +system in various operational profiles. As shown in Fig. 3a, +the specific test we use from this group focuses on the +system flying continuously in a figure-8 pattern. The main +performance metric for the test is the test duration. +B. Navigation +We have designed two main types of navigation tests, each +with several profiles defined based on the type of movement +(horizontal, vertical, or both) and the type of confinement +(horizontal, vertical, or both). As shown in Fig. 3b, navigation +through confined spaces involves traversal into and out of a +continuously confined space, with tests for hallway, tunnel, +stairwell, and shaft. Navigation through apertures involves +transient traversal through an opening, with tests for doorway +and window. Each navigation environment is characterized +according to the dimensions of the confined space or aperture, +lighting, surface textures, and the presence of obstructions on +either side of the confined space or aperture. The main metrics +of performance are efficacy and average navigation time. +C. Room Clearing +In this test method, the system performs a visual inspection +of an example room whose walls, floor, and ceiling are out- +fitted with visual acuity targets which contain nested Landolt +C symbols of decreasing size. As shown in Fig. 3c, the test +was performed under two conditions: with and without using +camera zoom. The main performance metrics are duration, +coverage, and average acuity. +D. Takeoff and Land/Perch +As illustrated in Fig. 3d, these tests evaluate the system’s +ability to takeoff and land or perch in various environments +that may be affected by stabilization issues or preventative +safety checks from the system. The conditions tested vary the +angle of the ground plane (flat, 5° and 10° pitch and roll) +and the presence of obstructions (1.2-2.4m overhead, 0.6-1.2m +lateral). The main performance metric is efficacy. +VI. RESULTS +Utilizing our framework outlined in Section III, we calculate +a performance score for each sUAS based upon the conditions +and performance metrics detailed below. As mentioned above, +our system provides a single score for each of the three +attributes, the EC, MC and HI axes, and utilizes those scores +to provide a single score for the entire test. It should be noted +that although we consider all three axes in our cFIS structure, +due to lack of data, we consider the lowest level (i.e., full +tele-operation) for the HI axis across all tests. The test-specific +details of the structure is discussed in corresponding sections. +Another factor to note is that for some of these experiments, +some sUAS were not available at the time of testing, due to the +sUAS being repaired, or other circumstances. In this case, we +attempt to remedy this by calculating a partial point achieved +by the sUAS. For situations where data for an entire test is + +(a) Runtime Endurance test showing a system performing a specified +movement +(b) Navigation tests (left to right, top to bottom): hallway, tunnel, +stairwell, shaft, door, window. +(c) Room clearing test showing a system inspecting surfaces for +visual acuity targets. +(d) Takeoff and land/perch tests showing variations in ground plane +angle (top row) and nearby obstructions (bottom row). +Fig. 3: Tests designed for the evaluation of sUAS contextual autonomy. +missing, we cannot fully evaluate the sUAS. However, we have +evaluated sUAS for individual tests for which the data was +recorded. Despite these edge cases, most sUAS had available +data, which was used in our evaluations. +As mentioned before, the proposed structure in Fig. 1, +was adapted to each specific test. The resulting cascaded FIS +are depicted in Fig. 6 each of which represents a cFIS for +a specific test. More specifically, each sub-figure shows the +inputs and outputs of each cFIS, as well as how the FIS +modules are connected. This is meant to provide a visual +aid, which can be useful to keep track of each FIS, as we +discuss the results. Associated membership functions for each +input, and the outputs, can be found in Table II. Some of +the FIS surfaces, which show the relationship between two +input values in an FIS, and the corresponding output value, are +shown in Fig. 4. Additionally, numerical results are reported +in Fig.5 and Table III. +A. Test Results +Runtime Endurance: The runtime endurance test is likely +the simplest of the tests performed, however, it is still a useful +test in gauging how a sUAS might perform in a real mission. +Fig. 3a illustrates the runtime endurance test design including +the navigation path and two stands. The adapted cFIS for +this test is shown in Fig. 6a with four inputs: number of +obstructions, number of crashes, light level and speed. +Despite the simplicity of this test, some sUAS did not +performed well largely due to slow speed. Our results indicate +that some sUAS may have trouble in portions of this real- +world mission which requires both speed and maneuverability. +(a) Final FIS Surface +(b) Through Aperture test EC +(c) Takeoff and Land/Perch EC +(d) Through Corridor MC +Fig. 4: FIS surfaces for different tests. +In should be noted that during this test only four sUAS were +available. +Navigation: In the navigation tests, due to the differences +between the through corridor and through apertures tests, two +slightly different cascaded FIS were designed. These can be +found in Fig. 6b and Fig. 6d. The input to the through corri- +dors cFIS includes area (cross section), light level, verticality, +coverage, number of crashes, and duration. The inputs to the +through apertures cFIS include area, light level, number of + + 10 degrees +10 degrees +5 degrees +5degree +1.2 mCutput_Score +10.5 +0 +0.5 +0.5 +MC +0 +ECOutput_Score +0.6 +0.4 +0.2 +0: +0 +2 +200 +400 +4 +600 +Area +Light0.7 +10.5 +0.3 +10 +10 +5 +5 +Pitch +0 +RollOutput_Score +0.5 +00 +2 +0.5 +Crashes +3 +CompletionXVariables +Description +MFs: Triangular{Low, Medium, High} +Area +Aperture/Hallway Cross-Section (m2) +[0, 0, 2.7] +[0.6, 3, 5.4] +[3.3, 6, 6] +Light +Ambient Light Level (Lux) +[0, 0, 337.5] +[75, 375, 675] +[412.5, 750, 750] +Vert +Verticality (°) +[0, 0, 37.5] +[7.5, 45, 82.5] +[52.5, 90, 90] +Crash +Number of Crashes +[0, 0, 1.25] +[0.5, 1.5, 2.5] +[1.75, 3, 3] +Rollovers +Number of Rollovers +[0, 0, 1.25] +[0.5, 1.5, 2.5] +[1.75, 3, 3] +Comp. % +Completion Percentage +[0, 0, 0.55] +[0.15, 0.6, 0.92] +[0.7, 1, 1] +Yaw/Pitch +Static Yaw/Pitch Angle (°) +[0, 0, 4.17] +[0.83, 5, 9.12] +[5.83, 10, 10] +VR +Static Vertical Obstruction (m) +[0.6, 0.6, 1.1] +[0.7, 1.2, 1.7] +[1.3, 1.8, 1.8] +LR +Static Lateral Obstruction (m) +[1.2, 1.2, 2.2] +[1.4, 2.4, 3.4] +[2.6, 3.6, 3.6] +Coverage +Coverage Percentage +[0, 0, 0.55] +[0.15, 0.6, 0.92] +[0.7, 1, 1] +Cs Detected +Landolt C Depth Detected +[0, 0, 50] +[10, 50, 90] +[50, 100, 100] +Duration +Duration of Test (Minutes) +[2.5, 2.5, 5.25] +[3.05, 5.25, 7.45] +[5.25, 8, 8] +Obs. +Number of Obstructions +[0, 0, 2.5] +[1, 3, 5] +[3.5, 6, 6] +Output Variable +Description +Sugeno MFs: Constant {Very Low to Very High} +Score +Combined Defuzzified Score +[0, 0.25, 0.5, 0.75, 1] +TABLE II: Membership Functions (MFs) for each input and output variable used in an FIS in the evaluation of these sUAS. +Fig. 5: Scores for each sUAS as a percentage of the maximum score +possible on the y-axis, with each test on the x-axis +crashes, and completion percentage. As shown above in Fig. 5, +each sUAS that performed the through apertures test, achieved +a maximum score, besides sUAS G, which performed slightly +worse, due to both issues in correctly traversing the aperture, +as well as being the only sUAS to suffer a crash during the test. +Next, for the through corridors test, As shown in Fig. 5, there +is more variance in the performance between each sUAS, with +UAS A performing the best, and UAS E performing the worst, +even though sUAS E tied for best in through apertures. This +is important, as there is more room for error while traversing +corridors, than there is traversing an aperture. +Takeoff and Land/Perch: For the takeoff and land/perch +test, the cFIS diagram can be found in Fig. 6e. The inputs +include pitch, yaw, number of crashes, completion percentage, +vertical and lateral obstruction, and number of rollovers. As +can be seen in Fig. 5 and Table III, sUAS A and sUAS G +perform best across both sections of the test and thus provide +the highest level of autonomy. Likewise, sUAS B performs the +worst in both portions of the test, showcasing a lower level of +autonomy, compared to the other sUAS. This evaluation allows +for the characterization of how an sUAS may perform during +portions of a mission which requires the system to takeoff or +land in a specified spot, of varying difficulty. +Room Clearing: Since the room clearing test is done in +a static environment, we included a time constraint in our +testing making the evaluation to focus on the performance +of the sUAS in regards to the Mission Complexity axis. The +designed cFIS can be seen in Fig. 6c. Inputs include light level, +number of obstructions, number of crashes, duration, coverage, +and landolt C depth detected. Results are found in Fig. 5, and +Table III. In this test, the strongest performer was UAS E; +however, most of the UAS performed closely to each other. +Surprisingly, a strong performance in the runtime endurance +test did not necessarily correlate to a strong performance in +this test. Both of these tests require a system with good maneu- +verability capabilities, but this test also requires a controller +which allows the user to visually identify different landmarks. +B. Final Results +UAS +T.C. +T.A. +Takeoff +Land +R.E. +R.C. +Predictive +Score +A +1.0 +1.0 +1.0 +1.0 +0.5 +0.76 +0.85 +B +0.90 +1.0 +0.71 +0.87 +0.76 +0.73 +0.82 +C +0.84 +1.0 +1.0 +0.87 +- +- +0.92 +D +- +- +0.75 +0.97 +0.65 +0.75 +0.77 +E +0.80 +1.0 +0.82 +0.89 +- +0.85 +0.87 +F +- +- +0.99 +0.91 +- +- +0.95 +G +0.83 +0.83 +1.0 +1.0 +0.5 +0.79 +0.80 +TABLE III: Scores of each sUAS for each test, as well as a weighted +multiple which allows for an overall evaluation of each sUAS +Contextual autonomy evaluations are concerned with the +performance of a system within an environment while per- +forming a specific task with a known level of complexity. +However, calculating an overall score that represents an av- +erage autonomy for a given system in a spectrum of tests +and environment is desirable. To combine the test scores into +a single score, we utilize a weighted product, with equal +weightings for each test, as we did previously in our non- +contextual evaluation [4]. The weighted product represented +as +P = +M +� +i +φwi +i , +(2) +where M is the number of individual tests, φi represents an +individual test score, and wi the weight assigned to that test. + +1.0 +T +T +0.9 +UASA +0.8 - +UAS B +UAS C +0.7 - +UAS D +UAS E +0.6 +UAS F +UAS G +0.5 +T.C. +T.A. +R.E. +R.C. +Takeoff +Land(a) Runtime Endurance (R.E.) test FIS +(b) Through Apertures (T.A.) test FIS +(c) Room Clearing (R.C.) test FIS +(d) Through Corridors (T.C.) test FIS +(e) Takeoff and Land/Perch test FIS +Fig. 6: Diagrams of each system of cascaded FIS utilized to +calculate scores for each test +has several benefits including that different test results can +be combined without requiring normalization or scaling. The +results for each sUAS are shown in Table III. It is important to +note that many of the sUAS perform better than the others in +some tests, but worse in others. One example of this is UAS +A, which has the fourth highest weighted multiple (overall +score), while performing the best in four out of six tests. This +is due to the sUAS’s relatively poor performance in both the +runtime endurance test, as well as in the room clearing test. +The use case of a singular score like this presents itself when +a user would like to know which sUAS is likely to provide +the most overall autonomy, across multiple tests and different +environment. +VII. CONCLUSIONS AND DISCUSSIONS +In this paper, we proposed a framework for evaluation of +contextual autonomy for robotic systems. Our framework con- +sists of a cascaded Fuzzy Inference System (cFIS) that com- +bines test results over three axes of evaluation (mission com- +plexity, environment complexity and human independence) +introduced by the ALFUS framework. We have designed +four tests with different mission complexity and environment +complexity levels and performed several experiments with +several sUAS, and we have shown that our modular framework +is adaptable to different tests. For future work, we plan to +extend our framework for performance evaluation. +To achieve this, a desired mission can be decomposed into +base tasks, such as takeoff/landing, traversing through environ- +ments/apertures, clearing rooms, and general maneuverability. +The user then can define a set of weights and calculate a +potential performance score of a sUAS for the target mission. +Unlike MPP [5], however, we suggest a method which is based +upon performance in set tasks, rather than a combination of +non-contextual attributes, and environmental factors. +ACKNOWLEDGMENT +This work is sponsored by the Department of the Army, U.S. +Army Combat Capabilities Development Command Soldier +Center, award number W911QY-18-2-0006. Approved for +public release #PR2022 88282 +REFERENCES +[1] P. J. Durst and W. Gray, “Levels of autonomy and autonomous system +performance assessment for intelligent unmanned systems,” Engineer- +ing research and development center Vicksburg Ms Geotechnical and +Structures lab, Tech. Rep., 2014. +[2] B. T. Clought, “Metrics, schmetrics! how the heck do you determine +a uav’s autonomy anyway?” Air Force Research Laboratory, Wright- +Pattterson Air Force Base, OH, Tech. Rep., August 2002. +[3] P. J. Durst, W. Gray, and M. Trentini, “A non-contextual model for +evaluating the autonomy level of intelligent unmanned ground vehicles,” +in Proceedings of the 2011 Ground Vehicle Systems Engineering and +Technology Symposium, 2011. +[4] B. Hertel, R. Donald, C. Dumas, and S. R. Ahmadzadeh, “Methods +for combining and representing non-contextual autonomy scores for +unmanned aerial systems,” International Conference on Automation, +Robotics, and Applications (ICARA), vol. 8th, pp. 145–149, 2022. +[5] P. Durst, W. Gray, A. Nikitenko, J. Caetano, M. Trentini, and R. King, +“A framework for predicting the mission-specific performance of au- +tonomous unmanned systems,” International Conference on Intelligent +Robots and Systems, pp. 1962–1969, 2014. +[6] T. B. Sheridan, “Automation, authority, and angst – revisited,” vol. 35, +September 1991, pp. 2–6. +[7] R. Parasuraman, T. B. Sheridan, and C. D. Wickens, “A model for types +and levels of human interaction with automation,” IEEE Transactions on +Systems, Man, and Cybernetics–Part A: Systems and Humans, vol. 30, +pp. 286–297, May 2000. +[8] F. Kendoul, “Towards a unified framework for uas autonomy and +technology readiness assessment (ATRA),” pp. 55–71, 2 2013. +[9] H.-M. Huang, “Autonomy levels for unmanned systems framework; +volume II: Framework models,” NIST Special Publication: Gaithersburg, +MD, USA, p. 30, 2007. +[10] A. Norton, R. Ahmadzadeh, K. Jerath, P. Robinette, J. Weitzen, T. Wick- +ramarathne, H. Yanco, M. Choi, R. Donald, B. Donoghue et al., +“Decisive test methods handbook: Test methods for evaluating suas in +subterranean and constrained indoor environments, version 1.1,” arXiv +preprint arXiv:2211.01801, 2022. + +Obs. +Crashes +Environmental +Mission +Complexity +Complexity +Light +FIS +FIS +Level +Speed +Output +Test FIS +ScoreArea +Crashes +Environmental +Mission +Complexity +Complexity +Light +FIS +FIS +Level +Comp. % +Output +Test FIS +ScoreCoverage +Cs +Detected +Crashes +Light +Environmental +Mission +Complexity +Complexity +Duration +FIS +FIS +Ohs. +Output +Test FIS +ScoreArea +Crashes +Environmental +Mission +Complexity +Complexity +Light +FIS +FIS +Level +Comp. % +Vert. +Output +Test FIS +ScoreCrashes +Pitch +Rollovers +Yaw +Environmental +Mission +Complexity +Complexity +Comp. % +LR +FIS +FIS +VR +Output +Test FIS +Score \ No newline at end of file diff --git a/1dE0T4oBgHgl3EQfuQGc/content/tmp_files/load_file.txt b/1dE0T4oBgHgl3EQfuQGc/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bbe3768f69ae4357f7ddecbf926d854c4393dde2 --- /dev/null +++ b/1dE0T4oBgHgl3EQfuQGc/content/tmp_files/load_file.txt @@ -0,0 +1,472 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf,len=471 +page_content='Contextual Autonomy Evaluation of Unmanned Aerial Vehicles in Subterranean Environments Ryan Donald Peter Gavriel Adam Norton S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Reza Ahmadzadeh PeARL lab and NERVE Center University of Massachusetts Lowell Lowell, USA Ryan Donald@student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='uml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='edu Peter Gavriel@uml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='edu Adam Norton@uml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='edu Reza Ahmadzadeh@uml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='edu Abstract—In this paper we focus on the evaluation of con- textual autonomy for robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' More specifically, we propose a fuzzy framework for calculating the autonomy score for a small Unmanned Aerial Systems (sUAS) for performing a task while considering task complexity and environmental factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Our framework is a cascaded Fuzzy Inference System (cFIS) composed of combination of three FIS which represent dif- ferent contextual autonomy capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' We performed several experiments to test our framework in various contexts, such as endurance time, navigation, take off/land, and room clearing, with seven different sUAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' We introduce a predictive measure which improves upon previous predictive measures, allowing for previous real-world task performance to be used in predicting future mission performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Index Terms—Contextual Autonomy, Unmanned Aerial Vehi- cles, Fuzzy Systems I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' INTRODUCTION In today’s world, robots are expected to become increasingly present by assisting humans in performing various tasks in different environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' While some robots have been designed for a single purpose, others can accomplish a variety of tasks with different levels of autonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Measuring robot autonomy is an important and ever evolving concept and existing meth- ods for evaluating robot autonomy can be categorized into two main families: contextual and non-contextual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' While the former methods consider mission and task-specific measures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=', ALFUS [1], ACL [2]), the latter only rely on implicit system capabilities and do not consider the mission and environment features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=', NCAP [3], [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Our study in this paper focuses on evaluating the contextual autonomy for small Unmanned Aerial Systems (sUAS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Exist- ing methods such as ALFUS [1] and MPP [5] share a similar shortcoming in that neither provides a simple implementation for use with real-world systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Another drawback of existing methods that our approach addresses is the lack of a consistent process for breaking down tasks into sub-tasks and combining scores calculated for sub-tasks into a unified score for the given task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' In this paper we propose a method for evaluating the contextual autonomy of sUAS based on a fuzzy interface that allows the operator to design and modify the evaluation system using linguistic reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' We designed four indoor tasks (endurance time, navigation, takeoff/land, and room clearing) and tested our interface in various experiments with seven different sUAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Our results show that the proposed approach calculates a contextual autonomy score that can be used to rank the systems for each context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' RELATED WORK Some of the first and more simplistic methods of categoriz- ing autonomous systems are the Levels of Automation (LOA) proposed by Sheridan [6] and its later expansion [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' LOA defines automation as “the full or partial replacement of a function previously carried out by the human operator” in a 1 to 10 range;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' 1 being full control by the human and 10 being full control by the computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' LOA does not accurately describe how outside factors can affect the autonomous capability of a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' While it could theoretically be applied to a robot, it would not be accurate as it fails to accommodate for differing degrees of difficulty in tasks, and environmental factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Another evaluation method is known as the Autonomy Control Levels (ACL) [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' ACL is designed for Unmanned Aerial Vehicles (UAV), and operates on a similar basis of uti- lizing autonomy levels from 0-10, with 0 being fully remotely controlled by a pilot, and 10 being a human-like system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' These levels closely resemble the 10 LOA, following the same concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' The ACL characterizes each system according to four metrics, which attempt to categorize different areas of autonomous behaviors for the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' In each of these, an autonomy level from 0-10 is given based upon these behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' This system has a similar drawback, in that it does not account for difficulties in the mission itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Another method is the Autonomy and Technological Readi- ness Assessment (ATRA) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' ATRA attempts to combine both the basic theory behind the Autonomy Level, and the Tech- nology Readiness Level (TRL) metric into one framework [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' TRL utilizes these two metrics in an attempt to evaluate the autonomy level provided by different technologies onboard the UAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' This is emphasized as a solution for the gap between existing theoretical work and technological advances in the UAS autonomy space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Autonomy Levels for Unmanned Systems (ALFUS) is a method for defining the autonomy of a system in terms of three different axes [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' ALFUS has a strong theoretical basis, but somewhat impractical in the real-world implementation due to the lack of maturity in some of these systems, as well arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='02603v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='RO] 6 Jan 2023 as the inability of most, if not all, available systems to reach the upper levels of the three axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' The three axes mentioned are known as the Mission Com- plexity (MC), Environmental Complexity (EC), and Human Independence (HI) axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Each one of these axes pertains itself to a different aspect of the contextual autonomy of a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' The MC axis pertains mostly towards the difficulty of the tasks and movements required of the system to complete the task (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' maneuvers, speed, searching).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Alternatively, the EC axis concerns itself with the difficulty in the performance of the task caused by environmental factors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Lighting, Obstacles, Enclosed Spaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Lastly, the HI axis is representative of the level of independence between the user and the system (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' task planning, task execution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Due to the ability to split the representation of a system’s autonomy into these three axes, it allows for the character- ization and evaluation of system’s autonomy in real world tests, including the impact that both the environment and the mission profile can have on the system’s autonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Our work in this paper is based off many of the ideas put forward through ALFUS, and we utilize it as a foundational part of our contextual autonomy evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' The Mission Performance Potential is proposed as a method for the evaluation of a unmanned system’s autonomous per- formance, as well as a predictor for future missions [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' This method provides a metric which represents the max- imum performance of a system in a given mission at a given autonomy level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Uniquely, this method includes both non-contextual autonomy metrics, and contextual autonomy metrics, and provides a single output prediction based on both types of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' One of the drawbacks of MPP is that it only provides a prediction of the performance of a system at a specified autonomy level for a specified mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' In other words, this does not evaluate how a real system performs, but rather the maximum potential for a system to perform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Our approach instead calculated the actual autonomy of a system based on actual data from real-world experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' 1: Our cascaded Fuzzy Inference System used for calcu- lating a contextual autonomy score for a performed task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' FRAMEWORK ALFUS’ summary model works with a set of metrics for each of its three axes, as well as a system of levels from 0 to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' These levels are based upon possible answers from those metrics, to provide a level evaluation of a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' As a generic framework, ALFUS tends to have a very broad, and somewhat open to interpretation, definition of metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' For instance, in the case of the EC axis, it ranges from a “simple environment,” to an “extreme environment.” However, the summary model describes the system in terms of an autonomy level for each axis, while the Contextual Autonomy Capability within ALFUS provides an actual score for each axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Due to the autonomy level evaluation, there is some ambiguity when characterizing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' This is one of the main concerns with ALFUS, in that while it does provide a strong theoretical background, the actual implementation of the ideas with real-world systems is not as clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' We utilize Takagi-Sugeno Fuzzy Inference Systems (FIS) as a means to combine different metrics in an evaluation of an sUAS which is both easy to use, and allows us to use some data which is either not easily defined numerically, or inherently qualitative about the environment, combined with standard quantitative metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Fuzzy inferences also allow for slight deviations in a metric to not cause a drastic change in the evaluation of that sUAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' We designed a set of tests with various mission and environment complexity levels (see Section V), and defined a fuzzy inference system for each test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Unlike MPP [5], our fuzzy inference systems are based on the three-axis model used in ALFUS, by creating an individual FIS for metrics associated with each axis (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=', MC, EC, HI), and an additional FIS which combines these three outputs into a single score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' This structure representing a cascaded FIS (cFIS) is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' For each test, the outcome of the FIS for all three axis is fed into a combining FIS that produces a final autonomy score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Each FIS in our cFIS is a Sugeno-type FIS with multiple inputs and one output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' For each input of an FIS, we consider three membership functions (MFs) labeled as low, medium, and high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Without loss of generality, we used triangular MFs, however, other types of MF can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' The input variables used in different tests and their corresponding MF parameters have been reported in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' The output of each Sugeno-type FIS has five singleton MFs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=', constant): very bad, bad, medium, good, very good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Our FIS’ use a triangular fuzzifier and a Sugeno defuzzifier (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=', weighted average output).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' For each FIS, we defined a rule base (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=', a set of linguistic rules).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' In the cFIS structure in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' 1, the defuzzified output of each FIS is a value in the range of [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' For the initial three FIS, 0 and 1 represent the lowest and highest complexity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' In the case of the final FIS, 0 and 1 represent the lowest and the highest autonomy, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' If we define the singleton value of each output function as zi, and the degree to which each output is weighted based upon the ruleset as wi, then the output final score can be calculated as follows: s = �N i=1 wizi �N i=1 wi (1) where N represents the number of rules in the rule base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Table I reports an example of the fuzzy ruleset we used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' The advantage of this system is that we can utilize many different Human Independence FIS Environmental Output Input Data Complexity TestFIS Score FIS Mission Complexity FISFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' 2: From left to right, top to bottom: Cleo Robotics Dronut, Flyability Elios 2, Lumenier Nighthawk 3, Parrot ANAFI USA GOV, Skydio X2D, Teal Drones Golden Eagle, Vantage Robotics Vesper types of data, and clearly define the ranges for each value, allowing the pilots performing the tests to provide feedback on the membership functions and rulesets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Mission Complexity Axis Low Medium High Environment Complexity Axis Low Very Bad Bad Medium Medium Bad Medium Good High Medium Good Very good TABLE I: Fuzzy Ruleset utilized in our final combinational FIS IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' UAS PLATFORMS Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' 2 illustrates seven sUAS platforms evaluated in our ex- periments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' The platforms include: the Cleo Robotics Dronut1, Flyability Elios 22, Lumenier Nighthawk 33, Parrot ANAFI USA GOV4, Skydio X2D5, Teal Drones Golden Eagle6, and Vantage Robotics Vesper7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' These platforms provide a wide ranging set of capabilities and use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' For instance, Parrot, Skydio X2D, Golden Eagle, and Vesper were developed for outdoor reconnaissance, whereas the Dronut and Elios 2 were developed for indoor reconnaissance and inspection, specif- ically in urban and industrial environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Previously, we have used the same set of sUAS for a non-contextual bench- marking [4], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' In our evaluations, we have anonymized the data by assigning the platforms labels A through G without any specific ordering or correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' TEST DESIGN To evaluate the contextual autonomy of our platforms, we have designed several tests across a spectrum of areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' The variables for which we collected data for each test is reported in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' In this section, we describe each test briefly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' 3 all tests have been designed for indoor environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' 1https://cleorobotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='com/ 2https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='flyability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='com/elios-2 3https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='lumenier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='com/ 4https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='parrot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='com/us/drones/anafi 5https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='skydio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='com/skydio-x2 6https://tealdrones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='com/suas-golden-eagle/ 7https://vantagerobotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='com/vesper/ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Runtime Endurance This family of tests focuses on the battery life of the system in various operational profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' 3a, the specific test we use from this group focuses on the system flying continuously in a figure-8 pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' The main performance metric for the test is the test duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Navigation We have designed two main types of navigation tests, each with several profiles defined based on the type of movement (horizontal, vertical, or both) and the type of confinement (horizontal, vertical, or both).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' 3b, navigation through confined spaces involves traversal into and out of a continuously confined space, with tests for hallway, tunnel, stairwell, and shaft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Navigation through apertures involves transient traversal through an opening, with tests for doorway and window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Each navigation environment is characterized according to the dimensions of the confined space or aperture, lighting, surface textures, and the presence of obstructions on either side of the confined space or aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' The main metrics of performance are efficacy and average navigation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Room Clearing In this test method, the system performs a visual inspection of an example room whose walls, floor, and ceiling are out- fitted with visual acuity targets which contain nested Landolt C symbols of decreasing size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' 3c, the test was performed under two conditions: with and without using camera zoom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' The main performance metrics are duration, coverage, and average acuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Takeoff and Land/Perch As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' 3d, these tests evaluate the system’s ability to takeoff and land or perch in various environments that may be affected by stabilization issues or preventative safety checks from the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' The conditions tested vary the angle of the ground plane (flat, 5° and 10° pitch and roll) and the presence of obstructions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='2-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='4m overhead, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='6-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='2m lateral).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' The main performance metric is efficacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' RESULTS Utilizing our framework outlined in Section III, we calculate a performance score for each sUAS based upon the conditions and performance metrics detailed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' As mentioned above, our system provides a single score for each of the three attributes, the EC, MC and HI axes, and utilizes those scores to provide a single score for the entire test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' It should be noted that although we consider all three axes in our cFIS structure, due to lack of data, we consider the lowest level (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=', full tele-operation) for the HI axis across all tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' The test-specific details of the structure is discussed in corresponding sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Another factor to note is that for some of these experiments, some sUAS were not available at the time of testing, due to the sUAS being repaired, or other circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' In this case, we attempt to remedy this by calculating a partial point achieved by the sUAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' For situations where data for an entire test is (a) Runtime Endurance test showing a system performing a specified movement (b) Navigation tests (left to right, top to bottom): hallway, tunnel, stairwell, shaft, door, window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' (c) Room clearing test showing a system inspecting surfaces for visual acuity targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' (d) Takeoff and land/perch tests showing variations in ground plane angle (top row) and nearby obstructions (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' 3: Tests designed for the evaluation of sUAS contextual autonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' missing, we cannot fully evaluate the sUAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' However, we have evaluated sUAS for individual tests for which the data was recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Despite these edge cases, most sUAS had available data, which was used in our evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' As mentioned before, the proposed structure in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' 1, was adapted to each specific test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' The resulting cascaded FIS are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' 6 each of which represents a cFIS for a specific test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' More specifically, each sub-figure shows the inputs and outputs of each cFIS, as well as how the FIS modules are connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' This is meant to provide a visual aid, which can be useful to keep track of each FIS, as we discuss the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Associated membership functions for each input, and the outputs, can be found in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Some of the FIS surfaces, which show the relationship between two input values in an FIS, and the corresponding output value, are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Additionally, numerical results are reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='5 and Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Test Results Runtime Endurance: The runtime endurance test is likely the simplest of the tests performed, however, it is still a useful test in gauging how a sUAS might perform in a real mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' 3a illustrates the runtime endurance test design including the navigation path and two stands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' The adapted cFIS for this test is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' 6a with four inputs: number of obstructions, number of crashes, light level and speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Despite the simplicity of this test, some sUAS did not performed well largely due to slow speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Our results indicate that some sUAS may have trouble in portions of this real- world mission which requires both speed and maneuverability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' (a) Final FIS Surface (b) Through Aperture test EC (c) Takeoff and Land/Perch EC (d) Through Corridor MC Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' 4: FIS surfaces for different tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' In should be noted that during this test only four sUAS were available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Navigation: In the navigation tests, due to the differences between the through corridor and through apertures tests, two slightly different cascaded FIS were designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' These can be found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' 6b and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' 6d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' The input to the through corri- dors cFIS includes area (cross section), light level, verticality, coverage, number of crashes, and duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' The inputs to the through apertures cFIS include area, light level, number of 10 degrees 10 degrees 5 degrees 5degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='2 mCutput_Score 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='5 MC 0 ECOutput_Score 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='2 0: 0 2 200 400 4 600 Area Light0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='7 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='3 10 10 5 5 Pitch 0 RollOutput_Score 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='5 00 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='5 Crashes 3 CompletionXVariables Description MFs: Triangular{Low, Medium, High} Area Aperture/Hallway Cross-Section (m2) [0, 0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='7] [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='6, 3, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='4] [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='3, 6, 6] Light Ambient Light Level (Lux) [0, 0, 337.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='5] [75, 375, 675] [412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='5, 750, 750] Vert Verticality (°) [0, 0, 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='5] [7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='5, 45, 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='5] [52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='5, 90, 90] Crash Number of Crashes [0, 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='25] [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='5] [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='75, 3, 3] Rollovers Number of Rollovers [0, 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='25] [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='5] [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='75, 3, 3] Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' % Completion Percentage [0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='55] [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='92] [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='7, 1, 1] Yaw/Pitch Static Yaw/Pitch Angle (°) [0, 0, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='17] [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='83, 5, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='12] [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='83, 10, 10] VR Static Vertical Obstruction (m) [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='6, 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='8] LR Static Lateral Obstruction (m) [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='2] [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='4, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='4] [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='6, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='6, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='6] Coverage Coverage Percentage [0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='55] [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='92] [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='7, 1, 1] Cs Detected Landolt C Depth Detected [0, 0, 50] [10, 50, 90] [50, 100, 100] Duration Duration of Test (Minutes) [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='5, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='25] [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='05, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='25, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='45] [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='25, 8, 8] Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Number of Obstructions [0, 0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='5] [1, 3, 5] [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='5, 6, 6] Output Variable Description Sugeno MFs: Constant {Very Low to Very High} Score Combined Defuzzified Score [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='75, 1] TABLE II: Membership Functions (MFs) for each input and output variable used in an FIS in the evaluation of these sUAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' 5: Scores for each sUAS as a percentage of the maximum score possible on the y-axis, with each test on the x-axis crashes, and completion percentage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' As shown above in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' 5, each sUAS that performed the through apertures test, achieved a maximum score, besides sUAS G, which performed slightly worse, due to both issues in correctly traversing the aperture, as well as being the only sUAS to suffer a crash during the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Next, for the through corridors test, As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' 5, there is more variance in the performance between each sUAS, with UAS A performing the best, and UAS E performing the worst, even though sUAS E tied for best in through apertures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' This is important, as there is more room for error while traversing corridors, than there is traversing an aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Takeoff and Land/Perch: For the takeoff and land/perch test, the cFIS diagram can be found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' 6e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' The inputs include pitch, yaw, number of crashes, completion percentage, vertical and lateral obstruction, and number of rollovers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' As can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' 5 and Table III, sUAS A and sUAS G perform best across both sections of the test and thus provide the highest level of autonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Likewise, sUAS B performs the worst in both portions of the test, showcasing a lower level of autonomy, compared to the other sUAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' This evaluation allows for the characterization of how an sUAS may perform during portions of a mission which requires the system to takeoff or land in a specified spot, of varying difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Room Clearing: Since the room clearing test is done in a static environment, we included a time constraint in our testing making the evaluation to focus on the performance of the sUAS in regards to the Mission Complexity axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' The designed cFIS can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' 6c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Inputs include light level, number of obstructions, number of crashes, duration, coverage, and landolt C depth detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Results are found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' 5, and Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' In this test, the strongest performer was UAS E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' however, most of the UAS performed closely to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Surprisingly, a strong performance in the runtime endurance test did not necessarily correlate to a strong performance in this test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Both of these tests require a system with good maneu- verability capabilities, but this test also requires a controller which allows the user to visually identify different landmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Final Results UAS T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Takeoff Land R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Predictive Score A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='85 B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='82 C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='84 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='92 D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='77 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='87 F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='95 G 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='83 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='80 TABLE III: Scores of each sUAS for each test, as well as a weighted multiple which allows for an overall evaluation of each sUAS Contextual autonomy evaluations are concerned with the performance of a system within an environment while per- forming a specific task with a known level of complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' However, calculating an overall score that represents an av- erage autonomy for a given system in a spectrum of tests and environment is desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' To combine the test scores into a single score, we utilize a weighted product, with equal weightings for each test, as we did previously in our non- contextual evaluation [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' The weighted product represented as P = M � i φwi i , (2) where M is the number of individual tests, φi represents an individual test score, and wi the weight assigned to that test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='0 T T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='9 UASA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='8 - UAS B UAS C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='7 - UAS D UAS E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='6 UAS F UAS G 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='5 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Takeoff Land(a) Runtime Endurance (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=') test FIS (b) Through Apertures (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=') test FIS (c) Room Clearing (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=') test FIS (d) Through Corridors (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=') test FIS (e) Takeoff and Land/Perch test FIS Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' 6: Diagrams of each system of cascaded FIS utilized to calculate scores for each test has several benefits including that different test results can be combined without requiring normalization or scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' The results for each sUAS are shown in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' It is important to note that many of the sUAS perform better than the others in some tests, but worse in others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' One example of this is UAS A, which has the fourth highest weighted multiple (overall score), while performing the best in four out of six tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' This is due to the sUAS’s relatively poor performance in both the runtime endurance test, as well as in the room clearing test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' The use case of a singular score like this presents itself when a user would like to know which sUAS is likely to provide the most overall autonomy, across multiple tests and different environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' CONCLUSIONS AND DISCUSSIONS In this paper, we proposed a framework for evaluation of contextual autonomy for robotic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Our framework con- sists of a cascaded Fuzzy Inference System (cFIS) that com- bines test results over three axes of evaluation (mission com- plexity, environment complexity and human independence) introduced by the ALFUS framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' We have designed four tests with different mission complexity and environment complexity levels and performed several experiments with several sUAS, and we have shown that our modular framework is adaptable to different tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' For future work, we plan to extend our framework for performance evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' To achieve this, a desired mission can be decomposed into base tasks, such as takeoff/landing, traversing through environ- ments/apertures, clearing rooms, and general maneuverability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' The user then can define a set of weights and calculate a potential performance score of a sUAS for the target mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Unlike MPP [5], however, we suggest a method which is based upon performance in set tasks, rather than a combination of non-contextual attributes, and environmental factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' 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ramarathne, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Yanco, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Choi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Donald, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Donoghue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=', “Decisive test methods handbook: Test methods for evaluating suas in subterranean and constrained indoor environments, version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='1,” arXiv preprint arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content='01801, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Crashes Environmental Mission Complexity Complexity Light FIS FIS Level Speed Output Test FIS ScoreArea Crashes Environmental Mission Complexity Complexity Light FIS FIS Level Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' % Output Test FIS ScoreCoverage Cs Detected Crashes Light Environmental Mission Complexity Complexity Duration FIS FIS Ohs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Output Test FIS ScoreArea Crashes Environmental Mission Complexity Complexity Light FIS FIS Level Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' % Vert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' Output Test FIS ScoreCrashes Pitch Rollovers Yaw Environmental Mission Complexity Complexity Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} +page_content=' % LR FIS FIS VR Output Test FIS Score' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfuQGc/content/2301.02603v1.pdf'} diff --git a/29AzT4oBgHgl3EQfDvqc/content/2301.00982v1.pdf b/29AzT4oBgHgl3EQfDvqc/content/2301.00982v1.pdf new file 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sha256:3df5c19b0a455eb3d58af2620d196e255a176c2a521091c1f251d36d6e469671 +size 8388653 diff --git a/2tE2T4oBgHgl3EQfNgbv/content/tmp_files/2301.03739v1.pdf.txt b/2tE2T4oBgHgl3EQfNgbv/content/tmp_files/2301.03739v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6f14563ebf9c62b844fe4faa75fac2ea1cefe4b3 --- /dev/null +++ b/2tE2T4oBgHgl3EQfNgbv/content/tmp_files/2301.03739v1.pdf.txt @@ -0,0 +1,850 @@ +Dowker Complexes and filtrations on self-relations +Dominic Desjardins Côté +January 11, 2023 +Abstract +Given a relation on X × Y , we can construct two abstract simplicial complexes +called Dowker complexes. +The geometric realizations of these simplicial complexes +are homotopically equivalent. We show that if two relations are conjugate, then they +have homotopically equivalent Dowker complexes. From a self-relation on X, this is a +directed graph, and we use the Dowker complexes to study their properties. We show +that if two relations are shift equivalent, then, at some power of the relation, their +Dowker complexes are homotopically equivalent. Finally, we define a new filtration +based on Dowker complexes with different powers of a relation. +Keywords : +Dowker complex, relation, filtration, graph theory, shift equivalence +1 +Introduction +We can use multivalued maps to study dynamical systems [17]. The idea is to use Conley index [7] +on upper semi-continuous multivalued maps. In applications, it can be hard to study a dynamical +system. We can use a model that seems to fit data, but it can be a challenge to find it. Another way +is to discretize the continuous space and use to multivalued maps to approximate the underlying +dynamical system [8] [30]. Another approach is to use combinatorial structures. To name a few, +we can use combinatorial vector fields from Forman [13] [18] [9]. Moreover, a generalization was +proposed by Mrozek called combinatorial multivector fields [23] [25]. Finally, others proposed to +use the distributive lattices to compute attractors on finite data [19] [20] [21]. +Multivalued maps can be restrictive. In [18], authors generalize them to partial multivalued +maps. But a partial multivalued map is equivalent to a relation. Some advancements were done +in [26] by using the Scymczak category of finite sets where objects are sets and morphisms are +relations. The Szymczak category [29] captures the essence of index pairs and index maps [7] which +is the core of the theory of Conley index. So one motivation of this paper is to continue to develop +the theory of relations, and it can be used to study dynamical system with finite data. +Our main object is a relation which is a subset of the cartesian product of two sets X and +Y . We can define two different abstract simplicial complexes on a relation. For the first simplicial +complex, we fixed a value y ∈ Y . For all elements in X, that they are related to y, they will span a +1 +arXiv:2301.03739v1 [math.CO] 10 Jan 2023 + +simplex together. For the second one, we reverse the role. We fixed a value x ∈ X. For all elements +in Y , that they are in relation with x, they will span a simplex together. They are called Dowker +complexes [10]. An important result is the Dowker’s Theorem. It says that the geometric realization +of these Dowker complexes are homotopically equivalent. The Dowker’s Theorem is quite useful in +applications. To name a few examples, we can use Dowker complexes to study signal coverage [14], +to find errors in relation of programs and files [1], to study the privacy of information [12] and in +social studies [3]. For the last one, this method is called Q-analysis which is developed by Atkins +[2]. The idea of Q-analysis is to study the q-connectivity and the q-tunnel of the Dowker complexes. +Our two main inspirations for definitions come from these articles [24] and [26]. +They are many contributions in this article. +First, conjugate relations have homotopically +equivalent Dowker complex. If two relations are shift equivalent with lag l, then at a certain power +their Dowker complexes are homotopically equivalent. We can define a filtration on relation based +on the Dowker complex at different powers of a finite self-relation. Moreover, this can be computed +in finite time. We can also compute the 0th homology of a high enough power with the connected +components of the graph induced by the relation. +This article goes as follows. In section 2, we remind some concepts and definitions on finite +relation, graph, simplicial complex, Dowker complexes and finally the famous Dowker’s Theorem. +In section 3, we define right morphism and left morphism. If there exists a right or left morphism +between two relations, then there is an inclusion from one of the Dowker complexes. Moreover, +we show that if there exists a conjugacy between two relations, then their Dowker complexes are +homotopically equivalent. In section 4, we generalize the idea of right and left morphism for mul- +tivalued right and multivalued left morphism. We show two important properties needed for the +definition of a filtration. The Dowker complex of a certain power of a relation is include in the +Dowker complex of the same relation with a higher power. For some finite relations at certain a +power j, every other Dowker complexes of the same relation at power higher than j are the same. +We call it the stabilization of the Dowker complex. We show that shift equivalence between rela- +tions have homotopically equivalent Dowker complex at some power. In section 5, we can define a +filtration on the Dowker complexes of different powers of a relation under some simple conditions by +using the two properties in section 4. We can use persistent homology on these filtration to extract +topological features of the Dowker complex. If a relation is acyclic, then we have that the number +of connected components of the graph associated to the relation up to a certain power is equal to +the dimension of the 0th homology. It can be generalized to the class of simple relations. We also +have a similar result for strongly connected relations. +2 +Preliminaries +2.1 +Finite Relations +Let X and Y be finite sets. We define a relation as a subset of X × Y . Let (x, y) ∈ R ⊂ X × Y , we +denote by xRy or by y ∈ R(x). We define the composition of relations as follows. Let R1 ⊂ X × Y +and R2 ⊂ Y × Z. +2 + +R2 ◦ R1 := {(x, z) ∈ R2 ◦ R1 | ∃y such that xR1y and yR2z}. +(1) +We define the inverse relation by swapping the sets of a relation. +R−1 := {(y, x) ∈ Y × X | y ∈ R(x)} +(2) +If a relation is a subset of X × X, then we say it’s a self-relation on X. We define the power of +a self-relation as follows : +Rn := +� +� +� +� +� +R ◦ Rn−1 +n > 0 +IdX +n = 0 +R−1 ◦ Rn+1 +n < 0 +The domain and the image for a relation R ⊂ X × Y are : +Dom R := {x ∈ X | ∃y such that (x, y) ∈ R} +(3) +Im R := {y ∈ Y | ∃x such that (x, y) ∈ R}. +(4) +We can see relations as partial multivalued maps. If Dom R = X, then we say that the relation +is a multivalued map. A relation is injective, if for all x1, x2 ∈ X, R(x1) = R(x2) implies that +x1 = x2. A relation is surjective if Im R = Y . Moreover, a map f : X → Y induces a relation +where (x, f(x)) ∈ R. Without ambiguity, we can compose maps and relations together to obtain a +new relation. +Definition 2.1. Let R be a self-relation on X. Let j be the least positive integer such that : +Rj = Rj+p for some p > 0. +We say that j is the index and the least p > 0 is the period. If j = 1, then R is periodic. A pair +(j, p) is the eventual period of R with index j and period p. +In other words, the period p will eventually be a period for R. +We sometimes use matrices with values in {0, 1} to represent relations. Let R ⊂ X × Y be a +relation with card(X) = m and card(Y ) = n. The matrix Mm×n have a value 1 at Mi,j if xiRyj +otherwise the value is 0. It can be called relation matrix, Boolean relation matrix, binary relation +matrix, binary Boolean matrix, (0, 1)-Boolean matrix and (0, 1)-matrix. For more information on +Boolean matrix theory, we refer to the book [22]. +We say that a self-relation R on X has a cycle at x if and only if there exists an n ∈ N such +that xRnx. We say R as a fixed point at x, if n = 1. If a relation has no cycle at x for all x with +period n > 1, then the relation is acyclic. A cycle is a sequence x1, x2, . . . , xn such that x1 = xn +and xiRxi+1. A self-relation R on X is simple if for any two cycles are either disjoints or equals. +3 + +2.2 +Graphs +In this subsection, we remind the definition of a graph and some notations. +Definition 2.2. A directed graph G is a pair (E, V ) where V is the set of vertices V and E is the +subset V × V the set of edges. +A relation can also be seen as a directed graph. If R is a self-relation on X, then X is the set +of vertices and the set of edges E = R. This graph has at most one directed edge from the vertex +A to the vertex B, and we also allow a self-loop on vertices. We note GR the graph induced by a +self-relation R. +Let x, y ∈ V . There is a (x, y)-path, if there exists a sequence of edges e1, e2, . . . , en ∈ E that +connect x to y without following the direction of edges. We can define an equivalence relation +on vertices of G. If there is a path between two vertices x and y, then x and y are in the same +class of equivalence. For a graph G, we say the number of connected components is the number of +class equivalences of the relation of paths. We say G is connected if there is only one connected +component. +If the sequence of edges of a (x, y)-path follows the direction of edges of the graph, then we +say it’s a (x, y)-walk. We can also define an equivalence relation with a walk between vertices. If +there is a walk from x to y and a walk from y to x, then x and y are in the same equivalence class. +This is the class of strongly connected components. For a graph G, we say the number of strongly +connected components is the number of class equivalence of the relation of walks. We say G is +strongly connected if there is only one strongly connected component. +2.3 +Simplicial Complexes and Dowker Complexes +In this subsection, we will discuss simplicial complex, Dowker complex and the Dowker’s Theorem. +For more information about simplicial complex, we suggest to read [27]. We do not present filtration +and persistent homology, but we refer to [11]. +An abstract simplicial complex is a set K that contains finite non-empty sets such as if A ∈ K, +then for all subsets of A are also in K. For further examples, we use geometric simplex. A geometric +n-simplex is the convex hull of a geometrically independent sets of vertices {v0, v1, . . . , vn} ∈ RN. +This is the set of x ∈ RN such as x = �n +i=0 tixi and 1 = �n +i=0 ti where ti ≥ 0 for all i. We denote +an n-simplex by [v0, v1, . . . , vn] is the simplex spanned by the vertices v0, v1, . . . , vn. Any simplex +spanned by the subsets of {v0, v1, . . . , vn} are called faces and denote by the symbol ≤. A simplicial +complex is a collection of simplices for all σ ∈ K, if τ ≤ σ then τ ∈ K and if σ1 ∩ σ2 = τ, then τ +is either the empty set or τ is a face of σ1 and σ2. We say that L ≤ K if L is a sub-complex of +K. A simplicial complex is contractible if its homology is equivalent to a point. Given an abstract +simplicial K, we can define a geometric simplicial complex and |K| call the geometric realization of +K. We call 0-simplices vertices and 1-simplices edges. The closure of a simplex σ is the set of all +the faces of the simplex. We denote it by cl(σ). We need one more definition related to simplicial +complexes. It will be useful in some proofs. +4 + +Definition 2.3. A simplicial complex K is edge-connected, if for any two vertices x and y there +is a sequence of edges e1, e2, . . . , en such that x ∈ e1, y ∈ en and cl(ei) ∩ cl(ei+1) ̸= ∅ for all +i = 1, 2, . . . , n − 1. +We have that the simplicial complex is connected if and only if it is edge-connected if and only +if H0 is dimension 1 [16]. +Now we explain how to construct abstract simplicial complexes from a relation which are called +Dowker complexes. Let R ⊂ X × Y be a relation and X, Y be two finite sets. There are two ways +to construct the Dowker complex from a relation. +Definition 2.4. Let R ⊂ X × Y be a finite relation and KR be the Dowker complex. A simplex +[x1, x2, . . . , xn] ∈ KR if and only if ∃y ∈ Y such as xiRy for all i = 1, 2, . . . , n. +We have an analogous construction. +Definition 2.5. Let R ⊂ X × Y be a finite relation and LR be the Dowker complex. A simplex +[y1, y2, . . . , ym] ∈ LR if and only if ∃x ∈ X such as xRyi for all i = 1, 2, . . . , n. +We denote [x1, x2, . . . xn] = σy ∈ KR if and only if xiRy for all i = 1, 2, . . . , n. We use y as an +index for σy to note that all vertices of σy are in R−1(y). We use the same notation for σx ∈ LR +but the vertices are in R(x). +By using the matrix notation, we can use rows and columns to build the simplices. The columns +are for KR and the rows are for LR. +Example 2.6. Let R ⊂ X × Y be a finite relation. +R := +� +��� +1 +0 +0 +0 +1 +0 +0 +1 +1 +0 +1 +0 +0 +0 +1 +1 +1 +0 +0 +0 +� +��� +(5) +The first column gives the 2-simplex [x1, x3, x4]. The third and the fourth column give the 0- +simplex [x2]. The second and the fifth column do not add new simplices. We obtain the simplicial +complex KR := {[x1, x3, x4], [x2]}. +The first row adds a 1-simplex [y1, y5] to LR. The second row gives a 1-simplex [y3, y4]. The +final row adds a 1-simplex [y1, y5]. We obtain the simplicial complex LR = {[y1, y5], [y3, y4], [y1, y2]}. +We obtain that |KR| and |LR| have two connected components and no higher dimension cycle. +The next theorem links to the homotopy between |KR| and |LR|. +Theorem 2.7 (Dowker’s Theorem). Let R ⊂ X × Y be a relation and let KR and LR be the +associated Dowker complexes. Then, the polyhedra |KR| and |LR| are homotopy equivalent. +5 + +(a) Geometric realization of +the Dowker complex KR. +(b) Geometric realization of +the Dowker complex LR. +Figure 1: These are geometric realizations of the Dowker complexe in Example 2.6. They +are homotopically equivalent. +In 1952, Dowker [10] has shown that KR and LR have isomorphic homology groups. In 1995, +Björner [4] has shown that |KR| and |LR| are homotopically equivalent, which is the more com- +monly use in the literature. In recent years, Dowker complexes were regained in popularity in the +community of topology data analysis. We can use them to do a filtration on weighted networks [6]. +In our cases, our filtration will be different and based on different powers of a self-relation. +3 +Left and Right Morphism +Let start with the definition of the graph homomorphism and next we define left and right morphisms +between relations. +Definition 3.1. Let R be a self-relation on X and R′ be a self-relation on Y . A map f : X → Y +is a graph homomorphism if the following condition is satisfied : +For every x1, x2 ∈ X such as x1Rx2 =⇒ f(x1)R′f(x2). +If f is bijective and its inverse is also a graph homomorphism, then f is a graph isomorphism. +We obtain that graph homomorphism keeps some information of the Dowker complex coming +from the first relation. +Lemma 3.2. Let f : X → Y be a graph homomorphism between R and R′. If f is injective, then +there exist a map p : KR �→ KR′. +6 + +Proof. Consider a n-simplex [x0, x1, . . . , xn] ∈ KR. Then, there exists α ∈ X such as xiRα for +all i = 0, 1, 2, . . . n. We have that f is a graph homomorphism. This implies that f(xi)R′f(α) +for all i = 0, 1, 2, . . . , n. Indeed, f is injective implies that [f(x0), f(x1), . . . , f(xn)] is also a n- +simplex in KR′. So we can construct a map p : KR �→ KR′ by sending a simplex [x0, x1, . . . , xn] to +[f(x0), f(x1), . . . , f(xn)]. By the previous argument, p is well defined and injective. +If we have a graph isomorphism between two relations, then the Dowker complexes remain +unchanged. This holds because graph isomorphisms are relabelling on the vertices of a graph. +Proposition 3.3. Let R1 be a self-relation on X and R2 be a self-relation on Y . If there exists a +graph isomorphism f between R1 and R2, then they have the same Dowker complexes up to the label +of vertices. +Proof. Graph homomorphisms f and f−1 are injective. By the Lemma 3.2, there exist two injective +maps p : KR �→ KR′ and p′ : KR′ �→ KR. So we have that KR and KR′ are the same up to the +label of vertices. +By similar arguments, we can show it for LR and LR′. +Graph homomorphisms are nice, but they can drastically change the Dowker complexes. So, +we defined a left morphism which it changes the source of an edge and a right morphism which it +changes the target of an edge. In this way, only one of the Dowker complexes will change from the +right morphism or the left morphism. +Definition 3.4. A right morphism f : (X, Y, R) → (X, Z, R′) is a map f : Y → Z such that for +every x ∈ X and y ∈ Y : +xRy =⇒ xR′f(y). +We obtain this simple Lemma which is very useful for later proofs. +Lemma 3.5. If there exists a right morphism f : (X, Y, R) → (X, Z, R′), then KR ≤ KR′. We +obtain the equality if f is a bijective map. +Proof. Let f : (X, Y, R) → (X, Z, R′) be a right morphism and [x1, x2, . . . , xn] ∈ KR. This implies +there exists a y ∈ Y such as xiRy for all i = 1, 2, . . . , n. We obtain that xiR′f(y) for all i. Finally, +we have [x1, x2, . . . , xn] ∈ KR′. +Now, we suppose that f is bijective. Let [x1, x2, . . . , xn] ∈ KR′. Then, there exists a z ∈ Z such +that xiR′z for all i = 1, 2, . . . , n. We have f−1(z) ∈ Y and f−1 is well defined because f is bijective. +Then, xiRf−1(z) for all i = 1, 2, . . . , n. We obtain that [x1, x2, . . . , xn] ∈ KR. +The idea of right morphism comes from the article [24]. The author only considered the right +morphism. But, in our case, we are also interesting of modifying the first set in the cartesian product +of a relation. +7 + +Definition 3.6. A left morphism g : (X, Z, R) → (Y, Z, R′) is a map g : X → Y such that for every +x ∈ X and z ∈ Z : +xRz =⇒ g(x)R′z. +We have an analogous Lemma for left morphism as the Lemma 3.5 for right morphism. +Lemma 3.7. If there exists a left morphism g : (X, Z, R) → (Y, Z, R), then LR ≤ LR′. We obtain +the equality if g is a bijective map. +With the definition of right and left morphism, we can easily show that if two relations are +conjugate, then there Dowker complexes are homotopically equivalent. We remind the definition of +conjugacy between relations before showing the proof. +Definition 3.8. Let R1 be a self-relation on X and R2 be a self-relation on Y . We say that R1 and +R2 are conjugate if there exists a bijective map ϕ : X → Y such as ϕ ◦ R1 = R2 ◦ ϕ. +Corollary 3.9. Let R be a self-relation on X and R′ be a self-relation on Y which are conjugate +by a bijective map ϕ : X → Y . Then, |KR|, |LR|, |LR′| and |KR′| are homotopy equivalent. +Proof. The map ϕ is bijective. It implies that KR = Kϕ◦R by Lemma 3.5 and LR′ = LR′◦ϕ by +Lemma 3.7. +By Dowker’s Theorem, we obtain that |KR| is homotopic equivalent to |LR′|, because Kϕ◦R = +KR′◦ϕ. +In [28], the author decides to combine the right and left morphism together. Let R ⊂ X × Y +and R′ ⊂ X′ × Y ′ be relations and f : X → X′ and g : Y → Y ′ be two maps. A pair (f, g) is +a morphism between relation R1 and R2 if for all x ∈ X, y ∈ Y such that xR1y it implies that +f(x)R2g(y). In [28], it is shown that the Dowker complex and (co)sheaf representation have nice +functoriality properties. In our case, it won’t be useful because we only need right or left morphism. +But we can see them as a pair (idX, f) where f is a right morphism and idX is the identity function +on X. +4 +Multi-right morphism and multi-left morphism +We want to work with multivalued maps. We generalize left and right morphism to multi-left and +multi-right morphism. +Definition 4.1. A multi-right morphism F : (X, Y, R) ⊸ (X, Z, R′) is a multivalued map F : Y ⊸ +Z such as for all x ∈ X, y ∈ Y : +xRy =⇒ xR′a for all a ∈ F(y). +8 + +We also obtain the same Lemma as before. +Lemma 4.2. Let R ⊂ X × Y and R′ ⊂ X × Z be relations. If there exist a multi-right morphism +F : (X, Y, R) ⊸ (X, Z, R′), then KR ≤ KR′. We obtain the equality if F is a bijective multivalued +map. +Proof. The proof is the same as Lemma 3.5. +Definition 4.3. A multi-left morphism G : (X, Z, R) ⊸ (Y, Z, R′) is a multivalued map G : X → Y +such for all x ∈ X, z ∈ Z : +xRz =⇒ aR′z for all a ∈ G(x). +Lemma 4.4. Let R ⊂ X × Z and R′ ⊂ Y × Z be relations. If there exist a multi-left morphism +G : (X, Z, R) ⊸ (Y, Z, R′), then LR ≤ LR′. We obtain the equality if G is a bijective multivalued +map. +We denote a multi-right morphism by mr-morphism and a multi-left morphism by ml-morphism. +Remarks 4.5. We remind that if a relation S ⊂ X×Y satisfies Dom S = X, then S is a well-defined +multivalued map. Moreover, for any relation R ⊂ Z ×X, we have that S : (Z, X, R) ⊸ (Z, Y, S ◦R) +is a well-defined mr-morphism. It is also true for ml-morphism. +The next corollary will be useful to define our filtrations. +Corollary 4.6. Let R be a self-relation on X. If Dom R = X, then KRn ≤ KRn+1. If Im R = X, +then LRn ≤ LRn+1. +Proof. Dom R = X implies that R is a multivalued map. Moreover, R : (X, X, Rn) ⊸ (X, X, Rn+1) +is well-defined mr-morphism. By Lemma 3.5, we have that KRn ≤ KRn+1. +In the same way, Im R = X implies that R−1 is a multivalued map. +We have that R−1 : +(X, X, Rn) ⊸ (X, X, Rn+1) is well-defined ml-morphism. By Lemma 3.7, we have that LRn ≤ +LRn+1. +Remark 4.7. The hypothesis Dom R = X in the previous corollary is important. Let us show that +by an example. If R is a self-relation on X, define by this matrix such as : +� +� +0 +1 +1 +0 +0 +1 +0 +0 +0 +� +� +(6) +Then, we obtain KR = {[x2, x3], [x2], [x3]}, KR2 = {[x3]} and KRn = ∅ for n > 2. +We don’t have KRn ⊂ KRn+1 for all n ∈ N>0. If the matrix is nilpotent, then KRn is an empty +set for an n ∈ N>0. +9 + +Using Corollary 4.6, we can show that the Dowker complexes of a relation stabilize at some +power n. +Corollary 4.8. Let R be a finite self-relation on X with an eventual period (j, p). If Dom R = X, +then, we have KRj = KRj+i for i ∈ N. If Im R = X, then LRj = LRj+i for i ∈ N. +Proof. By Corollary 4.6, we have the sequence : +KRj ≤ KRj+1 ≤ . . . ≤ KRj+p−1 ≤ KRj+p. +(7) +But p is the period of R, hence Rj = Rj+p implies that KRj = KRj+p. By (7), we obtain +KRj = KRj+i for i ∈ N. A similar proof can be done for LRj = LRj+i for i ∈ N. +We remind the definition of shift equivalence between two relations and we show the assump- +tion that relations are shift equivalences implies that theirs Dowker complexes are homotopically +equivalent at some power for each relation. +Definition 4.9. Let R1 be finite self-relation on X and R2 be finite self-relation on Y . R1 and R2 +are shift equivalent with a lag l, if there exists two relations S ⊂ X × Y and T ⊂ Y × X such as : +R1 ◦ T = T ◦ R2 +S ◦ R1 = R2 ◦ S +T ◦ S = Rl +1 +S ◦ T = Rl +2 +We say it is a strong shift equivalence if l = 1. +Corollary 4.10. Let R1 be finite self-relation on X with Dom R1 = X = Im R1 and R2 be finite +self-relation on Y with Dom R2 = Y = Im R2. Let (jp, p) be an eventual period of R1 and (jq, q) +be an eventual period of R2. Without loss of generality, we suppose that jp ≥ jq. If R1 and R2 are +shift equivalent with lag l, then |KR +jp +1 |, |KR +jq +2 |, |LR +jp +1 | and |LR +jq +2 | are homotopy equivalent. +Proof. If R1◦T = T ◦R2 then Rn +1 ◦T = T ◦Rn +2 is also true for n ∈ N. Moreover, we have Dom S = X +and Dom T = Y , because Dom R1 = X, Dom R2 = Y , T ◦ S = Rl +1 and S ◦ T = Rl +2. So we have +that S and R are well defined multivalued maps. +We want to show that KR +jp +1 += KS◦R +jp +1 +and LR +jq +2 += LR +jq +2 ◦S. We are going to use T and S as +mr-morphism with R1 and ml-morphism with R2. +We have that S : (X, X, Rjp +1 ) ⊸ (X, Y, S ◦ Rjp +1 ) and T : (X, Y, S ◦ Rjp +1 ) ⊸ (X, X, T ◦ S ◦ Rjp +1 ) +are well-defined mr-morphisms. We have T ◦ S = Rl +1. It implies that T ◦ S ◦ Rjp +1 = Rl+jp +1 +. We +obtain KR +jp +1 ≤ KS◦R +jp +1 ≤ KT◦S◦R +jp +1 = KR +l+jp +1 +. The eventual period of R1 is (jp, p). It implies that +KR +jp +1 = KR +jp+l +1 +. We obtain that KR +jp +1 = KS◦R +jp +1 . +We can see that S : (Y, Y, Rjp +2 ) ⊸ (X, Y, Rjp +2 ◦ S) and T : (X, Y, Rjp +2 ◦ S) ⊸ (Y, Y, Rjp +2 ◦ S ◦ T) +are well-defined ml-morphisms. We have S ◦ T = Rl +2. It implies that Rjp +2 ◦ S ◦ T = Rjp+l +2 +. We +10 + +(a) The graph GR1. +(b) The graph GR2. +Figure 2: Theses are the graphs from Example 4.11. +obtain LR +jp +2 ≤ LR +jp +2 ◦S ≤ LR +jp +2 ◦S◦T = LR +jp+l +2 +. We have LR +jq +2 = LR +jp+l +2 +because jp ≥ jq. We obtain +LR +jp +2 = LR +jp +2 ◦S. +Finally, we have KR +jp +1 = KS◦R +jp +1 = KR +jp +2 ◦S and LR +jp +2 = LR +jp +2 ◦S = LS◦R +jp +1 . By Dowker’s Theorem, +we obtain that |KR +jp +1 |, |KR +jq +2 |, |LR +jp +1 | and |LR +jq +2 | are homotopy equivalent. +Example 4.11. Let X be a finite set with 8 points and Y be a finite set with 3 points. Let R1 be +a self-relation on X and R2 be a self-relation on X defined by those graphs in Figure 2. R1 has an +eventual period (3, 3) and R2 has an eventual period (1, 3). We see in Figures 3(a), (b) and (d) that +the Dowker complexes are not homotopically equivalent. But, in Figures 3 (c) and (d), the Dowker +complexes with relations at power 3 are homotopically equivalent. +There is an interesting proposition from [15] that we can use for strongly connected relations. +We remind that an indecomposable Boolean matrix is a relation which is strongly connected and J +is a square matrix where all the entries are equals to 1. +Proposition 4.12 (Proposition 4.3 in [15] ). Every indecomposable Boolean matrix with positive +trace is strong shift equivalent to J. +We can easily compute the Dowker complexes of J. It is a (n−1)-simplex where n is the number +of rows of J for both Dowker complexes. Finally, we obtain that the Dowker complexes of a strongly +connected self-relation at a power high enough are contractible if the trace is strictly positive. +11 + +Y3(a) Dowker complex of |KR1|. +(b) Dowker complex of |LR1|. +(c) Dowker complexes of |KR3 +1| and +|LR3 +1|. +(d) +Dowker +complexes +|KR2|, |LR2|, |KR3 +2| and |LR3 +2|. +Figure 3: Theses are the different Dowker complexes from Example 4.11. +12 + +5 +Filtrations of Dowker complexes +For this section, we suppose that a R is a finite self-relation on X. From Corollary 4.6, if Dom R = +X, then we have KRi ≤ KRi+1 for all i ≥ 0. We have an inclusion and we get this filtration : +KR �→ KR2 �→ . . . �→ KRi �→ KRi+1 �→ . . . +(8) +In the same way with Im R = X, we have LRi ≤ LRi+1. We obtain another filtration : +LR �→ LR2 �→ . . . �→ LRi �→ LRi+1 �→ . . . +(9) +From Corollary 4.8, the Dowker complexes stabilize at a certain power. This means we can +compute the filtration (8) and (9) in finite time. For our filtrations, we start at i = 1, but we could +also start with the i = 0. We have that R0 = idX and R : (X, X, idX) ⊸ (X, X, R) is a well-defined +mr-morphism, if Dom R = X. Then, the filtration (8) becomes : +KidX = KR0 �→ KR �→ KR2 �→ . . . �→ KRi �→ KRi+1 �→ . . . +(10) +And for the filtration (9) by applying similar arguments, we obtain : +LidX = LR0 �→ LR �→ LR2 �→ . . . �→ LRi �→ LRi+1 �→ . . . +(11) +The homology of KidX and Lidx is the homology of n = card(X) points. In some cases, we +might want to start the filtration at i = 0 or i = 1. +We remind that, by Dowker’s Theorem, |KRi| and |LRi| are homotopically equivalent for all +i ∈ N. We obtain the same bar code representation for filtrations (8) and (9). +Example 5.1. Let R1 be a self-relation on X given by the graph in the Figure 4(a). R1 has 9 nodes +and is acyclic. The eventual period is (3, 1). We obtain the bar code at Figure 4(b). It has one +generator of H1 with the interval [1, 2] and we had 3 generators of H0 that die early and 1 generator +of H0 that survive to infinity. +Example 5.2. Let R2 be a self-relation on X given by the graph in the Figure 4(c). R2 has 10 +nodes and is simple. The eventual period is (3, 4). We obtain the bar code at the Figure 4(d). It has +4 generators of H0 that die at time 2 and 2 other generators that survive to infinity. +For the next results, we compute the 0th homology of the Dowker complexes for different types +of relation. +If R is a finite acyclic self-relation, then it has an eventual period p = 1 and j ∈ N>0 such as +Rj = Ri for all i ≥ j. So we denote this relation Rj by R∞, because it converges to a relation when +i → ∞. +13 + +(a) The graph of the acyclic rela- +tion from Example 5.1. +(b) The associated bar code of the rela- +tion from Example 5.1. We use the fil- +tration with KRi +1. The first bar in orange +is a generator in H1 and the others four +bars in blue are generators in H0. +(c) The graph of the relation with +multiple cycles from Example 5.2. +(d) The associated bar code of the rela- +tion from Example 5.2. We use the fil- +tration with KRi +2. The six bars in blue +are generators in H0. +Figure 4: In these figures, we have the graph on the left and the associated bar code diagram +on the right for Examples 5.1 and 5.2. Dashed lines in bar code mean it goes to infinity. +14 + +4.0 +T +3.5 +- +3.0 +T +2.5 +2.0 +1.5 +1.0 +- +0.5 +0.0 +1.0 +1.5 +2.0 +2.5 +0'E +3.5 +4.0 +4.5 +5.0X5 +X105 . +4 +3 - +1 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +5.0X3Definition 5.3. We say that x ∈ X is a minimum for a self-relation R, if there exists no y ∈ X +such as x ̸= y and xR∞y. We denote the set Ux := {y ∈ X | yR∞x}. +We say that x ∈ X is a maximum for a self-relation R, if there exists no y ∈ X such as x ̸= y +and yR∞x. We denote the set Dx := {y ∈ X | xR∞y}. +The maximums and minimums of an acyclic relation are important, because they are responsible +for the maximal simplices of KR∞ and LR∞. +Lemma 5.4. Let R be a finite acyclic self-relation on X with Dom R = X. Then, the maximal +simplices of KR∞ are given by the minimums of R∞. +Proof. We have that Dom R = X it implies that Dom R∞ = X. Then, for all x ∈ X there exists +z ∈ X such that xR∞z and z is a minimum. Let σy = [x1, x2, . . . , xn] ∈ KR∞ be an arbitrary +simplex. We have an y ∈ X such as xiR∞y for all i = 1, 2, . . . , n. By the first argument, there +exists a minimum z ∈ X such as yR∞z. Therefore, we have xiR∞yR∞z. Then, σy ⊂ σz. +We can do a similar result with LR∞ by using the maximums of R∞, if Im R = X. +Theorem 5.5. Let R an acyclic finite self-relation on X with Dom R = X. +number of connected components of GR = dim H0(KR∞) = dim H0(LR∞) +(12) +Proof. First, by Dowker’s Theorem, we have dim H0(KR∞) = dim H0(LR∞). We suppose that GR is +connected and show that dim H0(KR∞) = 1. More precisely, we show that KR∞ is edge-connected. +We have that Dom R = X implies that for all x ∈ X,[x] ∈ KR∞. +Let x, x′ ∈ X. There exists y1 ∈ X a minimum such as xR∞y1 and y1R∞y1. This implies +that e1 = [x, y1] ∈ KR∞. We also have that there exists yn ∈ X a minimum such as x′R∞yn and +ynR∞yn. This implies that en = [x′, yn] ∈ KR∞. +Since GR is connected, there exists a (y1, yn)-path of finite length. We denote this sequence by +y1, z1, z2, z3, . . . , zm, yn. Without loss of generality, we take the shortest path. There exist a i such +as zi ∈ Uy1 and zi+1 /∈ Uy1. First, we have e2 = [y1, zi] ∈ KR∞ and ziRzi+1. There exists y2 ̸= y1 a +minimum such as zi+1R∞y2. This implies that ziR∞y2 and we have the edge e3 = [zi, y2] ∈ KR∞. +We can repeat this process with the (y2, yn)-path until we obtain a sequence of edges that connect +the vertex [x] and [x′]. We obtain that KR∞ is edge-connected. +Now suppose that GR is not connected. Let H be a connected component of GR. Then, for all +x ∈ H and for all y /∈ H, we have that x /∈ R(y) and y /∈ R(x). It implies that for each connected +component gives a single generator for H0(KR∞). +We can construct a map j : cc(GR) → H0(KR∞) that sends the connected components of GR to +the generators of H0(KR∞). By the previous argument, we can make this map j injective . For any +generator g in H0(KR∞), there exists a x ∈ X such as g is homologous to [x] because Dom R = X. +This implies there exists a H ∈ cc(GR) such as x ∈ H. We obtain that the map j is bijective and +the equality (12). +15 + +Figure 5: The graph of R4 +2 from Example 5.2. GR4 +2 has 2 connected components. +We can show a similar proof for simple relations. +Theorem 5.6. Let R be a finite simple self-relation on X with Dom R = X and (j, p) be an eventual +period. Assume that GR is connected. There exists a r ∈ N such that : +dim H0(KRj) = dim H0(LRj) = number of connected components of GRr. +(13) +Proof. We can find q big enough so that Rq is acyclic, because R is a simple relation. We choose a i ∈ +N such as iq > j. We fix r = iq. We also have that Rr is also acyclic. By Corollary 4.8, we have that +KRj = KRr. By Theorem 5.5, we know that dim H0(KRj) = number of connected components of Rr. +If GR has more than one connected component, we apply this theorem for each connected +component of GR by using similar arguments as the proof of Theorem 5.5. From preceding results, +if Dom R ̸= X but Im R = X, we can redo the proofs with LRj. Another approach is to use R−1, +because Dom R−1 = X. +Remark 5.7. In Example 5.2, it is a simple relation. We have R4 +2 is acyclic. The graph of R4 +2 is +shown in Figure 5. It has two connected components and the bar code from Figure 4(b) has 2 bars +goes to infinity. It is expected from Theorem 5.6. +We have shown earlier in Corollary 4.10 and Proposition 4.12 that a strongly connected self- +relation R is shift equivalent to a matrix J, if tr(R) > 0. But we will like to have a result for any +strongly connected relations. But, first we need some definition and other results from other papers. +Let gcd(a, b) be the great common divisor of a and b. We define : +q = gcd(n1, n2, n3, . . .) +(14) +where ni is the length of a cycle and i ∈ I is the set of all different cycles from R. +We obtain this proposition : +Proposition 5.8 (Proposition 6.12 in [26]). Let R be a strongly connected self-relation on X, (j, p) +be the eventual period and q defined by (14). We have q|j. +16 + +Let’s define a new equivalence relation ∼q for a strongly connected self-relation R. We say that +x ∼q y, if for each (x, y)-walk has length equal to 0 modulo q. It is an equivalence relation. +Proposition 5.9 (Proposition 6.16 in [26]). Let R be a strongly connected self-relation on X. Let q +defined as (14). Then, ∼q is an equivalence relation in X with exactly q distinct equivalence classes. +We need one more Lemma before showing our final result. +Lemma 5.10 (Lemma 6.25 in [26]). Let R be a strongly connected self-relation on X and (j, p) an +eventual period. Then, +x ∼q x′ =⇒ Rj(x) = Rj(x′). +(15) +We are going to show that the number of class equivalence of ∼q is equal to dim(H0(KRj)) for +a strongly connected self-relation with eventual period (j, p). +Theorem 5.11. Let R be a finite self-relation on X with an eventual period (j, p), R is strongly +connected, q defined by (14). Then, we have : +number of [x]∼q = q = dim(H0(KRj)) = dim(H0(LRj)). +(16) +Proof. We show that for any x, y ∈ X, if x ∼q y, then x and y are edge-connected and if x ̸∼q y, +then x and y are not edge-connected. +First, we suppose that x ∼q y. By Lemma 5.10, we have that Rj(x) = Rj(y) ̸= ∅. There exists +a z ∈ Rj(x). It implies that [x, z] and [y, z] are in KRj. So, each vertex in the same equivalence +class is edge-connected. +Now, we suppose that, x ̸∼q y. There exists a (x, y)-walk of length n modulo q where n ̸= 0. +Let show that Rj(x) ∩ Rj(y) = ∅ . Let’s suppose there exists a z ∈ Rj(x) ∩ Rj(y). This implies +there exists a (x, z)-walk of length j and a (y, z)-walk of length j. But, from Proposition 5.8, q|j. +This implies x ∼q z and y ∼q z. But ∼q is an equivalence relation. We obtain that x ∼q y which is +a contradiction. We obtain that x ̸∼q y implies Rj(x) ∩ Rj(y) = ∅. We obtain that if x ∼q y then +they are edge-connected. But, if x ̸∼q y, then they are not edge-connected. There is q different +equivalence classes. The proof is complete. +For the case of R is an arbitrary relation, it is harder to find its homology H0. Also, for higher +dimensions of the homology groups, it’s hard to tell what happens. Further investigations are needed +for both cases. +Now, we return to the filtrations defined earlier. There are two other types of filtrations that we +can use. If R is a self-relation on X with Dom R = X = Im R, then we can use both filtrations. But +we obtain the same bar codes for both. That holds because, for each i ∈ N, |KRi| is homotopically +equivalent to |LRi| for any self-relation. We might need to come with other types of filtration. We +suggest two other types of filtration. +It will be interesting to use a zigzag filtration [5] with KRi and LRi by alternating them. It will +probably depend on the relation. Further investigations are needed. +17 + +We will present an interesting bi-filtration with KRm and LRn. We have that, if KRm ⊂ KRm+1 +and LRn ⊂ LRn+1, then KRm ∩LRn ⊂ KRm+1 ∩LRn and KRm ∩LRn ⊂ KRm ∩LRn+1 for all m, n ∈ N. +We obtain this bi-filtration : +... +... +. . . +KRm ∩ LRn +KRm+1 ∩ LRn +. . . +. . . +KRm ∩ LRn+1 +KRm+1 ∩ LRn+1 +. . . +... +... +The computation of the bi-filtration is also finite. Because the relation R is finite and Dom R = +X = Im R. We obtain an eventual period (j, p). In the bi-filtration, there are, at maximum, j2 +different simplicial complexes to compute. +One may ask why the intersection is a good idea to consider. Let’s explain it in more details. +Let R be a self-relation on X with Dom R = X = Im R and m, n ∈ N. Let σ ∈ KRm ∩ LRn where +σ = [x1, x2, . . . , xd]. Then, there exists xω ∈ X such that xiRmxω and there exists xα ∈ X such +that xαRnxi for all i. Another way to see this is, for each xi ∈ σ, there exists a (xω, xα)-walk of +length m+n passing through xi. We can subdivide this (xω, xα)-walk into a (xω, xi)-walk of length +m and a (xi, xα)-walk of length n. So, by only the existence of a simplex σ in KRm ∩ LRn, each +vertex of σ, it has a walk with a common starting point and a common ending point of same length +going through the vertex. It will be interesting to study these Dowker complexes when m tends to +infinity, n tends to infinity or both. +6 +Conclusion +In summary, we used the Dowker complexes to study some properties of self-relation. First, we +defined the right morphism and left morphism. We also generalized it to the case of multivalued +maps called multi-right morphism and multi-left morphism. The existence of a right or multi-right +morphism between R1 and R2 implies that the KR1 is included in KR2. Similarly, the existence of a +left or a multi-left morphism between R1 and R2 implies that LR1 is included in LR2. We have shown +that two relations are conjugate implies that they have homotopically equivalent Dowker complexes. +We have also shown that if two relations are shift equivalent, then their Dowker complexes are +homotopically equivalent at some power of the relations. We were interested in self-relation which +is equivalent of a type of directed graph. We have obtained two nice properties. If R is finite and +Dom R = X = Im R, then we have that KRi ≤ KRi+1 and LRi ≤ LRi+1 for all i ∈ N. Moreover, +there exists a j ∈ N such as KRj = KRi+j and LRj = LRi+j for all i ∈ N. With these two properties, +we defined two filtrations with KRi �→ KRi+1 and LRi �→ LRi+1. Also, the filtration ends at some +finite time. Finally, we proved some results about the 0th homology for some types of self-relation +at some power. We also proposed the intersection filtration and the zigzag filtration. +18 + +We have put some foundations to study directed graph using Dowker complexes. Moreover, we +think it might be a useful tool to study the dynamics of finite data define by a directed graph of +Dowker complexes. Let R be a self-relation with an eventual period (j, p). The positive or forward +invariant is given by the existence of a simplex in KRj when j converges to infinity. The negative +or backward invariant is given by the existence of a simplex in Lj +R when j converges to infinity. +But, by the stabilization of Dowker complexes, we can compute it in finite time. Finally, if we want +to study the invariant of R, this is given by the existence of a simplex in KRj ∩ LRj which is the +intersection of the forward and the backward invariant. But further investigations are needed. The +idea is to use the structure of the Dowker complexes to encode the dynamics of finite data. +19 + +References +[1] K. Ambrose, S. Huntsman, M. Robinson, and M. Yutin. Topological differential testing. CoRR, +abs/2003.00976, 2020. +[2] R. Atkin. 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Annals of Mathematics, pages 84–95, 1952. +[11] H. Edelsbrunner and J. L. Harer. +Computational Topology : An Introduction. +American +Mathematical Society, 2010. +[12] M. Erdmann. Topology of privacy: Lattice structures and information bubbles for inference +and obfuscation. arXiv, 2017. +[13] R. Forman. Combinatorial vector fields and dynamical systems. Mathematische Zeitschrift, +228:629–681, 1998. +[14] R. Ghrist, D. Lipsky, J. Derenick, and A. Speranzon. Topological landmark-based navigation +and mapping. +https://www2.math.upenn.edu/~ghrist/preprints/landmarkvisibility. +pdf. 2012. +[15] K. Hang Kim and F. W. Roush. On strong shift equivalence over a boolean semiring. Ergod. +Th. and Dynam. Sys., 6:81–97, 1986. +[16] T. Kaczynski, K. Mischaikow, and M. Mrozek. Computational Homology. Springer, 2004. +[17] T. Kaczynski and M. Mrozek. Conley index for discrete multivalued dynamical systems. Topol- +ogy and Its Appl., 65:83–96, 1995. +[18] T. Kaczynski, M. Mrozek, and T. Wanner. Towards a formal tie between combinatorial and +classical vector field dynamics. Journal of Computational Dynamics, 3(1):17–50, 2016. +[19] W. D. Kalies, K. Mischaukow, and R. C.A.M Vandervorst. Lattice structures for attractors i. +Journal of Computational Dynamics, 1(2):307–338, 2014. +20 + +[20] W. D. Kalies, K. Mischaukow, and R. C.A.M Vandervorst. Lattice structures for attractors ii. +Foundations of Computational Mathematics, 16:1151–1191, 2016. +[21] W. D. Kalies, K. Mischaukow, and R. C.A.M Vandervorst. Lattice structures for attractors iii. +Journal of Dynamics and Differential Equations, 34:1729–1768, 2022. +[22] K. H. Kim. Boolean Matrix Theory and Applications, volume (Monographs and textbooks in +pure and applied mathematics, v. 70). New York:Dekker, 1982. +[23] M. Lipiński, J. Kubica, M. Mrozek, and T. Wanner. Conley-Morse-Forman theory for general- +ized combinatorial multivector fields on finite topological spaces. arXiv:1911.12698 [math.DS], +pages 1–44, 2020. +[24] G. Minian Elias. The geometry of relations. Order, 27:213–224, 2010. +[25] M. Mrozek, R. Srzednicki, Thorpe J., and Th. Wanner. Combinatorial vs classical dynamics : +Recurrence. Commun. Nonlinear Sci. Numer. Simul., 108(106226):1–30, 2022. +[26] Marian Mrozek and Mateusz Przybylski. The szymczak functor on the category of finite sets +and finite relations. arXiv, 2022. +[27] J. R. Munkres. Elements of Algebraic Topology. Addison-Weslay, Cambridge, 1984. +[28] M. Robinson. Cosheaf representations of relations and Dowker complexes. Journal of Applied +and Computational Topology, 6:27–63, 2022. +[29] A. Szymczak. The Conley index for disccrete dynamical system. Topology Appl., 66:215–240, +1995. +[30] A. Szymczak. Index Pairs : From Dynamics to Combinatorics and Back. Ph.D. thesis, Georgia +Inst. Tech., 1999. +21 + diff --git a/2tE2T4oBgHgl3EQfNgbv/content/tmp_files/load_file.txt b/2tE2T4oBgHgl3EQfNgbv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..afc64e1ea9c486e39aa94b8e2a3e5cad51dabb5d --- /dev/null +++ b/2tE2T4oBgHgl3EQfNgbv/content/tmp_files/load_file.txt @@ -0,0 +1,984 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf,len=983 +page_content='Dowker Complexes and filtrations on self-relations Dominic Desjardins Côté January 11, 2023 Abstract Given a relation on X × Y , we can construct two abstract simplicial complexes called Dowker complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' The geometric realizations of these simplicial complexes are homotopically equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We show that if two relations are conjugate, then they have homotopically equivalent Dowker complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' From a self-relation on X, this is a directed graph, and we use the Dowker complexes to study their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We show that if two relations are shift equivalent, then, at some power of the relation, their Dowker complexes are homotopically equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Finally, we define a new filtration based on Dowker complexes with different powers of a relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Keywords : Dowker complex, relation, filtration, graph theory, shift equivalence 1 Introduction We can use multivalued maps to study dynamical systems [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' The idea is to use Conley index [7] on upper semi-continuous multivalued maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' In applications, it can be hard to study a dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We can use a model that seems to fit data, but it can be a challenge to find it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Another way is to discretize the continuous space and use to multivalued maps to approximate the underlying dynamical system [8] [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Another approach is to use combinatorial structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' To name a few, we can use combinatorial vector fields from Forman [13] [18] [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Moreover, a generalization was proposed by Mrozek called combinatorial multivector fields [23] [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Finally, others proposed to use the distributive lattices to compute attractors on finite data [19] [20] [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Multivalued maps can be restrictive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' In [18], authors generalize them to partial multivalued maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' But a partial multivalued map is equivalent to a relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Some advancements were done in [26] by using the Scymczak category of finite sets where objects are sets and morphisms are relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' The Szymczak category [29] captures the essence of index pairs and index maps [7] which is the core of the theory of Conley index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' So one motivation of this paper is to continue to develop the theory of relations, and it can be used to study dynamical system with finite data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Our main object is a relation which is a subset of the cartesian product of two sets X and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We can define two different abstract simplicial complexes on a relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' For the first simplicial complex, we fixed a value y ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' For all elements in X, that they are related to y, they will span a 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='03739v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='CO] 10 Jan 2023 simplex together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' For the second one, we reverse the role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We fixed a value x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' For all elements in Y , that they are in relation with x, they will span a simplex together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' They are called Dowker complexes [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' An important result is the Dowker’s Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' It says that the geometric realization of these Dowker complexes are homotopically equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' The Dowker’s Theorem is quite useful in applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' To name a few examples, we can use Dowker complexes to study signal coverage [14], to find errors in relation of programs and files [1], to study the privacy of information [12] and in social studies [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' For the last one, this method is called Q-analysis which is developed by Atkins [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' The idea of Q-analysis is to study the q-connectivity and the q-tunnel of the Dowker complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Our two main inspirations for definitions come from these articles [24] and [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' They are many contributions in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' First, conjugate relations have homotopically equivalent Dowker complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' If two relations are shift equivalent with lag l, then at a certain power their Dowker complexes are homotopically equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We can define a filtration on relation based on the Dowker complex at different powers of a finite self-relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Moreover, this can be computed in finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We can also compute the 0th homology of a high enough power with the connected components of the graph induced by the relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' This article goes as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' In section 2, we remind some concepts and definitions on finite relation, graph, simplicial complex, Dowker complexes and finally the famous Dowker’s Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' In section 3, we define right morphism and left morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' If there exists a right or left morphism between two relations, then there is an inclusion from one of the Dowker complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Moreover, we show that if there exists a conjugacy between two relations, then their Dowker complexes are homotopically equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' In section 4, we generalize the idea of right and left morphism for mul- tivalued right and multivalued left morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We show two important properties needed for the definition of a filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' The Dowker complex of a certain power of a relation is include in the Dowker complex of the same relation with a higher power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' For some finite relations at certain a power j, every other Dowker complexes of the same relation at power higher than j are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We call it the stabilization of the Dowker complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We show that shift equivalence between rela- tions have homotopically equivalent Dowker complex at some power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' In section 5, we can define a filtration on the Dowker complexes of different powers of a relation under some simple conditions by using the two properties in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We can use persistent homology on these filtration to extract topological features of the Dowker complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' If a relation is acyclic, then we have that the number of connected components of the graph associated to the relation up to a certain power is equal to the dimension of the 0th homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' It can be generalized to the class of simple relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We also have a similar result for strongly connected relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' 2 Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='1 Finite Relations Let X and Y be finite sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We define a relation as a subset of X × Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let (x, y) ∈ R ⊂ X × Y , we denote by xRy or by y ∈ R(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We define the composition of relations as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let R1 ⊂ X × Y and R2 ⊂ Y × Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' 2 R2 ◦ R1 := {(x, z) ∈ R2 ◦ R1 | ∃y such that xR1y and yR2z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' (1) We define the inverse relation by swapping the sets of a relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' R−1 := {(y, x) ∈ Y × X | y ∈ R(x)} (2) If a relation is a subset of X × X, then we say it’s a self-relation on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We define the power of a self-relation as follows : Rn := � � � � � R ◦ Rn−1 n > 0 IdX n = 0 R−1 ◦ Rn+1 n < 0 The domain and the image for a relation R ⊂ X × Y are : Dom R := {x ∈ X | ∃y such that (x, y) ∈ R} (3) Im R := {y ∈ Y | ∃x such that (x, y) ∈ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' (4) We can see relations as partial multivalued maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' If Dom R = X, then we say that the relation is a multivalued map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' A relation is injective, if for all x1, x2 ∈ X, R(x1) = R(x2) implies that x1 = x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' A relation is surjective if Im R = Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Moreover, a map f : X → Y induces a relation where (x, f(x)) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Without ambiguity, we can compose maps and relations together to obtain a new relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let R be a self-relation on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let j be the least positive integer such that : Rj = Rj+p for some p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We say that j is the index and the least p > 0 is the period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' If j = 1, then R is periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' A pair (j, p) is the eventual period of R with index j and period p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' In other words, the period p will eventually be a period for R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We sometimes use matrices with values in {0, 1} to represent relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let R ⊂ X × Y be a relation with card(X) = m and card(Y ) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' The matrix Mm×n have a value 1 at Mi,j if xiRyj otherwise the value is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' It can be called relation matrix, Boolean relation matrix, binary relation matrix, binary Boolean matrix, (0, 1)-Boolean matrix and (0, 1)-matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' For more information on Boolean matrix theory, we refer to the book [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We say that a self-relation R on X has a cycle at x if and only if there exists an n ∈ N such that xRnx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We say R as a fixed point at x, if n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' If a relation has no cycle at x for all x with period n > 1, then the relation is acyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' A cycle is a sequence x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' , xn such that x1 = xn and xiRxi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' A self-relation R on X is simple if for any two cycles are either disjoints or equals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='2 Graphs In this subsection, we remind the definition of a graph and some notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' A directed graph G is a pair (E, V ) where V is the set of vertices V and E is the subset V × V the set of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' A relation can also be seen as a directed graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' If R is a self-relation on X, then X is the set of vertices and the set of edges E = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' This graph has at most one directed edge from the vertex A to the vertex B, and we also allow a self-loop on vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We note GR the graph induced by a self-relation R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let x, y ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' There is a (x, y)-path, if there exists a sequence of edges e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' , en ∈ E that connect x to y without following the direction of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We can define an equivalence relation on vertices of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' If there is a path between two vertices x and y, then x and y are in the same class of equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' For a graph G, we say the number of connected components is the number of class equivalences of the relation of paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We say G is connected if there is only one connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' If the sequence of edges of a (x, y)-path follows the direction of edges of the graph, then we say it’s a (x, y)-walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We can also define an equivalence relation with a walk between vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' If there is a walk from x to y and a walk from y to x, then x and y are in the same equivalence class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' This is the class of strongly connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' For a graph G, we say the number of strongly connected components is the number of class equivalence of the relation of walks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We say G is strongly connected if there is only one strongly connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='3 Simplicial Complexes and Dowker Complexes In this subsection, we will discuss simplicial complex, Dowker complex and the Dowker’s Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' For more information about simplicial complex, we suggest to read [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We do not present filtration and persistent homology, but we refer to [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' An abstract simplicial complex is a set K that contains finite non-empty sets such as if A ∈ K, then for all subsets of A are also in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' For further examples, we use geometric simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' A geometric n-simplex is the convex hull of a geometrically independent sets of vertices {v0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' , vn} ∈ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' This is the set of x ∈ RN such as x = �n i=0 tixi and 1 = �n i=0 ti where ti ≥ 0 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We denote an n-simplex by [v0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' , vn] is the simplex spanned by the vertices v0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' , vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Any simplex spanned by the subsets of {v0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' , vn} are called faces and denote by the symbol ≤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' A simplicial complex is a collection of simplices for all σ ∈ K, if τ ≤ σ then τ ∈ K and if σ1 ∩ σ2 = τ, then τ is either the empty set or τ is a face of σ1 and σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We say that L ≤ K if L is a sub-complex of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' A simplicial complex is contractible if its homology is equivalent to a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Given an abstract simplicial K, we can define a geometric simplicial complex and |K| call the geometric realization of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We call 0-simplices vertices and 1-simplices edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' The closure of a simplex σ is the set of all the faces of the simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We denote it by cl(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We need one more definition related to simplicial complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' It will be useful in some proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' 4 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' A simplicial complex K is edge-connected, if for any two vertices x and y there is a sequence of edges e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' , en such that x ∈ e1, y ∈ en and cl(ei) ∩ cl(ei+1) ̸= ∅ for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' , n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We have that the simplicial complex is connected if and only if it is edge-connected if and only if H0 is dimension 1 [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Now we explain how to construct abstract simplicial complexes from a relation which are called Dowker complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let R ⊂ X × Y be a relation and X, Y be two finite sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' There are two ways to construct the Dowker complex from a relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let R ⊂ X × Y be a finite relation and KR be the Dowker complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' A simplex [x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' , xn] ∈ KR if and only if ∃y ∈ Y such as xiRy for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We have an analogous construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let R ⊂ X × Y be a finite relation and LR be the Dowker complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' A simplex [y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' , ym] ∈ LR if and only if ∃x ∈ X such as xRyi for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We denote [x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' xn] = σy ∈ KR if and only if xiRy for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We use y as an index for σy to note that all vertices of σy are in R−1(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We use the same notation for σx ∈ LR but the vertices are in R(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' By using the matrix notation, we can use rows and columns to build the simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' The columns are for KR and the rows are for LR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let R ⊂ X × Y be a finite relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' R := � ��� 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 � ��� (5) The first column gives the 2-simplex [x1, x3, x4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' The third and the fourth column give the 0- simplex [x2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' The second and the fifth column do not add new simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We obtain the simplicial complex KR := {[x1, x3, x4], [x2]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' The first row adds a 1-simplex [y1, y5] to LR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' The second row gives a 1-simplex [y3, y4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' The final row adds a 1-simplex [y1, y5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We obtain the simplicial complex LR = {[y1, y5], [y3, y4], [y1, y2]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We obtain that |KR| and |LR| have two connected components and no higher dimension cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' The next theorem links to the homotopy between |KR| and |LR|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='7 (Dowker’s Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let R ⊂ X × Y be a relation and let KR and LR be the associated Dowker complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Then, the polyhedra |KR| and |LR| are homotopy equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' 5 (a) Geometric realization of the Dowker complex KR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' (b) Geometric realization of the Dowker complex LR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Figure 1: These are geometric realizations of the Dowker complexe in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' They are homotopically equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' In 1952, Dowker [10] has shown that KR and LR have isomorphic homology groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' In 1995, Björner [4] has shown that |KR| and |LR| are homotopically equivalent, which is the more com- monly use in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' In recent years, Dowker complexes were regained in popularity in the community of topology data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We can use them to do a filtration on weighted networks [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' In our cases, our filtration will be different and based on different powers of a self-relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' 3 Left and Right Morphism Let start with the definition of the graph homomorphism and next we define left and right morphisms between relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let R be a self-relation on X and R′ be a self-relation on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' A map f : X → Y is a graph homomorphism if the following condition is satisfied : For every x1, x2 ∈ X such as x1Rx2 =⇒ f(x1)R′f(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' If f is bijective and its inverse is also a graph homomorphism, then f is a graph isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We obtain that graph homomorphism keeps some information of the Dowker complex coming from the first relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let f : X → Y be a graph homomorphism between R and R′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' If f is injective, then there exist a map p : KR �→ KR′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' 6 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Consider a n-simplex [x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' , xn] ∈ KR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Then, there exists α ∈ X such as xiRα for all i = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We have that f is a graph homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' This implies that f(xi)R′f(α) for all i = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Indeed, f is injective implies that [f(x0), f(x1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' , f(xn)] is also a n- simplex in KR′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' So we can construct a map p : KR �→ KR′ by sending a simplex [x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' , xn] to [f(x0), f(x1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' , f(xn)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' By the previous argument, p is well defined and injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' If we have a graph isomorphism between two relations, then the Dowker complexes remain unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' This holds because graph isomorphisms are relabelling on the vertices of a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let R1 be a self-relation on X and R2 be a self-relation on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' If there exists a graph isomorphism f between R1 and R2, then they have the same Dowker complexes up to the label of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Graph homomorphisms f and f−1 are injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' By the Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='2, there exist two injective maps p : KR �→ KR′ and p′ : KR′ �→ KR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' So we have that KR and KR′ are the same up to the label of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' By similar arguments, we can show it for LR and LR′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Graph homomorphisms are nice, but they can drastically change the Dowker complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' So, we defined a left morphism which it changes the source of an edge and a right morphism which it changes the target of an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' In this way, only one of the Dowker complexes will change from the right morphism or the left morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' A right morphism f : (X, Y, R) → (X, Z, R′) is a map f : Y → Z such that for every x ∈ X and y ∈ Y : xRy =⇒ xR′f(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We obtain this simple Lemma which is very useful for later proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' If there exists a right morphism f : (X, Y, R) → (X, Z, R′), then KR ≤ KR′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We obtain the equality if f is a bijective map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let f : (X, Y, R) → (X, Z, R′) be a right morphism and [x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' , xn] ∈ KR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' This implies there exists a y ∈ Y such as xiRy for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We obtain that xiR′f(y) for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Finally, we have [x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' , xn] ∈ KR′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Now, we suppose that f is bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let [x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' , xn] ∈ KR′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Then, there exists a z ∈ Z such that xiR′z for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We have f−1(z) ∈ Y and f−1 is well defined because f is bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Then, xiRf−1(z) for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We obtain that [x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' , xn] ∈ KR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' The idea of right morphism comes from the article [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' The author only considered the right morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' But, in our case, we are also interesting of modifying the first set in the cartesian product of a relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' 7 Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' A left morphism g : (X, Z, R) → (Y, Z, R′) is a map g : X → Y such that for every x ∈ X and z ∈ Z : xRz =⇒ g(x)R′z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We have an analogous Lemma for left morphism as the Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='5 for right morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' If there exists a left morphism g : (X, Z, R) → (Y, Z, R), then LR ≤ LR′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We obtain the equality if g is a bijective map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' With the definition of right and left morphism, we can easily show that if two relations are conjugate, then there Dowker complexes are homotopically equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We remind the definition of conjugacy between relations before showing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let R1 be a self-relation on X and R2 be a self-relation on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We say that R1 and R2 are conjugate if there exists a bijective map ϕ : X → Y such as ϕ ◦ R1 = R2 ◦ ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let R be a self-relation on X and R′ be a self-relation on Y which are conjugate by a bijective map ϕ : X → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Then, |KR|, |LR|, |LR′| and |KR′| are homotopy equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' The map ϕ is bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' It implies that KR = Kϕ◦R by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='5 and LR′ = LR′◦ϕ by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' By Dowker’s Theorem, we obtain that |KR| is homotopic equivalent to |LR′|, because Kϕ◦R = KR′◦ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' In [28], the author decides to combine the right and left morphism together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let R ⊂ X × Y and R′ ⊂ X′ × Y ′ be relations and f : X → X′ and g : Y → Y ′ be two maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' A pair (f, g) is a morphism between relation R1 and R2 if for all x ∈ X, y ∈ Y such that xR1y it implies that f(x)R2g(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' In [28], it is shown that the Dowker complex and (co)sheaf representation have nice functoriality properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' In our case, it won’t be useful because we only need right or left morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' But we can see them as a pair (idX, f) where f is a right morphism and idX is the identity function on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' 4 Multi-right morphism and multi-left morphism We want to work with multivalued maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We generalize left and right morphism to multi-left and multi-right morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' A multi-right morphism F : (X, Y, R) ⊸ (X, Z, R′) is a multivalued map F : Y ⊸ Z such as for all x ∈ X, y ∈ Y : xRy =⇒ xR′a for all a ∈ F(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' 8 We also obtain the same Lemma as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let R ⊂ X × Y and R′ ⊂ X × Z be relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' If there exist a multi-right morphism F : (X, Y, R) ⊸ (X, Z, R′), then KR ≤ KR′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We obtain the equality if F is a bijective multivalued map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' The proof is the same as Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' A multi-left morphism G : (X, Z, R) ⊸ (Y, Z, R′) is a multivalued map G : X → Y such for all x ∈ X, z ∈ Z : xRz =⇒ aR′z for all a ∈ G(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let R ⊂ X × Z and R′ ⊂ Y × Z be relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' If there exist a multi-left morphism G : (X, Z, R) ⊸ (Y, Z, R′), then LR ≤ LR′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We obtain the equality if G is a bijective multivalued map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We denote a multi-right morphism by mr-morphism and a multi-left morphism by ml-morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Remarks 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We remind that if a relation S ⊂ X×Y satisfies Dom S = X, then S is a well-defined multivalued map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Moreover, for any relation R ⊂ Z ×X, we have that S : (Z, X, R) ⊸ (Z, Y, S ◦R) is a well-defined mr-morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' It is also true for ml-morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' The next corollary will be useful to define our filtrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let R be a self-relation on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' If Dom R = X, then KRn ≤ KRn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' If Im R = X, then LRn ≤ LRn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Dom R = X implies that R is a multivalued map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Moreover, R : (X, X, Rn) ⊸ (X, X, Rn+1) is well-defined mr-morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='5, we have that KRn ≤ KRn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' In the same way, Im R = X implies that R−1 is a multivalued map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We have that R−1 : (X, X, Rn) ⊸ (X, X, Rn+1) is well-defined ml-morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='7, we have that LRn ≤ LRn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' The hypothesis Dom R = X in the previous corollary is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let us show that by an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' If R is a self-relation on X, define by this matrix such as : � � 0 1 1 0 0 1 0 0 0 � � (6) Then, we obtain KR = {[x2, x3], [x2], [x3]}, KR2 = {[x3]} and KRn = ∅ for n > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We don’t have KRn ⊂ KRn+1 for all n ∈ N>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' If the matrix is nilpotent, then KRn is an empty set for an n ∈ N>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' 9 Using Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='6, we can show that the Dowker complexes of a relation stabilize at some power n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let R be a finite self-relation on X with an eventual period (j, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' If Dom R = X, then, we have KRj = KRj+i for i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' If Im R = X, then LRj = LRj+i for i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' By Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='6, we have the sequence : KRj ≤ KRj+1 ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' ≤ KRj+p−1 ≤ KRj+p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' (7) But p is the period of R, hence Rj = Rj+p implies that KRj = KRj+p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' By (7), we obtain KRj = KRj+i for i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' A similar proof can be done for LRj = LRj+i for i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We remind the definition of shift equivalence between two relations and we show the assump- tion that relations are shift equivalences implies that theirs Dowker complexes are homotopically equivalent at some power for each relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let R1 be finite self-relation on X and R2 be finite self-relation on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' R1 and R2 are shift equivalent with a lag l, if there exists two relations S ⊂ X × Y and T ⊂ Y × X such as : R1 ◦ T = T ◦ R2 S ◦ R1 = R2 ◦ S T ◦ S = Rl 1 S ◦ T = Rl 2 We say it is a strong shift equivalence if l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let R1 be finite self-relation on X with Dom R1 = X = Im R1 and R2 be finite self-relation on Y with Dom R2 = Y = Im R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let (jp, p) be an eventual period of R1 and (jq, q) be an eventual period of R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Without loss of generality, we suppose that jp ≥ jq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' If R1 and R2 are shift equivalent with lag l, then |KR jp 1 |, |KR jq 2 |, |LR jp 1 | and |LR jq 2 | are homotopy equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' If R1◦T = T ◦R2 then Rn 1 ◦T = T ◦Rn 2 is also true for n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Moreover, we have Dom S = X and Dom T = Y , because Dom R1 = X, Dom R2 = Y , T ◦ S = Rl 1 and S ◦ T = Rl 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' So we have that S and R are well defined multivalued maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We want to show that KR jp 1 = KS◦R jp 1 and LR jq 2 = LR jq 2 ◦S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We are going to use T and S as mr-morphism with R1 and ml-morphism with R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We have that S : (X, X, Rjp 1 ) ⊸ (X, Y, S ◦ Rjp 1 ) and T : (X, Y, S ◦ Rjp 1 ) ⊸ (X, X, T ◦ S ◦ Rjp 1 ) are well-defined mr-morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We have T ◦ S = Rl 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' It implies that T ◦ S ◦ Rjp 1 = Rl+jp 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We obtain KR jp 1 ≤ KS◦R jp 1 ≤ KT◦S◦R jp 1 = KR l+jp 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' The eventual period of R1 is (jp, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' It implies that KR jp 1 = KR jp+l 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We obtain that KR jp 1 = KS◦R jp 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We can see that S : (Y, Y, Rjp 2 ) ⊸ (X, Y, Rjp 2 ◦ S) and T : (X, Y, Rjp 2 ◦ S) ⊸ (Y, Y, Rjp 2 ◦ S ◦ T) are well-defined ml-morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We have S ◦ T = Rl 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' It implies that Rjp 2 ◦ S ◦ T = Rjp+l 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We 10 (a) The graph GR1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' (b) The graph GR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Figure 2: Theses are the graphs from Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' obtain LR jp 2 ≤ LR jp 2 ◦S ≤ LR jp 2 ◦S◦T = LR jp+l 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We have LR jq 2 = LR jp+l 2 because jp ≥ jq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We obtain LR jp 2 = LR jp 2 ◦S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Finally, we have KR jp 1 = KS◦R jp 1 = KR jp 2 ◦S and LR jp 2 = LR jp 2 ◦S = LS◦R jp 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' By Dowker’s Theorem, we obtain that |KR jp 1 |, |KR jq 2 |, |LR jp 1 | and |LR jq 2 | are homotopy equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let X be a finite set with 8 points and Y be a finite set with 3 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let R1 be a self-relation on X and R2 be a self-relation on X defined by those graphs in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' R1 has an eventual period (3, 3) and R2 has an eventual period (1, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We see in Figures 3(a), (b) and (d) that the Dowker complexes are not homotopically equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' But, in Figures 3 (c) and (d), the Dowker complexes with relations at power 3 are homotopically equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' There is an interesting proposition from [15] that we can use for strongly connected relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We remind that an indecomposable Boolean matrix is a relation which is strongly connected and J is a square matrix where all the entries are equals to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='12 (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='3 in [15] ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Every indecomposable Boolean matrix with positive trace is strong shift equivalent to J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We can easily compute the Dowker complexes of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' It is a (n−1)-simplex where n is the number of rows of J for both Dowker complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Finally, we obtain that the Dowker complexes of a strongly connected self-relation at a power high enough are contractible if the trace is strictly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' 11 Y3(a) Dowker complex of |KR1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' (b) Dowker complex of |LR1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' (c) Dowker complexes of |KR3 1| and |LR3 1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' (d) Dowker complexes |KR2|, |LR2|, |KR3 2| and |LR3 2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Figure 3: Theses are the different Dowker complexes from Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' 12 5 Filtrations of Dowker complexes For this section, we suppose that a R is a finite self-relation on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' From Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='6, if Dom R = X, then we have KRi ≤ KRi+1 for all i ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We have an inclusion and we get this filtration : KR �→ KR2 �→ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' �→ KRi �→ KRi+1 �→ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' (8) In the same way with Im R = X, we have LRi ≤ LRi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We obtain another filtration : LR �→ LR2 �→ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' �→ LRi �→ LRi+1 �→ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' (9) From Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='8, the Dowker complexes stabilize at a certain power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' This means we can compute the filtration (8) and (9) in finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' For our filtrations, we start at i = 1, but we could also start with the i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We have that R0 = idX and R : (X, X, idX) ⊸ (X, X, R) is a well-defined mr-morphism, if Dom R = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Then, the filtration (8) becomes : KidX = KR0 �→ KR �→ KR2 �→ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' �→ KRi �→ KRi+1 �→ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' (10) And for the filtration (9) by applying similar arguments, we obtain : LidX = LR0 �→ LR �→ LR2 �→ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' �→ LRi �→ LRi+1 �→ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' (11) The homology of KidX and Lidx is the homology of n = card(X) points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' In some cases, we might want to start the filtration at i = 0 or i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We remind that, by Dowker’s Theorem, |KRi| and |LRi| are homotopically equivalent for all i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We obtain the same bar code representation for filtrations (8) and (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let R1 be a self-relation on X given by the graph in the Figure 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' R1 has 9 nodes and is acyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' The eventual period is (3, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We obtain the bar code at Figure 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' It has one generator of H1 with the interval [1, 2] and we had 3 generators of H0 that die early and 1 generator of H0 that survive to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let R2 be a self-relation on X given by the graph in the Figure 4(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' R2 has 10 nodes and is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' The eventual period is (3, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We obtain the bar code at the Figure 4(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' It has 4 generators of H0 that die at time 2 and 2 other generators that survive to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' For the next results, we compute the 0th homology of the Dowker complexes for different types of relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' If R is a finite acyclic self-relation, then it has an eventual period p = 1 and j ∈ N>0 such as Rj = Ri for all i ≥ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' So we denote this relation Rj by R∞, because it converges to a relation when i → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' 13 (a) The graph of the acyclic rela- tion from Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' (b) The associated bar code of the rela- tion from Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We use the fil- tration with KRi 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' The first bar in orange is a generator in H1 and the others four bars in blue are generators in H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' (c) The graph of the relation with multiple cycles from Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' (d) The associated bar code of the rela- tion from Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We use the fil- tration with KRi 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' The six bars in blue are generators in H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Figure 4: In these figures, we have the graph on the left and the associated bar code diagram on the right for Examples 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Dashed lines in bar code mean it goes to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' 14 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='0 T 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='0 T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content="5 0'E 3." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='0X5 X105 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' 4 3 - 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='0X3Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We say that x ∈ X is a minimum for a self-relation R, if there exists no y ∈ X such as x ̸= y and xR∞y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We denote the set Ux := {y ∈ X | yR∞x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We say that x ∈ X is a maximum for a self-relation R, if there exists no y ∈ X such as x ̸= y and yR∞x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We denote the set Dx := {y ∈ X | xR∞y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' The maximums and minimums of an acyclic relation are important, because they are responsible for the maximal simplices of KR∞ and LR∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let R be a finite acyclic self-relation on X with Dom R = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Then, the maximal simplices of KR∞ are given by the minimums of R∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We have that Dom R = X it implies that Dom R∞ = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Then, for all x ∈ X there exists z ∈ X such that xR∞z and z is a minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let σy = [x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' , xn] ∈ KR∞ be an arbitrary simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We have an y ∈ X such as xiR∞y for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' By the first argument, there exists a minimum z ∈ X such as yR∞z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Therefore, we have xiR∞yR∞z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Then, σy ⊂ σz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We can do a similar result with LR∞ by using the maximums of R∞, if Im R = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let R an acyclic finite self-relation on X with Dom R = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' number of connected components of GR = dim H0(KR∞) = dim H0(LR∞) (12) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' First, by Dowker’s Theorem, we have dim H0(KR∞) = dim H0(LR∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We suppose that GR is connected and show that dim H0(KR∞) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' More precisely, we show that KR∞ is edge-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We have that Dom R = X implies that for all x ∈ X,[x] ∈ KR∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let x, x′ ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' There exists y1 ∈ X a minimum such as xR∞y1 and y1R∞y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' This implies that e1 = [x, y1] ∈ KR∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We also have that there exists yn ∈ X a minimum such as x′R∞yn and ynR∞yn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' This implies that en = [x′, yn] ∈ KR∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Since GR is connected, there exists a (y1, yn)-path of finite length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We denote this sequence by y1, z1, z2, z3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' , zm, yn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Without loss of generality, we take the shortest path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' There exist a i such as zi ∈ Uy1 and zi+1 /∈ Uy1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' First, we have e2 = [y1, zi] ∈ KR∞ and ziRzi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' There exists y2 ̸= y1 a minimum such as zi+1R∞y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' This implies that ziR∞y2 and we have the edge e3 = [zi, y2] ∈ KR∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We can repeat this process with the (y2, yn)-path until we obtain a sequence of edges that connect the vertex [x] and [x′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We obtain that KR∞ is edge-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Now suppose that GR is not connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let H be a connected component of GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Then, for all x ∈ H and for all y /∈ H, we have that x /∈ R(y) and y /∈ R(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' It implies that for each connected component gives a single generator for H0(KR∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We can construct a map j : cc(GR) → H0(KR∞) that sends the connected components of GR to the generators of H0(KR∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' By the previous argument, we can make this map j injective .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' For any generator g in H0(KR∞), there exists a x ∈ X such as g is homologous to [x] because Dom R = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' This implies there exists a H ∈ cc(GR) such as x ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We obtain that the map j is bijective and the equality (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' 15 Figure 5: The graph of R4 2 from Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' GR4 2 has 2 connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We can show a similar proof for simple relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let R be a finite simple self-relation on X with Dom R = X and (j, p) be an eventual period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Assume that GR is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' There exists a r ∈ N such that : dim H0(KRj) = dim H0(LRj) = number of connected components of GRr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' (13) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We can find q big enough so that Rq is acyclic, because R is a simple relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We choose a i ∈ N such as iq > j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We fix r = iq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We also have that Rr is also acyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' By Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='8, we have that KRj = KRr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='5, we know that dim H0(KRj) = number of connected components of Rr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' If GR has more than one connected component, we apply this theorem for each connected component of GR by using similar arguments as the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' From preceding results, if Dom R ̸= X but Im R = X, we can redo the proofs with LRj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Another approach is to use R−1, because Dom R−1 = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' In Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='2, it is a simple relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We have R4 2 is acyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' The graph of R4 2 is shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' It has two connected components and the bar code from Figure 4(b) has 2 bars goes to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' It is expected from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We have shown earlier in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='10 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='12 that a strongly connected self- relation R is shift equivalent to a matrix J, if tr(R) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' But we will like to have a result for any strongly connected relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' But, first we need some definition and other results from other papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let gcd(a, b) be the great common divisor of a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We define : q = gcd(n1, n2, n3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=') (14) where ni is the length of a cycle and i ∈ I is the set of all different cycles from R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We obtain this proposition : Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='8 (Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='12 in [26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let R be a strongly connected self-relation on X, (j, p) be the eventual period and q defined by (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We have q|j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' 16 Let’s define a new equivalence relation ∼q for a strongly connected self-relation R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We say that x ∼q y, if for each (x, y)-walk has length equal to 0 modulo q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' It is an equivalence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='9 (Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='16 in [26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let R be a strongly connected self-relation on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let q defined as (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Then, ∼q is an equivalence relation in X with exactly q distinct equivalence classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We need one more Lemma before showing our final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='10 (Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='25 in [26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let R be a strongly connected self-relation on X and (j, p) an eventual period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Then, x ∼q x′ =⇒ Rj(x) = Rj(x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' (15) We are going to show that the number of class equivalence of ∼q is equal to dim(H0(KRj)) for a strongly connected self-relation with eventual period (j, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let R be a finite self-relation on X with an eventual period (j, p), R is strongly connected, q defined by (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Then, we have : number of [x]∼q = q = dim(H0(KRj)) = dim(H0(LRj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' (16) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We show that for any x, y ∈ X, if x ∼q y, then x and y are edge-connected and if x ̸∼q y, then x and y are not edge-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' First, we suppose that x ∼q y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='10, we have that Rj(x) = Rj(y) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' There exists a z ∈ Rj(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' It implies that [x, z] and [y, z] are in KRj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' So, each vertex in the same equivalence class is edge-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Now, we suppose that, x ̸∼q y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' There exists a (x, y)-walk of length n modulo q where n ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let show that Rj(x) ∩ Rj(y) = ∅ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let’s suppose there exists a z ∈ Rj(x) ∩ Rj(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' This implies there exists a (x, z)-walk of length j and a (y, z)-walk of length j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' But, from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='8, q|j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' This implies x ∼q z and y ∼q z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' But ∼q is an equivalence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We obtain that x ∼q y which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We obtain that x ̸∼q y implies Rj(x) ∩ Rj(y) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We obtain that if x ∼q y then they are edge-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' But, if x ̸∼q y, then they are not edge-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' There is q different equivalence classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' For the case of R is an arbitrary relation, it is harder to find its homology H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Also, for higher dimensions of the homology groups, it’s hard to tell what happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Further investigations are needed for both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Now, we return to the filtrations defined earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' There are two other types of filtrations that we can use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' If R is a self-relation on X with Dom R = X = Im R, then we can use both filtrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' But we obtain the same bar codes for both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' That holds because, for each i ∈ N, |KRi| is homotopically equivalent to |LRi| for any self-relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We might need to come with other types of filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We suggest two other types of filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' It will be interesting to use a zigzag filtration [5] with KRi and LRi by alternating them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' It will probably depend on the relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Further investigations are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' 17 We will present an interesting bi-filtration with KRm and LRn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We have that, if KRm ⊂ KRm+1 and LRn ⊂ LRn+1, then KRm ∩LRn ⊂ KRm+1 ∩LRn and KRm ∩LRn ⊂ KRm ∩LRn+1 for all m, n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We obtain this bi-filtration : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' KRm ∩ LRn KRm+1 ∩ LRn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' KRm ∩ LRn+1 KRm+1 ∩ LRn+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' The computation of the bi-filtration is also finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Because the relation R is finite and Dom R = X = Im R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We obtain an eventual period (j, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' In the bi-filtration, there are, at maximum, j2 different simplicial complexes to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' One may ask why the intersection is a good idea to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let’s explain it in more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let R be a self-relation on X with Dom R = X = Im R and m, n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let σ ∈ KRm ∩ LRn where σ = [x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' , xd].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Then, there exists xω ∈ X such that xiRmxω and there exists xα ∈ X such that xαRnxi for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Another way to see this is, for each xi ∈ σ, there exists a (xω, xα)-walk of length m+n passing through xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We can subdivide this (xω, xα)-walk into a (xω, xi)-walk of length m and a (xi, xα)-walk of length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' So, by only the existence of a simplex σ in KRm ∩ LRn, each vertex of σ, it has a walk with a common starting point and a common ending point of same length going through the vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' It will be interesting to study these Dowker complexes when m tends to infinity, n tends to infinity or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' 6 Conclusion In summary, we used the Dowker complexes to study some properties of self-relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' First, we defined the right morphism and left morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We also generalized it to the case of multivalued maps called multi-right morphism and multi-left morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' The existence of a right or multi-right morphism between R1 and R2 implies that the KR1 is included in KR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Similarly, the existence of a left or a multi-left morphism between R1 and R2 implies that LR1 is included in LR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We have shown that two relations are conjugate implies that they have homotopically equivalent Dowker complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We have also shown that if two relations are shift equivalent, then their Dowker complexes are homotopically equivalent at some power of the relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We were interested in self-relation which is equivalent of a type of directed graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We have obtained two nice properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' If R is finite and Dom R = X = Im R, then we have that KRi ≤ KRi+1 and LRi ≤ LRi+1 for all i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Moreover, there exists a j ∈ N such as KRj = KRi+j and LRj = LRi+j for all i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' With these two properties, we defined two filtrations with KRi �→ KRi+1 and LRi �→ LRi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Also, the filtration ends at some finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Finally, we proved some results about the 0th homology for some types of self-relation at some power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' We also proposed the intersection filtration and the zigzag filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' 18 We have put some foundations to study directed graph using Dowker complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Moreover, we think it might be a useful tool to study the dynamics of finite data define by a directed graph of Dowker complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Let R be a self-relation with an eventual period (j, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' The positive or forward invariant is given by the existence of a simplex in KRj when j converges to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' The negative or backward invariant is given by the existence of a simplex in Lj R when j converges to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' But, by the stabilization of Dowker complexes, we can compute it in finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Finally, if we want to study the invariant of R, this is given by the existence of a simplex in KRj ∩ LRj which is the intersection of the forward and the backward invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' But further investigations are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' The idea is to use the structure of the Dowker complexes to encode the dynamics of finite data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' 19 References [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Ambrose, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Huntsman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Robinson, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Yutin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Topological differential testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' CoRR, abs/2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content='00976, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Futures, Heinemann, 10(6), 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Björner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Topological methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Handb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=', 2:1819–1872, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' [5] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Carlsson and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' de Silva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Zigzag persistence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Found Comput Math, 10, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Chowdhurry and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Mémoli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' A functorial Dowker theorem and persistent homology of asymmetric networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Journal of Applied and Computational Topology, 2:115–175, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' [7] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Conley.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' A rigorous numerical method for the global analy- sis of infinite-dimensional discrete dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Applied Dynamical Systems, 3(2):117–160, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' [9] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' Desjardins Côté.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} +page_content=' From finite vector field data to 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+page_content=' 21' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQfNgbv/content/2301.03739v1.pdf'} diff --git a/3tFAT4oBgHgl3EQfERy2/vector_store/index.pkl b/3tFAT4oBgHgl3EQfERy2/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..61ce750e27c2954de052f53ee149d4f37c913339 --- /dev/null +++ b/3tFAT4oBgHgl3EQfERy2/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:bef6153b8b6aacc6e9f04a30d478798323b02d2c9e31cb7f18182a3a29b9f73e +size 181842 diff --git a/4NE2T4oBgHgl3EQfjwdR/content/tmp_files/2301.03971v1.pdf.txt b/4NE2T4oBgHgl3EQfjwdR/content/tmp_files/2301.03971v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a51d1f88f57d6111b2c46bf5cf6f43867b1eca67 --- /dev/null +++ b/4NE2T4oBgHgl3EQfjwdR/content/tmp_files/2301.03971v1.pdf.txt @@ -0,0 +1,1291 @@ +Unsupervised Mandarin-Cantonese Machine Translation +Megan Dare, Valentina Fajardo Diaz, Averie Ho Zoen So, Yifan Wang, Shibingfeng Zhang +Summer Semester Software Project 2022 +Language Science and Technology, Saarland University +{mdare,valenfd,averieso,yifwang,zhangshi@coli.uni-saarland.de} +Abstract +Advancements in unsupervised machine trans- +lation have enabled the development of ma- +chine translation systems that can translate be- +tween languages for which there is not an +abundance of parallel data available. We ex- +plored unsupervised machine translation be- +tween Mandarin Chinese and Cantonese. De- +spite the vast number of native speakers of +Cantonese, there is still no large-scale corpus +for the language, due to the fact that Can- +tonese is primarily used for oral communica- +tion. +The key contributions of our project +include: 1. +The creation of a new corpus +containing approximately 1 million Cantonese +sentences, and 2. +A large-scale compari- +son across different model architectures, tok- +enization schemes, and embedding structures. +Our best model trained with character-based +tokenization and a Transformer architecture +achieved a character-level BLEU of 25.1 when +translating from Mandarin to Cantonese and of +24.4 when translating from Cantonese to Man- +darin. In this paper we discuss our research +process, experiments, and results. +1 +Introduction +In recent years, neural machine translation has +gained massive research interests. Most of these +studies (e.g. Bahdanau et al. 2014; Luong et al. +2015; Wu et al. 2016; Vaswani et al. 2017) focus +on the construction of neural machine translation +systems leveraging parallel bilingual corpora. Nev- +ertheless, such an approach is not feasible for many +language pairs due to the scarcity of resources for +such pairs, as is the case for Cantonese and Man- +darin. The study of automatic translation between +these two languages faces the same problem: to +the best of our knowledge, despite the vast number +of native speakers of both languages, there is still +no large-scale Mandarin-Cantonese parallel corpus. +In addition, monolingual corpora for Cantonese are +hard to collect as it is a low-resource language that +is mainly used for only oral communication. +Currently, only a few studies have been done +on Cantonese-Mandarin translation, among which +some compare various low-resource models for +this language pair. However, these studies nor- +mally focus on a comparison between one or two +model types. Based on our motivation of imple- +menting and training a Cantonese-Mandarin trans- +lation model and current state of research, we set +our goal as building a robust model trained on +a more diverse dataset, which can help improve +communication between Cantonese and Mandarin +speakers. Additionally, we seek to compare vari- +ous model architectures, tokenization schemes, and +embedding structures to conduct a comprehensive +understanding on which settings may lead to the +best performance for the Cantonese-Mandarin lan- +guage pair. +After a close analysis of the current state of re- +search and the available resources, we propose to +develop a Cantonese-Mandarin machine translation +system that is capable of conducting translation in +both directions. The training of the system involves +only Mandarin and Cantonese monolingual corpora +collected from Wikipedia and various websites. +Our work also makes contributions to the Can- +tonese language NLP field by collecting Cantonese +textual data and building a public large-scale mono- +lingual corpus, which did not exist until now. +In addition, considering the similarity between +Cantonese and Mandarin, our translation system +will provide a foundation for further development +regarding machine translation tasks that center +around language pairs composed of two similar +languages. +2 +Background +2.1 +Cantonese and Chinese: an overview +Cantonese is one of the most widely spoken va- +rieties of Chinese other than Mandarin Chinese +(Matthews and Yip, 2013). It is estimated to have +arXiv:2301.03971v1 [cs.CL] 10 Jan 2023 + +more than 55 million native speakers, with large +populations found in southern China provinces +Guangdong and Guangxi, as well as regions includ- +ing Hong Kong and Macau, it is also commonly +spoken in overseas Cantonese communities in Sin- +gapore, Malaysia, North America and Australia as +a result of emigration (Matthews and Yip, 2013). +While numerous NLP applications have been +developed for Mandarin Chinese, little has been +developed for Cantonese. One reason for this is the +limited linguistic resources that have been collected +for Cantonese. Primarily a spoken language and a +non-standard variety, written Cantonese is not tra- +ditionally used or taught in schools. Instead, Can- +tonese speakers typically learn to read and write +in standard Chinese through education, so there is +no language barrier for Cantonese speakers when +interacting with computer applications designed in +standard Chinese. +On the other hand, with the availability of the +internet and the rise of social media, Cantonese is +much more commonly used and written online in +recent years, which can be seen as an indicator for +a market in Cantonese NLP applications. +It is important to note that this phenomenon +might only be applicable to Hong Kong Cantonese, +and not other variants such as the one in Guang- +dong province. More recent discussions about Can- +tonese, such as Bauer (2018), make a point to dis- +tinguish between the Hong Kong Cantonese variant +and the others, since the use of Cantonese is on the +rise in Hong Kong, while declining in provinces +within mainland China. Not only has this led to +Hong Kong being named “the Cantonese-speaking +capital of the world" (Bolton, 2011, p.64), but also +the rise of written Cantonese locally and subse- +quently, the Cantonese text data that are available +online, which are of the Hong Kong variant of Can- +tonese. +2.2 +Linguistic Differences between Cantonese +and Mandarin +Despite the common misconception that Chinese +dialects share the same grammar, Cantonese and +Mandarin are different at phonological, lexical and +syntactic levels, and are not mutually intelligible +(Matthews and Yip, 2013). Some suggests it is +more accurate describe Cantonese as a distinct +language of the Chinese language family (Snow, +2004). For the rest of this section, we describe +some features that differ between Mandarin and +Hong Kong Cantonese. +2.2.1 +Writing Systems +To anyone who can read Chinese, the most notable +visual variation in written Chinese is the writing +system - Traditional or Simplified Chinese. The +two systems are equivalent to each other, and have +one-to-one correspondence for each character. The +following is some examples of traditional / sim- +plified characters: “open" 開/开, “talk" 話/话 and +“book" 書/书. The usage of either system is primar- +ily due to regional difference, with mainland China +using the simplified system, while Hong Kong and +Taiwan use the traditional system. +2.2.2 +Lexical and Syntactic comparisons +Vocabulary difference is the main barrier which +prevents Mandarin speakers from understanding +Cantonese (Snow, 2004), it is also the aspect which +is the most distinguishable between Cantonese and +Mandarin. +According to Snow (2004), written +Cantonese in formal domains can contain around +10-15% Cantonese-only characters, while this per- +centage in informal domains can go up to 25-40%. +Notably, the vocabulary that differ are some of +the most frequent words, including many func- +tion words, as seen in Table 1. +Syntactically, +Meaning +Cantonese +Mandarin +possessive marker +ge3 +的de +perfective marker +zo2 +了le +pronoun pluralizer +dei6 +們mén +negator +唔m4 +不bù +is (copula) +係hai6 +是shì +this +呢ne1 +這zhè +Table 1: Examples of lexical difference between Can- +tonese and Mandarin from Snow (2004, p.49). Can- +tonese romanizations follow the Jyutping system. +Cantonese and Mandarin are broadly similar but +with some differences that are often overlooked +(Matthews and Yip, 2013). Some common differ- +ences are in terms of word order, including indi- +rect object and comparative constructions (Snow, +2004): +Indirect object construction: +Cantonese: +我俾錢佢ngo5 bei2 cin4 keoi5 +(I + give + money + he) +Mandarin: +我給他錢wó gˇei t¯a qían +2 + +(I + give + he + money) +‘I give him money’ +Comparative construction: +Cantonese: +我高過佢ngo5 gou1 gwo3 keoi5 +(I + tall + more than + he) +Mandarin: +我比他高wó bˇı t¯a g¯ao +(I + compared to + he + tall) +‘I’m taller than him.’ +2.2.3 +Challenges Unique to Cantonese NLP +Firstly, there exists a certain degree of variabil- +ity in written Cantonese since it was never stan- +dardised. As such, some words can be written +with completely different characters yet have the +same meanings and pronunciations. For example, +“like" can be written as 中意or 鍾意(read: zung1 +ji31), “still" can be written as 仲or 重(read: zung6) +(Matthews and Yip, 2013). Additionally, when +some Cantonese words cannot be represented by +existing Chinese characters, they could be written +in a romanized form, such as the comparative (eg. +“-er" in “cheaper") can be written with “D", as well +as a non-romanized form (read: di1) (Snow, 2004; +Matthews and Yip, 2013). +Secondly, code-switching to English is a com- +mon phenomena in Cantonese, which is not a +feature in standard Chinese or Mandarin. Code- +switching in Hong Kong Cantonese is mostly in- +trasentential (below clause level) (Li, 2000), for +example: +我今朝9點有個meeting。 +ngo5 dei6 gam1 ziu1 gau2 dim2 jau5 go3 +MEETING +‘We have a meeting at 9am today.’ +3 +Related Work +3.1 +Unsupervised Machine Translation +Unsupervised machine translation with no parallel +data is a challenging task that has attracted many +interests. The presence of cross-lingual embed- +dings (Mikolov et al., 2013; Artetxe et al., 2016, +2017a, 2018a,b; Conneau et al., 2017) provides +prior knowledge for machine translation systems +and makes it possible to train a machine transla- +tion model in an unsupervised way. Artetxe et al. +(2017b) and Lample et al. (2017) are the first at- +tempts to explore the possibility of constructing +1romanizations according to the Jyutping system. +a neural machine translation system using only +monolingual corpora from both source and target +languages. The proposed system is based on an +encoder-decoder architecture with attention mecha- +nism (Bahdanau et al., 2014), trained with a denois- +ing auto-encoding task (Vincent et al., 2008) and a +back-translation task (Sennrich et al., 2015). The +encoder is shared by both the source and target lan- +guages, so that sentences from both languages can +be mapped to a common latent space, while each +language has its own decoder to reconstruct en- +coded sentences back into its own language space. +Cross-lingual embeddings are leveraged as an ini- +tialization for the system, providing additional lex- +ical level information. Such a structural property +allows the translation model to be bi-directional, +that is, the same model can be employed in both the +L1-to-L2 translation task and the L2-to-L1 transla- +tion task. +This approach is extended in Lample et al. (2018) +by applying a transformer model and using sub- +word level tokenization methods. Attention-only +structures provide higher model capacity, and sub- +word level tokenization method Byte Pair Encod- +ing (BPE) reduce the size of vocabulary and helps +solving problems in translation. Addition- +ally, they re-exploited the potential of statistical +approaches in unsupervised machine translation +tasks. A phrase-based machine translation model +initialized with an automatically populated phrase +table and language model is trained by iterative +back-translation. Results of the experiment show +that a statistical approach can reach similar perfor- +mance or even outperform neural systems when the +data is scarce, as the neural model tends to over- +fit the corpora, and thus does not generalize well. +Together with Singh and Singh (2020), they show +that unsupervised approaches can be used to con- +struct machine translation systems for low-source +languages (e.g., Urdu, Romanian, Manipuri). +In recent years, pre-trained language models +have become popular due to their competitive +ability of representing and generating natural lan- +guages learned from transfer learning on large- +scale self-supervised datasets. Lample and Con- +neau Lample and Conneau (2019) take their work +one step further by pre-training both the encoder +and decoder in their model using a cross-lingual +language model (XLM). They then fine-tune the +pre-trained model to an unsupervised neural ma- +chine translation model following the training pro- +3 + +cess described in Lample et al. (2018). The pre- +training stage results in a sharp BLEU score in- +crease over previous benchmarks for unsupervised +machine translation. +Unsupervised machine translation methods are +also applied in dialectal machine translation tasks, +where the similarity and commonality between lan- +guages can be leveraged. Farhan et al. (2020) uses +common words between Arabic dialects as anchor +points to steer projections of surrounding words be- +tween two dialects, creating a more accurate map- +ping between source and target words. In this way, +they construct an unsupervised machine translation +system with a BLEU score of 32.14, which is re- +markably high compared with the highest BLEU +score obtained in the supervised setting (48.25). +3.2 +Mandarin-Cantonese Machine +Translation +Due to the scarcity of available datasets, Cantonese +language is always under-researched in NLP tasks. +This issue is even more severe in machine trans- +lation tasks, which usually requires large amount +of parallel data. For this reason, many researches +on Cantonese-Mandarin machine translation are +intended to collect more data or to fully exploit the +limited data in a semi-supervised or unsupervised +way. +Hei Yi Mak and Tan Lee (2021) construct a +large-scale Cantonese-Mandarin parallel dataset +by mining parallel sentences from Mandarin and +Cantonese Wikipedia. They apply a similarity- +based sentence alignment approach and use sen- +tence pairs with high confidence score as parallel +sentences. In this way, they end up with a paral- +lel corpus of about 100,000 sentences. They also +fine-tune a pre-trained language model using the +collected data and obtain a competitive translation +system that outperforms Baidu Fanyi, a commonly +used translator in China. +Concurrently, some efforts have been made to +create unsupervised Cantonese-Mandarin transla- +tion systems. (Wan et al., 2020) handles Cantonese- +Mandarin translation as a dialect translation prob- +lem. which attempts to exploit the commonality +between two language dialects. On the basis of +(Lample et al., 2018)’s transformer model, they +make use of pivot-private embeddings and layer +coordination to better utilize the similarity and dif- +ference between the two languages. Trained on +two large monolingual datasets of 20 million collo- +quial sentences for each Mandarin and Cantonese, +their model reaches an improvement of up to 12 +BLEU score for Cantonese to Mandarin, and 5 +BLEU from Mandarin to Cantonese compared to +their baseline transformer model. +There have been other works relying on pre- +trained cross-lingual language models (XLM). In +Wong and Tsai (2022), the authors initialize the +encoder and decoder with XLM as described in +(Lample and Conneau, 2019), while using pivot- +private embeddings rather than cross-lingual em- +beddings. Using this enriched structure, they are +able to achieve slight BLEU score improvements +over previous XLM models. +4 +Corpus Construction +While existing Cantonese corpora are scarce, and +usually collected for linguistic purposes which is +smaller in scale and of a specific demographic (eg. +Wong et al. 2017; Luke and Wong 2015), text data +is available on the internet due to Cantonese being +the common language used on social media. This +also led to a rise in Cantonese writing in tradition- +ally more formal domains such as advertisements, +online news and subtitles. +Therefore, we aim for the corpus to span across +various domains for a comprehensive collection of +modern Cantonese usage. Secondly, since standard +Chinese is also commonly used among Cantonese +speakers in online settings, in the data selection pro- +cess, we aim to avoid sources which use standard +Chinese. Lastly, in our pre-processing, we preserve +some unique features in Cantonese such as code- +switching in English. Detailed data statistics of the +corpus is available on the Github repository. +As we focus on collecting data for Cantonese, +note that we simply use the Chinese Wikipedia +for Mandarin data, since there is already a large +amount of data available just from one source. +4.1 +Data Collection +The Cantonese data available from various sources +on the internet are either readily downloadable (for +Wikipedia, corpus and dictionary) or are scraped +by us (for Instagram, subtitles and articles). Due to +structural differences in the various websites, scrap- +ing functions are individually written for each of +the three classes of sources. In general, the script +moves recursively over the website domain and +extracts any text in each web page. The scraping +script is available on our GitHub repository. Fig- +4 + +ure 1 shows the distribution in data domain of the +Cantonese training dataset, which contains only +monolingual data sources. +4.1.1 +Monolingual Data +Cantonese Wikipedia +The largest source of data +available was Cantonese Wikipedia, which was +downloaded from Wikimedia dump2, then pure +text data is obtained with WikiExtractor (Attardi, +2015). Cantonese Wikipedia amounts to 690k lines +of text, making up 70% of the Cantonese corpus +overall. +Corpus +As mentioned, there is a small number +of open source Cantonese corpora collected for aca- +demic purposes, mainly transcribed from spoken +Cantonese. Additionally, there is another corpus +which contains scraped text data. Existing corpora +add up to 95k lines of Cantonese text, with the ma- +jority coming from Openrice restraurant reviews +(78k). +• openrice-senti3: scraped restaurant reviews +from popular Hong Kong website OpenRice +(https://www.openrice.com/zh/ +hongkong). +• HK Cantonese Corpus4 (Wong et al., 2017): +manually +transcribed +oral +conversations +recorded between 1997-1998, includes spon- +taneous speech as well as radio programmes. +• tatoeba5: a website which contains crowd- +sourced sentences and their translations in +many languages, including Cantonese. +Instagram +Due to its popularity in Hong Kong, +the domains from Instagram can be varied, ranging +from blogs, advertisements, news and governmen- +tal organisations. We scrape posts and comments +via imginn.org from 14 accounts, 5 of which +are categorised as news, the others are categorised +as non-news. Instagram comments make up the +second largest source of Cantonese data with 108k +lines (11%), while Instagram news are 58k lines +and Instagram non-news 30k lines. +Subtitles +Cantonese YouTube6 +is a crowd- +sourced compilation of youtube videos with spo- +ken Cantonese subtitles. It is a voluntary effort +2https://dumps.wikimedia.org/zh_yuewiki/20220601 +3https://github.com/toastynews/openrice-senti +4https://github.com/fcbond/hkcancor +5https://tatoeba.org/en +6https://docs.google.com/spreadsheets/d/1CmN8GPalrb4 +5YFIPrWgh7GRYyoUhnizEOImY6kAW82w +Figure 1: Distribution of data domain in the Cantonese +training set (monolingual data only). +from Cantonese learners, and each video is manu- +ally tagged with “Written Cantonese" or “Standard +Written Chinese", which allows us to filter for only +Cantonese videos. We are able to scrape directly +from Youtube with the help of the Youtube Tran- +script API7. There are 1,620 lines. +Articles +We scrape blog articles written by vari- +ous authors in Cantonese from the freelancer plat- +form https://handstopmouthstop.com. +There are 6,531 lines from the website. +4.1.2 +Parallel Data +As the experiments described in the future sections +are unsupervised, parallel data is not included in +the training set. They are only used for the test set. +Corpus +Cantonese-HK and Chinese-HK Uni- +versal Dependencies Treebank8(Luke and Wong, +2015): manually transcribed and annotated film +subtitles and legislative proceedings of Hong Kong, +in both Cantonese and Mandarin. There are 1,004 +parallel sentences from this corpus. +Dictionary +Kaifangcidian9 +is +an +online +Cantonese-Chinese dictionary which comes with +parallel sentences for each lexical entry. There are +13,004 parallel sentences from the dictionary. +Subtitles +Kongjisubtitles 10 is a Cantonese sub- +title team that specialises in “kongji"(meaning +“Hong Kong words" in romanized Cantonese) and +focuses on subtitling Thai online series. Since +7https://github.com/jdepoix/youtube-transcript-api +8https://github.com/UniversalDependencies/UD_Cantonese- +HK +9https://kaifangcidian.com/han/yue/ +10https://sites.google.com/view/lihkg-kongjisubtitles +5 + +instagram comments +restaurantreviews +11% +8% +instagram news +6% +instagram non-news +3% +2% +corpus +1% +subtitles & articles +70% +wikipediasome of the same videos also have Mandarin subti- +tles, we align them based on the timestamps of the +videos. This amounts to 77,479 lines of parallel +data. +4.2 +Pre-processing +Our data is scraped from different resources and +inevitably contains noise. The following tools are +leveraged for the pre-processing of collected data: +Sentence Cutter +Sentence cutter cuts each text +into sentences. The cutting points are punctuation +marks such as 。.!? that defines the end of a sen- +tence. +Mandarin-Cantonese Filter +Due to the fact that +most Cantonese speakers are also native in Man- +darin, Mandarin text is normally present in Can- +tonese data scraped from social media. Mandarin- +Cantonese Filter aims to determine whether a sen- +tence is written in Mandarin or Cantonese by calcu- +lating the number of language-specific characters. +This tool is involved only in the pre-processing of +Cantonese data. +Cantonese-specific characters are: , 唔, 係, , 啦, +, 既, 咁, 佢, , 冇, 仲, , 乜, 噉, 咪, 咩, 俾, 呢, , 黎, , +喂, 喇, 喎, 睇 +Mandarin-specific characters are: 是, 的, 他, 她, +沒, 也, 看, 說, 在,说 +Foreign Text Filter +Text written in foreign lan- +guages such as Russian, Japanese and Korean +abounds in collected data. +Foreign Text Filter +serves to filter out all sentences that are not writ- +ten in Chinese characters. If the Chinese charac- +ters contributes to less than 5% of sentence’s total +length, the sentence is removed. +url, emoji, hashtag Remover +This tool serves +to remove url, emoji, and hashtag from sentence +using regular expression. +Jieba Tokenizer +Jieba 11 is a Mandarin NLP li- +brary. In our project, we used Jieba tokenizer to +pre-process our Mandarin data. +PyCantonese Tokenizer +PyCantonese 12 is a +Cantonese NLP library. In our project, we used Py- +Cantonese tokenizer to pre-process our Cantonese +data. +We did not include any Mandarin data from so- +cial media in our dataset, considering that data +11https://github.com/fxsjy/jieba +12https://pycantonese.org/ +(a) Mandarin corpus +(b) Cantonese corpus +Figure 2: Distribution of sentence length. +scraped from social media is always full of noises +and Mandarin data from Wikipedia is already abun- +dant for our task. We included Cantonese data +scraped from social media since Cantonese data +from Wikipedia is not sufficient. +4.2.1 +Overall Data Statistics +After pre-processing, there are 912,258 lines of +monolingual Cantonese data and 16M lines of +monolingual Mandarin data. In terms of domains, +the Cantonese corpus has 70% data from Wikipedia +while the Mandarin corpus is 100% Wikipedia. Fig- +ure 2 shows that the distribution of sentence length +in Cantonese and Mandarin are broadly similar af- +ter pre-processing. +5 +Methodology +As shown in Figure 3, we follow a standard un- +supervised machine translation architecture with +a shared encoder and language-specific decoders +in our experiment. Models are trained on a de- +6 + +1e6 +1.4 +1.2 +1.0 +frequency +0.8 +0.6 +0.4 +0.2 +0.0 +0 +5 +10 +15 +20 +25 +30 +sentencelength(punctuationincluded)60000 +50000 +40000 +frequency +30000 +20000 +10000 +0 +0 +5 +10 +15 +20 +25 +30 +sentencelength(punctuationincluded)Figure 3: General architecture of the unsupervised machine translation systems in this experiment. A shared +encoder maps sentences from L1/L2 to a common latent space, then a language-specific decoder reconstructs the +encoded sentence back into its own language space. The model is trained by a denoising auto-encoding task and a +back-translation task. +noising auto-encoding task and an on-the-fly back- +translation task. To have an overall study of how +different setups affect the model performance, we +make three sets of comparisons: +1. Model architectures. +2. Cross-lingual embeddings. +3. Tokenization methods. +5.1 +Model Architectures +In this experiment, we compare an RNN-based +attention model and a transformer model. +• RNN-based model: We adopt the architecture +from (Artetxe et al., 2017b): Both encoder +and decoder have 2-layer bidirectional GRU +(Cho et al., 2014), Luong’s attention (Luong +et al., 2015) is applied to align the source sen- +tence and translation. Input sentences are con- +verted to 512-dimensional cross-lingual em- +beddings. Considering the relatively lower ca- +pacity, the cross-lingual embeddings are fixed +during training. +• Transformer model: Following (Lample et al., +2018), we use 4-layer encoder and decoder +with 3-layer sharing parameters for both Can- +tonese and Mandarin sides. +When gener- +ating translations, the decoder starts with a +language-specific token, specifying +the language it is operating with. The embed- +ding matrices are trainable during the training +process. +5.2 +Cross-lingual Embeddings +Cross-lingual embeddings can be learned in various +different ways. In our experiments we compare the +following three approaches: +• Mapping: It has been extensively studied how +to map monolingual word embeddings into +a cross-lingual space.(Mikolov et al., 2013; +Artetxe et al., 2016, 2017a, 2018a,b; Conneau +et al., 2017) In this project, we use Vecmap +13 by Artexte to obtain cross-lingual embed- +dings from monolingual ones. In particular, +we adopt the “identical” setting, where the +shared vocabulary in two languages can be +used as anchors to learn the mapping. This +approach is applied to RNN-based models. +• Learning from concatenated data: Another +setup is to learn embeddings on the concatena- +tion of source and target corpora in a monolin- +gual way. As embeddings are learned in the +context of both languages, the resultant em- +beddings can be seen as cross-lingual. This +approach is applied on both RNN-based mod- +els and transformer models. +• Pivot-private embeddings: We also experi- +ment with 512-dimensional pivot-private em- +beddings which consists of a 256-dimensional +cross-lingual embedding learned on the con- +catenated dataset and a 256-dimensional pri- +vate embedding, which is learned on two +monolingual datasets separately. +This ap- +proach is assumed to be able to capture the +commonality between both languages and pre- +serve language-specific characteristics as well +(Wan et al., 2020). We adopt this approach on +transformer models. +5.3 +Tokenization Methods +We are also interested whether byte-pair encod- +ing helps training Cantonese-Mandarin translation +systems, so we compare it to a character-level tok- +enization method. +13https://github.com/artetxem/vecmap +7 + +L1 decoder +L1output +Sharedencoder(L1/L2) +L2 decoder +L2output +Cross-lingual +embeddings +L1/L2 input• Word-level tokenization: As a baseline, we do +no further tokenization on the collected data +which is separated by words using Jieba and +PyCantonese. In this setting, a total number +of 80K/1M unique words are present in the +Cantonese/Mandarin corpora respectively. +• Character-level tokenization: Since Mandarin +and Cantonese are both analytic languages, +character-level tokenization is a valid option +to tokenize sentences. This results in 8K/14K +unique tokens in Cantonese/Mandarin training +data respectively. +• Byte-pair encoding: We also use byte-pair +encoding to obtain a vocabulary of 50K sub- +words on word-tokenized datasets. The em- +beddings of sub-words are learned using meth- +ods described above. +6 +Experiments and Results +In this section, we describe the experiments we +conducted and the results of both automatic and hu- +man evaluation. Our code and relevant repositories +are publicly available online 14. +6.1 +Task Setup +6.1.1 +Baseline Model +Due to the large overlap in vocabulary between +Mandarin and Cantonese and the lack of compli- +cated morphology in both languages, for our base- +line model we take advantage of these character- +istics by evaluating Mandarin sentences as if they +were a translation into Cantonese, and visa-versa. +This method is carried out by simply converting +both Mandarin and Cantonese evaluation datasets +to the same character set using OpenCC 15 (our +experiments used the Traditional Chinese (Hong +Kong variant) character set) and evaluating the +BLEU score directly. +6.1.2 +RNN-based Experiments +In order to improve upon the baseline model perfor- +mance, we train several models using Artetxe’s +RNN+Attention-based architecture for unsuper- +vised machine translation 16. The primary objec- +tive, aside from improving BLEU scores over the +baseline, is to identify which settings (e.g. tok- +enization scheme and embedding training method) +14https://github.com/meganndare/cantonese-nlp +15https://github.com/BYVoid/OpenCC +16https://github.com/artetxem/undreamt +lead to the best model performance. As detailed in +the methodology section we experiment with word, +character, and byte-pair encoding (BPE) tokeniza- +tion, as well as cross-lingual embeddings obtained +by learning a mapping into cross-lingual space, and +by concatenation and training a skip-gram model. +Additionally, for the BPE-tokenized models we +have experimented with learning the BPE tokens +separately for each language, or jointly. +6.1.3 +Balanced Dataset Experiments +One characteristic of our full training dataset is that +it is imbalanced (1 million Cantonese sentences +versus 16 million Mandarin sentences). This is +due to the abundance of Mandarin text data and +the scarcity of Cantonese text data available. As +a result, we were curious to understand whether +having an imbalanced dataset negatively affects +our training results. To this end we conducted an +experiment using what we refer to as our ’Balanced +Dataset’. To create the set, Mandarin sentences are +chosen at random to be removed from the training +set until a downsampled version of approximately +the same size as the Cantonese training set was ob- +tained, that also preserves the sentence length dis- +tribution of the original Mandarin training set. We +then compare the performance of models trained +using the balanced dataset to those trained using the +full set, utilizing some simple baseline settings for +comparison, namely word and character-tokenized +models. +6.1.4 +Transformer Experiments +Guided by advancements in neural network model +architectures over the past several years, we are +interested in how using a transformer architecture +would impact our results. For the transformer ex- +periments we leveraged Facebook Research’s Un- +supervised Neural Machine Translation Model 17 +for training. Using the results from our RNN-based +models, we primarily focused on character and +BPE tokenization schemes, and have also experi- +mented with a more complex cross-lingual embed- +ding type called pivot-private embeddings. Due to +differences in implementation between the RNN +and Transformer-based models, we were unable +to train Vecmap embeddings for this set of experi- +ments. +17https://github.com/facebookresearch/UnsupervisedMT +8 + +Model Name +Can>Man Char BLEU +Man>Can Char BLEU +Baseline (Character Conversion) Model +13.3 +13.2 +RNN (Word Tok + Vecmap Embed) +13.1 +14.9 +RNN (Char Tok + Vecmap Embed) +19.8 +22.5 +RNN (Char Tok + Concat Embed) +19.4 +20.3 +RNN (BPE Tok learned separately + Vecmap Embed) +18.0 +18.8 +RNN (BPE Tok learned jointly + Vecmap Embed) +19.3 +19.5 +RNN (Balanced Dataset + Word Tok + Vecmap Embed) +6.2 +11.5 +RNN (Balanced Dataset + Char Tok + Vecmap Embed) +17.1 +20.4 +Transformer (Char Tok + Concat Embed)** +24.4 +25.1 +Transformer (Char Tok + Pivot-Private Embed) +21.2 +20.5 +Transformer (BPE Tok learned jointly + Concat Embed) +20.2 +17.4 +Table 2: Overview of all automatic evaluation results. All BLEU (Bilingual Evaluation Understudy) metric +scores are calculated at the character-level. Best-performing model indicated by **. +6.2 +Results +6.2.1 +Automatic Evaluation +Model Architectures +The first metric that our +study sought to investigate was the varying per- +formances of Mandarin-Cantonese unsupervised +machine translation based on the underlying neu- +ral network architecture, namely an RNN-based +architecture versus a Transformer architecture. We +observed that the transformer model led to higher +BLEU scores when other factors are constant. This +can be observed in the RNN (Char Tok + Con- +cat Embed) versus Transformer (Char Tok + Con- +cat Embed) models, where Cantonese-to-Mandarin +translation yielded 19.4 versus 24.4, respectively; +and Mandarin-to-Cantonese yielded 20.3 versus +25.1, respectively. In fact, our highest performing +model from the study was trained on a Transformer +architecture. +Cross-lingual +Embeddings +The +study +also +makes comparisons between different types of +cross-lingual embeddings. +Of primary interest +are training monolingual embeddings and map- +ping them to a shared cross-lingual space using +Vecmap (as detailed in the Methodology section), +and learning embeddings from the concatenated +data. In a comparison between RNN (Char Tok ++ Vecmap Embed) and RNN (Char Tok + Con- +cat Embed) models, we can see that the mapping- +based cross-lingual embeddings have outperformed +the concatenation-based technique, yielding a +Cantonese-to-Mandarin BLEU of 19.8 and 19.4, +respectively; and a Mandarin-to-Cantonese BLEU +of 22.5 and 20.3, respectively. +In addition to mapping-based and concatenation- +based cross-lingual embeddings, we also had time +to run one experiment on pivot-private embeddings +(as detailed in the Methodology section). By com- +paring the Transformer (Char Tok + Concat Em- +bed) and Transformer (Char Tok + Pivot-Private +Embed) models, we observe that concatenation- +based embeddings outperform pivot-private em- +beddings, with a Cantonese-to-Mandarin BLEU +of 24.4 versus 21.2, and a Mandarin-to-Cantonese +BLEU of 25.1 to 20.5, respectively. +Tokenization Methods +Our study additionally +makes a comparison between different types of +tokenization methods: word, character, and BPE- +tokenized models. Word-tokenization always per- +forms the worst, in all cases aside from one (see +RNN (Word Tok + Vecmap Embed) Mandarin-to- +Cantonese results in Table 2), models trained with +word-tokenized training data did not outperform +even the Baseline (Character Conversion) Model +in which no neural network was trained. +While BPE-tokenized data tends to perform very +well for languages with an alphabet system, such +as French or English, we did not observe a such +a strong result in the models trained using BPE- +tokenized data for the Mandarin-Cantonese lan- +guage pair. We experimented by learning BPE +token vocabularies both separately and jointly, ob- +serving a slight performance improvement when +learned jointly. However, neither BPE setting could +outperform our character-tokenized models (see Ta- +ble 2 for two results that lead to this conclusion: +RNN (Char Tok + Vecmap Embed) versus RNN +9 + +(BPE Tok learned jointly + Vecmap Embed), as well +as Transformer (Char Tok + Concat Embed) versus +Transformer (BPE Tok learned jointly + Concat +Embed)). +Balanced Dataset +We conclude that neither +word nor character-tokenized models trained on +the balanced dataset outperformed models trained +using the full training dataset. Thus, it is advanta- +geous to use as much data as possible for model +training, even if the two languages have an uneven +amount of sentences. +6.2.2 +Human Evaluation +We conduct human evaluation on the Transformer +(Char Tok + Concat Embed) model output in order +to assess the extent to which our translation system +would be useful to Cantonese and Mandarin speak- +ers respectively. Considering that Cantonese speak- +ers can understand Standard Chinese, a translation +system from Mandarin to Cantonese should aim +for localisation and fluency in Cantonese, while not +losing the original meaning of the sentence. On the +other hand, the primary purpose of a Cantonese- +to-Mandarin translation system is to facilitate Can- +tonese comprehension for Mandarin speakers. For +these diverging purposes in our translation direc- +tions, we manually evaluate each translation direc- +tion with separate criteria, which is explained in +the following sections. +Procedure +100 lines from the test set are selected +for evaluation, identical for both translation di- +rections. One native speaker of each target lan- +guage evaluates for that direction only (i.e. Can- +tonese speaker evaluates Mandarin to Cantonese +sentences, and visa-versa). During evaluation, the +evaluator has access to the original input and the +target output. The evaluation decision is binary for +both criteria, the evaluator can only choose either +YES or NO. In the example sentences below, Man- +darin features are highlighted in orange, Cantonese +features are highlighted in teal and ungrammatical +features are highlighted in red. +Cantonese to Mandarin +System outputs are +evaluated against the criteria concerning whether +the output helps Mandarin speakers understand +Cantonese text. 34% were found helpful for un- +derstanding Cantonese text, 61% were found not +helpful, 5% sentences are discarded because the +original text in Cantonese is already perfectly com- +prehensible for Mandarin speaker. +Mandarin to Cantonese +System outputs are +evaluated against the criteria “Does the system out- +put contribute to Cantonese fluency / localisation?". +It is found to be the case for 47% of the sentences, +false for 52% of the sentences with 1%sentences +discarded since the input and target were identical. +(1)-(4) are examples of the system output for the +Mandarin to Cantonese direction. In (1), the out- +put is evaluated as helpful even though it has not +completely transformed all Mandarin features into +Cantonese ones, however, the components with the +highest semantic value (拍拖dating and 散break +up) are in Cantonese where it was originally in +Mandarin. Compared to (3), where the output still +retains mostly Mandarin and has no Cantonese fea- +tures. Comparing (2) and (4), they both have some +grammatical errors (in red), but the impact of such +error in (2) is less significant to the overall meaning +of the sentence, while in (4) the overall sentence is +incomprehensible. +Examples of output that is helpful: +(1) +Mandarin reference (source): +身邊有兩位好朋友，交往了三年， +就那樣分手了。 +Cantonese reference (target): +身邊有兩位好友，拍三年拖，就噉散 +。 +System output: +身邊有兩位好友，拍了三年拖，就這 +樣散了。 +Sentence meaning: I have two friends +who had been dating for three years, +and they broke up just like that. +(2) +Mandarin reference (source): +別這麼犟，快點向媽認錯。 +Cantonese reference (target): +咪咁硬頸，快同亞媽認錯。 +System output: +否“硬頸，快些和亞媽認錯。 +Sentence meaning: Don’t be so stubborn, +apologize to your mother at once. +Examples of output that is not helpful: +(3) +Mandarin reference (source): +別小看他，他已經有了三項發明。 +Cantonese reference (target): +10 + +咪睇小佢，佢已經有三項發明。 +System output: +否看小她，她已經有了三項發明。 +Sentence meaning: Don’t underestimate +him, he already has three inventions. +(4) +Mandarin reference (source): +給海關沒收了那些東西。 +Cantonese reference (target): +畀海關執。 +System output: +給海關執了那麼。 +Sentence meaning: The things that were +confiscated by customs. +7 +Discussion +Our Mandarin-Cantonese machine translation +project displays the differences between two to- +kenization methods (character-level and byte pair +encoding), with an outcome different than expected +regarding byte pair encoding. A possible reason +for this may be that such a big vocabulary size can +lead to worse embeddings, taking into account the +size of our corpus. +One of our approaches was down-sampling the +full dataset into a balanced one, from which we +expected a higher BLEU score compared to when +using the full dataset. However, this had the op- +posite effect on the BLEU score and it ended up +being lower than in the previous occasions. This +is perhaps due to the fact that 1 million sentences +is just simply not enough data for a machine to +become ’fluent’ in a language. +As further work, we propose that this project +can be extended by combining out best architec- +ture, best tokenization and best embedding training +method (transformer + character + mapping), by de- +veloping a cross-lingual mapping for embeddings +that is compatible with a transformer network in +order to confirm whether it does lead to higher +results. +In addition, other options worth exploring would +be the grammatical similarity between Cantonese +and Mandarin and developing an statistical ma- +chine translation model. +8 +Summary and conclusion +The aim of implementing a Cantonese-Mandarin +MT-model was accomplished by: +• Creating a large-scale corpus out of several +online sources such as Wikipedia, scraped +Instagram comments, YouTube subtitles and +restaurant reviews. +• Implementing and training several Cantonese- +Mandarin translation models while studying +the effects of different tokenization strategies, +such as character-level and byte-pair encod- +ing. While BPE was expected to outperform +character-level tokenization, this was not the +case in our experiments. +The outcomes of this project showed that overall, +in 61% of the cases, the outcome translation was +not useful to help Mandarin speakers understand +Cantonese text. As far as what fluency concerns, +in 52 out of 100 cases, the system’s output did not +show any contribution. +Further work and research is essential in order to +reach good percentages of performance and fluency +in such a machine translation model. This project +has contributed a large Cantonese dataset that was +not available before as it is now. +We hope that with this project we moved one +step forward into a direction that has been studied +for some years now, contributing to further devel- +opments and advancement. +References +Mikel Artetxe, Gorka Labaka, and Eneko Agirre. 2016. +Learning principled bilingual mappings of word em- +beddings while preserving monolingual invariance. +In Proceedings of the 2016 conference on empiri- +cal methods in natural language processing, pages +2289–2294. +Mikel Artetxe, Gorka Labaka, and Eneko Agirre. +2017a. Learning bilingual word embeddings with +(almost) no bilingual data. +In Proceedings of the +55th Annual Meeting of the Association for Compu- +tational Linguistics (Volume 1: Long Papers), pages +451–462. +Mikel Artetxe, Gorka Labaka, and Eneko Agirre. +2018a. 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In 2021 5th International Conference on +Natural Language Processing and Information Re- +trieval (NLPIR), pages 81–87. +12 + diff --git a/4NE2T4oBgHgl3EQfjwdR/content/tmp_files/load_file.txt b/4NE2T4oBgHgl3EQfjwdR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ab2084f9958b7b92a2b7e41b95ee9c431346cbd7 --- /dev/null +++ b/4NE2T4oBgHgl3EQfjwdR/content/tmp_files/load_file.txt @@ -0,0 +1,562 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf,len=561 +page_content='Unsupervised Mandarin-Cantonese Machine Translation Megan Dare, Valentina Fajardo Diaz, Averie Ho Zoen So, Yifan Wang, Shibingfeng Zhang Summer Semester Software Project 2022 Language Science and Technology, Saarland University {mdare,valenfd,averieso,yifwang,zhangshi@coli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='uni-saarland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='de} Abstract Advancements in unsupervised machine trans- lation have enabled the development of ma- chine translation systems that can translate be- tween languages for which there is not an abundance of parallel data available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' We ex- plored unsupervised machine translation be- tween Mandarin Chinese and Cantonese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' De- spite the vast number of native speakers of Cantonese, there is still no large-scale corpus for the language, due to the fact that Can- tonese is primarily used for oral communica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' The key contributions of our project include: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' The creation of a new corpus containing approximately 1 million Cantonese sentences, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' A large-scale compari- son across different model architectures, tok- enization schemes, and embedding structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Our best model trained with character-based tokenization and a Transformer architecture achieved a character-level BLEU of 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='1 when translating from Mandarin to Cantonese and of 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='4 when translating from Cantonese to Man- darin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' In this paper we discuss our research process, experiments, and results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 1 Introduction In recent years, neural machine translation has gained massive research interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Most of these studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Bahdanau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Luong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 2017) focus on the construction of neural machine translation systems leveraging parallel bilingual corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Nev- ertheless, such an approach is not feasible for many language pairs due to the scarcity of resources for such pairs, as is the case for Cantonese and Man- darin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' The study of automatic translation between these two languages faces the same problem: to the best of our knowledge, despite the vast number of native speakers of both languages, there is still no large-scale Mandarin-Cantonese parallel corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' In addition, monolingual corpora for Cantonese are hard to collect as it is a low-resource language that is mainly used for only oral communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Currently, only a few studies have been done on Cantonese-Mandarin translation, among which some compare various low-resource models for this language pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' However, these studies nor- mally focus on a comparison between one or two model types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Based on our motivation of imple- menting and training a Cantonese-Mandarin trans- lation model and current state of research, we set our goal as building a robust model trained on a more diverse dataset, which can help improve communication between Cantonese and Mandarin speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Additionally, we seek to compare vari- ous model architectures, tokenization schemes, and embedding structures to conduct a comprehensive understanding on which settings may lead to the best performance for the Cantonese-Mandarin lan- guage pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' After a close analysis of the current state of re- search and the available resources, we propose to develop a Cantonese-Mandarin machine translation system that is capable of conducting translation in both directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' The training of the system involves only Mandarin and Cantonese monolingual corpora collected from Wikipedia and various websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Our work also makes contributions to the Can- tonese language NLP field by collecting Cantonese textual data and building a public large-scale mono- lingual corpus, which did not exist until now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' In addition, considering the similarity between Cantonese and Mandarin, our translation system will provide a foundation for further development regarding machine translation tasks that center around language pairs composed of two similar languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 2 Background 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='1 Cantonese and Chinese: an overview Cantonese is one of the most widely spoken va- rieties of Chinese other than Mandarin Chinese (Matthews and Yip, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' It is estimated to have arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='03971v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='CL] 10 Jan 2023 more than 55 million native speakers, with large populations found in southern China provinces Guangdong and Guangxi, as well as regions includ- ing Hong Kong and Macau, it is also commonly spoken in overseas Cantonese communities in Sin- gapore, Malaysia, North America and Australia as a result of emigration (Matthews and Yip, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' While numerous NLP applications have been developed for Mandarin Chinese, little has been developed for Cantonese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' One reason for this is the limited linguistic resources that have been collected for Cantonese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Primarily a spoken language and a non-standard variety, written Cantonese is not tra- ditionally used or taught in schools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Instead, Can- tonese speakers typically learn to read and write in standard Chinese through education, so there is no language barrier for Cantonese speakers when interacting with computer applications designed in standard Chinese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' On the other hand, with the availability of the internet and the rise of social media, Cantonese is much more commonly used and written online in recent years, which can be seen as an indicator for a market in Cantonese NLP applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' It is important to note that this phenomenon might only be applicable to Hong Kong Cantonese, and not other variants such as the one in Guang- dong province.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' More recent discussions about Can- tonese, such as Bauer (2018), make a point to dis- tinguish between the Hong Kong Cantonese variant and the others, since the use of Cantonese is on the rise in Hong Kong, while declining in provinces within mainland China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Not only has this led to Hong Kong being named “the Cantonese-speaking capital of the world" (Bolton, 2011, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='64), but also the rise of written Cantonese locally and subse- quently, the Cantonese text data that are available online, which are of the Hong Kong variant of Can- tonese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='2 Linguistic Differences between Cantonese and Mandarin Despite the common misconception that Chinese dialects share the same grammar, Cantonese and Mandarin are different at phonological, lexical and syntactic levels, and are not mutually intelligible (Matthews and Yip, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Some suggests it is more accurate describe Cantonese as a distinct language of the Chinese language family (Snow, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' For the rest of this section, we describe some features that differ between Mandarin and Hong Kong Cantonese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='1 Writing Systems To anyone who can read Chinese, the most notable visual variation in written Chinese is the writing system - Traditional or Simplified Chinese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' The two systems are equivalent to each other, and have one-to-one correspondence for each character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' The following is some examples of traditional / sim- plified characters: “open" 開/开, “talk" 話/话 and “book" 書/书.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' The usage of either system is primar- ily due to regional difference, with mainland China using the simplified system, while Hong Kong and Taiwan use the traditional system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='2 Lexical and Syntactic comparisons Vocabulary difference is the main barrier which prevents Mandarin speakers from understanding Cantonese (Snow, 2004), it is also the aspect which is the most distinguishable between Cantonese and Mandarin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' According to Snow (2004), written Cantonese in formal domains can contain around 10-15% Cantonese-only characters, while this per- centage in informal domains can go up to 25-40%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Notably, the vocabulary that differ are some of the most frequent words, including many func- tion words, as seen in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Syntactically, Meaning Cantonese Mandarin possessive marker ge3 的de perfective marker zo2 了le pronoun pluralizer dei6 們mén negator 唔m4 不bù is (copula) 係hai6 是shì this 呢ne1 這zhè Table 1: Examples of lexical difference between Can- tonese and Mandarin from Snow (2004, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Can- tonese romanizations follow the Jyutping system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Cantonese and Mandarin are broadly similar but with some differences that are often overlooked (Matthews and Yip, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Some common differ- ences are in terms of word order, including indi- rect object and comparative constructions (Snow, 2004): Indirect object construction: Cantonese: 我俾錢佢ngo5 bei2 cin4 keoi5 (I + give + money + he) Mandarin: 我給他錢wó gˇei t¯a qían 2 (I + give + he + money) ‘I give him money’ Comparative construction: Cantonese: 我高過佢ngo5 gou1 gwo3 keoi5 (I + tall + more than + he) Mandarin: 我比他高wó bˇı t¯a g¯ao (I + compared to + he + tall) ‘I’m taller than him.’ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='3 Challenges Unique to Cantonese NLP Firstly, there exists a certain degree of variabil- ity in written Cantonese since it was never stan- dardised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' As such, some words can be written with completely different characters yet have the same meanings and pronunciations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' For example, “like" can be written as 中意or 鍾意(read: zung1 ji31), “still" can be written as 仲or 重(read: zung6) (Matthews and Yip, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Additionally, when some Cantonese words cannot be represented by existing Chinese characters, they could be written in a romanized form, such as the comparative (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' “-er" in “cheaper") can be written with “D", as well as a non-romanized form (read: di1) (Snow, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Matthews and Yip, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Secondly, code-switching to English is a com- mon phenomena in Cantonese, which is not a feature in standard Chinese or Mandarin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Code- switching in Hong Kong Cantonese is mostly in- trasentential (below clause level) (Li, 2000), for example: 我今朝9點有個meeting。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' ngo5 dei6 gam1 ziu1 gau2 dim2 jau5 go3 MEETING ‘We have a meeting at 9am today.’ 3 Related Work 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='1 Unsupervised Machine Translation Unsupervised machine translation with no parallel data is a challenging task that has attracted many interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' The presence of cross-lingual embed- dings (Mikolov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Artetxe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=', 2016, 2017a, 2018a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Conneau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=', 2017) provides prior knowledge for machine translation systems and makes it possible to train a machine transla- tion model in an unsupervised way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Artetxe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' (2017b) and Lample et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' (2017) are the first at- tempts to explore the possibility of constructing 1romanizations according to the Jyutping system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' a neural machine translation system using only monolingual corpora from both source and target languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' The proposed system is based on an encoder-decoder architecture with attention mecha- nism (Bahdanau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=', 2014), trained with a denois- ing auto-encoding task (Vincent et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=', 2008) and a back-translation task (Sennrich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' The encoder is shared by both the source and target lan- guages, so that sentences from both languages can be mapped to a common latent space, while each language has its own decoder to reconstruct en- coded sentences back into its own language space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Cross-lingual embeddings are leveraged as an ini- tialization for the system, providing additional lex- ical level information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Such a structural property allows the translation model to be bi-directional, that is, the same model can be employed in both the L1-to-L2 translation task and the L2-to-L1 transla- tion task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' This approach is extended in Lample et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' (2018) by applying a transformer model and using sub- word level tokenization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Attention-only structures provide higher model capacity, and sub- word level tokenization method Byte Pair Encod- ing (BPE) reduce the size of vocabulary and helps solving problems in translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Addition- ally, they re-exploited the potential of statistical approaches in unsupervised machine translation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' A phrase-based machine translation model initialized with an automatically populated phrase table and language model is trained by iterative back-translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Results of the experiment show that a statistical approach can reach similar perfor- mance or even outperform neural systems when the data is scarce, as the neural model tends to over- fit the corpora, and thus does not generalize well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Together with Singh and Singh (2020), they show that unsupervised approaches can be used to con- struct machine translation systems for low-source languages (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=', Urdu, Romanian, Manipuri).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' In recent years, pre-trained language models have become popular due to their competitive ability of representing and generating natural lan- guages learned from transfer learning on large- scale self-supervised datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Lample and Con- neau Lample and Conneau (2019) take their work one step further by pre-training both the encoder and decoder in their model using a cross-lingual language model (XLM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' They then fine-tune the pre-trained model to an unsupervised neural ma- chine translation model following the training pro- 3 cess described in Lample et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' The pre- training stage results in a sharp BLEU score in- crease over previous benchmarks for unsupervised machine translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Unsupervised machine translation methods are also applied in dialectal machine translation tasks, where the similarity and commonality between lan- guages can be leveraged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Farhan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' (2020) uses common words between Arabic dialects as anchor points to steer projections of surrounding words be- tween two dialects, creating a more accurate map- ping between source and target words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' In this way, they construct an unsupervised machine translation system with a BLEU score of 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='14, which is re- markably high compared with the highest BLEU score obtained in the supervised setting (48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='2 Mandarin-Cantonese Machine Translation Due to the scarcity of available datasets, Cantonese language is always under-researched in NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' This issue is even more severe in machine trans- lation tasks, which usually requires large amount of parallel data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' For this reason, many researches on Cantonese-Mandarin machine translation are intended to collect more data or to fully exploit the limited data in a semi-supervised or unsupervised way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Hei Yi Mak and Tan Lee (2021) construct a large-scale Cantonese-Mandarin parallel dataset by mining parallel sentences from Mandarin and Cantonese Wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' They apply a similarity- based sentence alignment approach and use sen- tence pairs with high confidence score as parallel sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' In this way, they end up with a paral- lel corpus of about 100,000 sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' They also fine-tune a pre-trained language model using the collected data and obtain a competitive translation system that outperforms Baidu Fanyi, a commonly used translator in China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Concurrently, some efforts have been made to create unsupervised Cantonese-Mandarin transla- tion systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' (Wan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=', 2020) handles Cantonese- Mandarin translation as a dialect translation prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' which attempts to exploit the commonality between two language dialects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' On the basis of (Lample et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=', 2018)’s transformer model, they make use of pivot-private embeddings and layer coordination to better utilize the similarity and dif- ference between the two languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Trained on two large monolingual datasets of 20 million collo- quial sentences for each Mandarin and Cantonese, their model reaches an improvement of up to 12 BLEU score for Cantonese to Mandarin, and 5 BLEU from Mandarin to Cantonese compared to their baseline transformer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' There have been other works relying on pre- trained cross-lingual language models (XLM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' In Wong and Tsai (2022), the authors initialize the encoder and decoder with XLM as described in (Lample and Conneau, 2019), while using pivot- private embeddings rather than cross-lingual em- beddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Using this enriched structure, they are able to achieve slight BLEU score improvements over previous XLM models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 4 Corpus Construction While existing Cantonese corpora are scarce, and usually collected for linguistic purposes which is smaller in scale and of a specific demographic (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Wong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Luke and Wong 2015), text data is available on the internet due to Cantonese being the common language used on social media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' This also led to a rise in Cantonese writing in tradition- ally more formal domains such as advertisements, online news and subtitles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Therefore, we aim for the corpus to span across various domains for a comprehensive collection of modern Cantonese usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Secondly, since standard Chinese is also commonly used among Cantonese speakers in online settings, in the data selection pro- cess, we aim to avoid sources which use standard Chinese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Lastly, in our pre-processing, we preserve some unique features in Cantonese such as code- switching in English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Detailed data statistics of the corpus is available on the Github repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' As we focus on collecting data for Cantonese, note that we simply use the Chinese Wikipedia for Mandarin data, since there is already a large amount of data available just from one source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='1 Data Collection The Cantonese data available from various sources on the internet are either readily downloadable (for Wikipedia, corpus and dictionary) or are scraped by us (for Instagram, subtitles and articles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Due to structural differences in the various websites, scrap- ing functions are individually written for each of the three classes of sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' In general, the script moves recursively over the website domain and extracts any text in each web page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' The scraping script is available on our GitHub repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Fig- 4 ure 1 shows the distribution in data domain of the Cantonese training dataset, which contains only monolingual data sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='1 Monolingual Data Cantonese Wikipedia The largest source of data available was Cantonese Wikipedia, which was downloaded from Wikimedia dump2, then pure text data is obtained with WikiExtractor (Attardi, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Cantonese Wikipedia amounts to 690k lines of text, making up 70% of the Cantonese corpus overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Corpus As mentioned, there is a small number of open source Cantonese corpora collected for aca- demic purposes, mainly transcribed from spoken Cantonese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Additionally, there is another corpus which contains scraped text data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Existing corpora add up to 95k lines of Cantonese text, with the ma- jority coming from Openrice restraurant reviews (78k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' openrice-senti3: scraped restaurant reviews from popular Hong Kong website OpenRice (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='openrice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='com/zh/ hongkong).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' HK Cantonese Corpus4 (Wong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=', 2017): manually transcribed oral conversations recorded between 1997-1998, includes spon- taneous speech as well as radio programmes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' tatoeba5: a website which contains crowd- sourced sentences and their translations in many languages, including Cantonese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Instagram Due to its popularity in Hong Kong, the domains from Instagram can be varied, ranging from blogs, advertisements, news and governmen- tal organisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' We scrape posts and comments via imginn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='org from 14 accounts, 5 of which are categorised as news, the others are categorised as non-news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Instagram comments make up the second largest source of Cantonese data with 108k lines (11%), while Instagram news are 58k lines and Instagram non-news 30k lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Subtitles Cantonese YouTube6 is a crowd- sourced compilation of youtube videos with spo- ken Cantonese subtitles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' It is a voluntary effort 2https://dumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='wikimedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='org/zh_yuewiki/20220601 3https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='com/toastynews/openrice-senti 4https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='com/fcbond/hkcancor 5https://tatoeba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='org/en 6https://docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='com/spreadsheets/d/1CmN8GPalrb4 5YFIPrWgh7GRYyoUhnizEOImY6kAW82w Figure 1: Distribution of data domain in the Cantonese training set (monolingual data only).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' from Cantonese learners, and each video is manu- ally tagged with “Written Cantonese" or “Standard Written Chinese", which allows us to filter for only Cantonese videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' We are able to scrape directly from Youtube with the help of the Youtube Tran- script API7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' There are 1,620 lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Articles We scrape blog articles written by vari- ous authors in Cantonese from the freelancer plat- form https://handstopmouthstop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' There are 6,531 lines from the website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='2 Parallel Data As the experiments described in the future sections are unsupervised, parallel data is not included in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' They are only used for the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Corpus Cantonese-HK and Chinese-HK Uni- versal Dependencies Treebank8(Luke and Wong, 2015): manually transcribed and annotated film subtitles and legislative proceedings of Hong Kong, in both Cantonese and Mandarin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' There are 1,004 parallel sentences from this corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Dictionary Kaifangcidian9 is an online Cantonese-Chinese dictionary which comes with parallel sentences for each lexical entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' There are 13,004 parallel sentences from the dictionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Subtitles Kongjisubtitles 10 is a Cantonese sub- title team that specialises in “kongji"(meaning “Hong Kong words" in romanized Cantonese) and focuses on subtitling Thai online series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Since 7https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='com/jdepoix/youtube-transcript-api 8https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='com/UniversalDependencies/UD_Cantonese- HK 9https://kaifangcidian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='com/han/yue/ 10https://sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='com/view/lihkg-kongjisubtitles 5 instagram comments restaurantreviews 11% 8% instagram news 6% instagram non-news 3% 2% corpus 1% subtitles & articles 70% wikipediasome of the same videos also have Mandarin subti- tles, we align them based on the timestamps of the videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' This amounts to 77,479 lines of parallel data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='2 Pre-processing Our data is scraped from different resources and inevitably contains noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' The following tools are leveraged for the pre-processing of collected data: Sentence Cutter Sentence cutter cuts each text into sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' The cutting points are punctuation marks such as 。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='.!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' that defines the end of a sen- tence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Mandarin-Cantonese Filter Due to the fact that most Cantonese speakers are also native in Man- darin, Mandarin text is normally present in Can- tonese data scraped from social media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Mandarin- Cantonese Filter aims to determine whether a sen- tence is written in Mandarin or Cantonese by calcu- lating the number of language-specific characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' This tool is involved only in the pre-processing of Cantonese data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Cantonese-specific characters are: , 唔, 係, , 啦, , 既, 咁, 佢, , 冇, 仲, , 乜, 噉, 咪, 咩, 俾, 呢, , 黎, , 喂, 喇, 喎, 睇 Mandarin-specific characters are: 是, 的, 他, 她, 沒, 也, 看, 說, 在,说 Foreign Text Filter Text written in foreign lan- guages such as Russian, Japanese and Korean abounds in collected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Foreign Text Filter serves to filter out all sentences that are not writ- ten in Chinese characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' If the Chinese charac- ters contributes to less than 5% of sentence’s total length, the sentence is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' url, emoji, hashtag Remover This tool serves to remove url, emoji, and hashtag from sentence using regular expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Jieba Tokenizer Jieba 11 is a Mandarin NLP li- brary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' In our project, we used Jieba tokenizer to pre-process our Mandarin data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' PyCantonese Tokenizer PyCantonese 12 is a Cantonese NLP library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' In our project, we used Py- Cantonese tokenizer to pre-process our Cantonese data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' We did not include any Mandarin data from so- cial media in our dataset, considering that data 11https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='com/fxsjy/jieba 12https://pycantonese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='org/ (a) Mandarin corpus (b) Cantonese corpus Figure 2: Distribution of sentence length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' scraped from social media is always full of noises and Mandarin data from Wikipedia is already abun- dant for our task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' We included Cantonese data scraped from social media since Cantonese data from Wikipedia is not sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='1 Overall Data Statistics After pre-processing, there are 912,258 lines of monolingual Cantonese data and 16M lines of monolingual Mandarin data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' In terms of domains, the Cantonese corpus has 70% data from Wikipedia while the Mandarin corpus is 100% Wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Fig- ure 2 shows that the distribution of sentence length in Cantonese and Mandarin are broadly similar af- ter pre-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 5 Methodology As shown in Figure 3, we follow a standard un- supervised machine translation architecture with a shared encoder and language-specific decoders in our experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Models are trained on a de- 6 1e6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='0 frequency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='0 0 5 10 15 20 25 30 sentencelength(punctuationincluded)60000 50000 40000 frequency 30000 20000 10000 0 0 5 10 15 20 25 30 sentencelength(punctuationincluded)Figure 3: General architecture of the unsupervised machine translation systems in this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' A shared encoder maps sentences from L1/L2 to a common latent space, then a language-specific decoder reconstructs the encoded sentence back into its own language space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' The model is trained by a denoising auto-encoding task and a back-translation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' noising auto-encoding task and an on-the-fly back- translation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' To have an overall study of how different setups affect the model performance, we make three sets of comparisons: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Model architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Cross-lingual embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Tokenization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='1 Model Architectures In this experiment, we compare an RNN-based attention model and a transformer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' RNN-based model: We adopt the architecture from (Artetxe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=', 2017b): Both encoder and decoder have 2-layer bidirectional GRU (Cho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=', 2014), Luong’s attention (Luong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=', 2015) is applied to align the source sen- tence and translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Input sentences are con- verted to 512-dimensional cross-lingual em- beddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Considering the relatively lower ca- pacity, the cross-lingual embeddings are fixed during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Transformer model: Following (Lample et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=', 2018), we use 4-layer encoder and decoder with 3-layer sharing parameters for both Can- tonese and Mandarin sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' When gener- ating translations, the decoder starts with a language-specific token, specifying the language it is operating with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' The embed- ding matrices are trainable during the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='2 Cross-lingual Embeddings Cross-lingual embeddings can be learned in various different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' In our experiments we compare the following three approaches: Mapping: It has been extensively studied how to map monolingual word embeddings into a cross-lingual space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' (Mikolov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Artetxe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=', 2016, 2017a, 2018a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Conneau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=', 2017) In this project, we use Vecmap 13 by Artexte to obtain cross-lingual embed- dings from monolingual ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' In particular, we adopt the “identical” setting, where the shared vocabulary in two languages can be used as anchors to learn the mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' This approach is applied to RNN-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Learning from concatenated data: Another setup is to learn embeddings on the concatena- tion of source and target corpora in a monolin- gual way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' As embeddings are learned in the context of both languages, the resultant em- beddings can be seen as cross-lingual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' This approach is applied on both RNN-based mod- els and transformer models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Pivot-private embeddings: We also experi- ment with 512-dimensional pivot-private em- beddings which consists of a 256-dimensional cross-lingual embedding learned on the con- catenated dataset and a 256-dimensional pri- vate embedding, which is learned on two monolingual datasets separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' This ap- proach is assumed to be able to capture the commonality between both languages and pre- serve language-specific characteristics as well (Wan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' We adopt this approach on transformer models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='3 Tokenization Methods We are also interested whether byte-pair encod- ing helps training Cantonese-Mandarin translation systems, so we compare it to a character-level tok- enization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 13https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='com/artetxem/vecmap 7 L1 decoder L1output Sharedencoder(L1/L2) L2 decoder L2output Cross-lingual embeddings L1/L2 input• Word-level tokenization: As a baseline, we do no further tokenization on the collected data which is separated by words using Jieba and PyCantonese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' In this setting, a total number of 80K/1M unique words are present in the Cantonese/Mandarin corpora respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Character-level tokenization: Since Mandarin and Cantonese are both analytic languages, character-level tokenization is a valid option to tokenize sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' This results in 8K/14K unique tokens in Cantonese/Mandarin training data respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Byte-pair encoding: We also use byte-pair encoding to obtain a vocabulary of 50K sub- words on word-tokenized datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' The em- beddings of sub-words are learned using meth- ods described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 6 Experiments and Results In this section, we describe the experiments we conducted and the results of both automatic and hu- man evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Our code and relevant repositories are publicly available online 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='1 Task Setup 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='1 Baseline Model Due to the large overlap in vocabulary between Mandarin and Cantonese and the lack of compli- cated morphology in both languages, for our base- line model we take advantage of these character- istics by evaluating Mandarin sentences as if they were a translation into Cantonese, and visa-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' This method is carried out by simply converting both Mandarin and Cantonese evaluation datasets to the same character set using OpenCC 15 (our experiments used the Traditional Chinese (Hong Kong variant) character set) and evaluating the BLEU score directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='2 RNN-based Experiments In order to improve upon the baseline model perfor- mance, we train several models using Artetxe’s RNN+Attention-based architecture for unsuper- vised machine translation 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' The primary objec- tive, aside from improving BLEU scores over the baseline, is to identify which settings (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' tok- enization scheme and embedding training method) 14https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='com/meganndare/cantonese-nlp 15https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='com/BYVoid/OpenCC 16https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='com/artetxem/undreamt lead to the best model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' As detailed in the methodology section we experiment with word, character, and byte-pair encoding (BPE) tokeniza- tion, as well as cross-lingual embeddings obtained by learning a mapping into cross-lingual space, and by concatenation and training a skip-gram model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Additionally, for the BPE-tokenized models we have experimented with learning the BPE tokens separately for each language, or jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='3 Balanced Dataset Experiments One characteristic of our full training dataset is that it is imbalanced (1 million Cantonese sentences versus 16 million Mandarin sentences).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' This is due to the abundance of Mandarin text data and the scarcity of Cantonese text data available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' As a result, we were curious to understand whether having an imbalanced dataset negatively affects our training results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' To this end we conducted an experiment using what we refer to as our ’Balanced Dataset’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' To create the set, Mandarin sentences are chosen at random to be removed from the training set until a downsampled version of approximately the same size as the Cantonese training set was ob- tained, that also preserves the sentence length dis- tribution of the original Mandarin training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' We then compare the performance of models trained using the balanced dataset to those trained using the full set, utilizing some simple baseline settings for comparison, namely word and character-tokenized models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='4 Transformer Experiments Guided by advancements in neural network model architectures over the past several years, we are interested in how using a transformer architecture would impact our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' For the transformer ex- periments we leveraged Facebook Research’s Un- supervised Neural Machine Translation Model 17 for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Using the results from our RNN-based models, we primarily focused on character and BPE tokenization schemes, and have also experi- mented with a more complex cross-lingual embed- ding type called pivot-private embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Due to differences in implementation between the RNN and Transformer-based models, we were unable to train Vecmap embeddings for this set of experi- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 17https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='com/facebookresearch/UnsupervisedMT 8 Model Name Can>Man Char BLEU Man>Can Char BLEU Baseline (Character Conversion) Model 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='3 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='2 RNN (Word Tok + Vecmap Embed) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='1 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='9 RNN (Char Tok + Vecmap Embed) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='8 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='5 RNN (Char Tok + Concat Embed) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='4 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='3 RNN (BPE Tok learned separately + Vecmap Embed) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='8 RNN (BPE Tok learned jointly + Vecmap Embed) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='3 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='5 RNN (Balanced Dataset + Word Tok + Vecmap Embed) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='2 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='5 RNN (Balanced Dataset + Char Tok + Vecmap Embed) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='1 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='4 Transformer (Char Tok + Concat Embed)** 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='4 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='1 Transformer (Char Tok + Pivot-Private Embed) 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='2 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='5 Transformer (BPE Tok learned jointly + Concat Embed) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='2 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='4 Table 2: Overview of all automatic evaluation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' All BLEU (Bilingual Evaluation Understudy) metric scores are calculated at the character-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Best-performing model indicated by **.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='2 Results 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='1 Automatic Evaluation Model Architectures The first metric that our study sought to investigate was the varying per- formances of Mandarin-Cantonese unsupervised machine translation based on the underlying neu- ral network architecture, namely an RNN-based architecture versus a Transformer architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' We observed that the transformer model led to higher BLEU scores when other factors are constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' This can be observed in the RNN (Char Tok + Con- cat Embed) versus Transformer (Char Tok + Con- cat Embed) models, where Cantonese-to-Mandarin translation yielded 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='4 versus 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='4, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' and Mandarin-to-Cantonese yielded 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='3 versus 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' In fact, our highest performing model from the study was trained on a Transformer architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Cross-lingual Embeddings The study also makes comparisons between different types of cross-lingual embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Of primary interest are training monolingual embeddings and map- ping them to a shared cross-lingual space using Vecmap (as detailed in the Methodology section), and learning embeddings from the concatenated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' In a comparison between RNN (Char Tok + Vecmap Embed) and RNN (Char Tok + Con- cat Embed) models, we can see that the mapping- based cross-lingual embeddings have outperformed the concatenation-based technique, yielding a Cantonese-to-Mandarin BLEU of 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='8 and 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='4, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' and a Mandarin-to-Cantonese BLEU of 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='5 and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' In addition to mapping-based and concatenation- based cross-lingual embeddings, we also had time to run one experiment on pivot-private embeddings (as detailed in the Methodology section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' By com- paring the Transformer (Char Tok + Concat Em- bed) and Transformer (Char Tok + Pivot-Private Embed) models, we observe that concatenation- based embeddings outperform pivot-private em- beddings, with a Cantonese-to-Mandarin BLEU of 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='4 versus 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='2, and a Mandarin-to-Cantonese BLEU of 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='1 to 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Tokenization Methods Our study additionally makes a comparison between different types of tokenization methods: word, character, and BPE- tokenized models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Word-tokenization always per- forms the worst, in all cases aside from one (see RNN (Word Tok + Vecmap Embed) Mandarin-to- Cantonese results in Table 2), models trained with word-tokenized training data did not outperform even the Baseline (Character Conversion) Model in which no neural network was trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' While BPE-tokenized data tends to perform very well for languages with an alphabet system, such as French or English, we did not observe a such a strong result in the models trained using BPE- tokenized data for the Mandarin-Cantonese lan- guage pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' We experimented by learning BPE token vocabularies both separately and jointly, ob- serving a slight performance improvement when learned jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' However, neither BPE setting could outperform our character-tokenized models (see Ta- ble 2 for two results that lead to this conclusion: RNN (Char Tok + Vecmap Embed) versus RNN 9 (BPE Tok learned jointly + Vecmap Embed), as well as Transformer (Char Tok + Concat Embed) versus Transformer (BPE Tok learned jointly + Concat Embed)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Balanced Dataset We conclude that neither word nor character-tokenized models trained on the balanced dataset outperformed models trained using the full training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Thus, it is advanta- geous to use as much data as possible for model training, even if the two languages have an uneven amount of sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='2 Human Evaluation We conduct human evaluation on the Transformer (Char Tok + Concat Embed) model output in order to assess the extent to which our translation system would be useful to Cantonese and Mandarin speak- ers respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Considering that Cantonese speak- ers can understand Standard Chinese, a translation system from Mandarin to Cantonese should aim for localisation and fluency in Cantonese, while not losing the original meaning of the sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' On the other hand, the primary purpose of a Cantonese- to-Mandarin translation system is to facilitate Can- tonese comprehension for Mandarin speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' For these diverging purposes in our translation direc- tions, we manually evaluate each translation direc- tion with separate criteria, which is explained in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Procedure 100 lines from the test set are selected for evaluation, identical for both translation di- rections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' One native speaker of each target lan- guage evaluates for that direction only (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Can- tonese speaker evaluates Mandarin to Cantonese sentences, and visa-versa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' During evaluation, the evaluator has access to the original input and the target output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' The evaluation decision is binary for both criteria, the evaluator can only choose either YES or NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' In the example sentences below, Man- darin features are highlighted in orange, Cantonese features are highlighted in teal and ungrammatical features are highlighted in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Cantonese to Mandarin System outputs are evaluated against the criteria concerning whether the output helps Mandarin speakers understand Cantonese text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 34% were found helpful for un- derstanding Cantonese text, 61% were found not helpful, 5% sentences are discarded because the original text in Cantonese is already perfectly com- prehensible for Mandarin speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Mandarin to Cantonese System outputs are evaluated against the criteria “Does the system out- put contribute to Cantonese fluency / localisation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' ".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' It is found to be the case for 47% of the sentences, false for 52% of the sentences with 1%sentences discarded since the input and target were identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' (1)-(4) are examples of the system output for the Mandarin to Cantonese direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' In (1), the out- put is evaluated as helpful even though it has not completely transformed all Mandarin features into Cantonese ones, however, the components with the highest semantic value (拍拖dating and 散break up) are in Cantonese where it was originally in Mandarin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Compared to (3), where the output still retains mostly Mandarin and has no Cantonese fea- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Comparing (2) and (4), they both have some grammatical errors (in red), but the impact of such error in (2) is less significant to the overall meaning of the sentence, while in (4) the overall sentence is incomprehensible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Examples of output that is helpful: (1) Mandarin reference (source): 身邊有兩位好朋友，交往了三年， 就那樣分手了。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Cantonese reference (target): 身邊有兩位好友，拍三年拖，就噉散 。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' System output: 身邊有兩位好友，拍了三年拖，就這 樣散了。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Sentence meaning: I have two friends who had been dating for three years, and they broke up just like that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' (2) Mandarin reference (source): 別這麼犟，快點向媽認錯。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Cantonese reference (target): 咪咁硬頸，快同亞媽認錯。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' System output: 否“硬頸，快些和亞媽認錯。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Sentence meaning: Don’t be so stubborn, apologize to your mother at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Examples of output that is not helpful: (3) Mandarin reference (source): 別小看他，他已經有了三項發明。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Cantonese reference (target): 10 咪睇小佢，佢已經有三項發明。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' System output: 否看小她，她已經有了三項發明。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Sentence meaning: Don’t underestimate him, he already has three inventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' (4) Mandarin reference (source): 給海關沒收了那些東西。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Cantonese reference (target): 畀海關執。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' System output: 給海關執了那麼。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Sentence meaning: The things that were confiscated by customs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 7 Discussion Our Mandarin-Cantonese machine translation project displays the differences between two to- kenization methods (character-level and byte pair encoding), with an outcome different than expected regarding byte pair encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' A possible reason for this may be that such a big vocabulary size can lead to worse embeddings, taking into account the size of our corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' One of our approaches was down-sampling the full dataset into a balanced one, from which we expected a higher BLEU score compared to when using the full dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' However, this had the op- posite effect on the BLEU score and it ended up being lower than in the previous occasions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' This is perhaps due to the fact that 1 million sentences is just simply not enough data for a machine to become ’fluent’ in a language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' As further work, we propose that this project can be extended by combining out best architec- ture, best tokenization and best embedding training method (transformer + character + mapping), by de- veloping a cross-lingual mapping for embeddings that is compatible with a transformer network in order to confirm whether it does lead to higher results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' In addition, other options worth exploring would be the grammatical similarity between Cantonese and Mandarin and developing an statistical ma- chine translation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 8 Summary and conclusion The aim of implementing a Cantonese-Mandarin MT-model was accomplished by: Creating a large-scale corpus out of several online sources such as Wikipedia, scraped Instagram comments, YouTube subtitles and restaurant reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Implementing and training several Cantonese- Mandarin translation models while studying the effects of different tokenization strategies, such as character-level and byte-pair encod- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' While BPE was expected to outperform character-level tokenization, this was not the case in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' The outcomes of this project showed that overall, in 61% of the cases, the outcome translation was not useful to help Mandarin speakers understand Cantonese text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' As far as what fluency concerns, in 52 out of 100 cases, the system’s output did not show any contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Further work and research is essential in order to reach good percentages of performance and fluency in such a machine translation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' This project has contributed a large Cantonese dataset that was not available before as it is now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' We hope that with this project we moved one step forward into a direction that has been studied for some years now, contributing to further devel- opments and advancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' References Mikel Artetxe, Gorka Labaka, and Eneko Agirre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Learning principled bilingual mappings of word em- beddings while preserving monolingual invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' In Proceedings of the 2016 conference on empiri- cal methods in natural language processing, pages 2289–2294.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Mikel Artetxe, Gorka Labaka, and Eneko Agirre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 2017a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Learning bilingual word embeddings with (almost) no bilingual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' In Proceedings of the 55th Annual Meeting of the Association for Compu- tational Linguistics (Volume 1: Long Papers), pages 451–462.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Mikel Artetxe, Gorka Labaka, and Eneko Agirre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 2018a.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Yonghui Wu, Mike Schuster, Zhifeng Chen, Quoc V Le, Mohammad Norouzi, Wolfgang Macherey, Maxim Krikun, Yuan Cao, Qin Gao, Klaus Macherey, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Google’s neural machine translation system: Bridging the gap between hu- man and machine translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' arXiv preprint arXiv:1609.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content='08144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Hei Yi Mak and Tan Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' Low-resource nmt: A case study on the written and spoken languages in hong kong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' In 2021 5th International Conference on Natural Language Processing and Information Re- trieval (NLPIR), pages 81–87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} +page_content=' 12' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfjwdR/content/2301.03971v1.pdf'} diff --git a/4tAyT4oBgHgl3EQf2Plc/content/tmp_files/2301.00747v1.pdf.txt b/4tAyT4oBgHgl3EQf2Plc/content/tmp_files/2301.00747v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..30120825778520a0fa967c31c67dcb27c65fcd33 --- /dev/null +++ b/4tAyT4oBgHgl3EQf2Plc/content/tmp_files/2301.00747v1.pdf.txt @@ -0,0 +1,1476 @@ +Time-domain observation of ballistic orbital-angular-momentum currents with giant relaxation +length in tungsten +Tom S. Seifert1,2, Dongwook Go3, Hiroki Hayashi4, Reza Rouzegar1,2, +Frank Freimuth3,5, Kazuya Ando4,6-7, Yuriy Mokrousov3,5, Tobias Kampfrath1,2 +1Freie Universität Berlin, 14195 Berlin, Germany +2Fritz Haber Institut der Max-Planck-Gesellschaft, 14195 Berlin, Germany +3Forschungszentrum Jülich, 52425 Jülich, Germany +4Department of Applied Physics and Physico-Informatics, Keio University, Yokohama 223-8522, Japan +5Johannes Gutenberg-Universität Mainz, 55099 Mainz, Germany +6Keio Institute of Pure and Applied Sciences, Keio University, Yokohama 223-8522, Japan +7Center for Spintronics Research Network, Keio University, Yokohama 223-8522, Japan + +Abstract +The emerging field of orbitronics exploits the electron orbital momentum 𝐿, which may allow magnetic- +information transfer with significantly higher density over longer distances in more materials than possible +with spin-polarized electrons. However, direct experimental observation of 𝐿 currents, their extended +propagation lengths and their conversion into charge currents has remained challenging. Here, we optically +trigger ultrafast angular-momentum transport in Ni|W|SiO2 thin-film stacks. The resulting terahertz charge- +current bursts exhibit a marked delay and width that grow linearly with W thickness. We consistently ascribe +these observations to a ballistic 𝐿 current from Ni through W with giant decay length (∼ 80 nm) and slow +velocity (∼ 0.1 nm/fs). At the W/SiO2 interface, the 𝐿 flow is converted into a charge current by the inverse +orbital Rashba-Edelstein effect. Our findings establish orbitronic materials with long-distance ballistic 𝐿 +transport as possible candidates for future ultrafast devices and an approach to discriminate Hall- and +Rashba-Edelstein-like conversion processes. + +Introduction +Spintronics research aims at utilizing the flow of spin angular momentum carried by electrons to transport +information and eventually manipulate magnetic order [1]. Actually, electrons have two distinct channels of +angular momentum: the electron spin 𝑆 and orbital angular momentum 𝐿. While 𝑆 is successfully exploited +in the field of spintronics to transport information by spin currents and to convert the latter into detectable +charge currents by spin-to-charge conversion (S2C) [2], 𝐿 has only recently gained attention in the field of +orbitronics. To make this fascinating concept compatible and competitive with conventional electronics [3, +4], the speed of spin-orbitronic functionalities needs to be scalable to terahertz (THz) rates [5]. +A first key advantage of 𝐿 over 𝑆 is that it can assume arbitrarily high values for one electron, which is +interesting for efficient manipulation of future orbitronic devices [1, 6, 7]. Second, 𝐿-to-charge conversion +(L2C) does not rely on spin-orbit interaction (SOI), which opens the orbitronic workbench to abundant light +metals [8]. Third, 𝐿-currents are predicted to propagate over increased lengths reaching almost 100 nm [9]. +Finally, 𝐿-induced torques should have a starkly different behavior compared to 𝑆-induced torques [10-14]. +Recent studies provided strong indications of 𝐿 transport and charge-to-𝐿-current conversion by the orbital +Hall effect (OHE) in a thin layer of a paramagnetic material (PM). The 𝑆 or 𝐿 accumulation resulting from an +in-plane charge current was interrogated by magnetooptic imaging [8] or by measuring the torque it exerted +on the magnetization of an adjacent thin-film ferromagnetic material (FM) [1, 9-24]. The FM was chosen to +be either susceptible to 𝑆 (e.g., Ni81Fe19, CoFeB) or 𝐿 accumulation (e.g., Ni). + +Unfortunately, it remains experimentally challenging to measure 𝐿 curents by L2C. First, it is difficult to +distinguish L2C by the OHE from L2C by an orbital Rashba-Edelstein effect (OREE) because both phenomena +obey identical macroscopic symmetries. Second and for the same reason, OHE and OREE are difficult to +separate from their S2C counterparts, i.e., from the spin Hall effect (SHE) and the spin-based Rashba- +Edelstein effect (SREE) [25]. Previous work, however, indicates different spatial propagation and relaxation +dynamics of 𝑆 and 𝐿 currents [9-11]. Therefore, an experimental approach such as THz emission +spectroscopy [26, 27], which monitors currents with femtosecond resolution, is perfectly suited to access the +possibly different ultrafast 𝐿/𝑆 propagation and conversion dynamics. +Here, we study ultrafast signatures of 𝑆 and 𝐿 transport from a FM into a PM that is launched by exciting +FM|PM stacks with a femtosecond laser pulse. L2C and S2C in the PM is measured by monitoring the emitted +THz pulse. Upon changing the FM from Ni to Ni81Fe19 (Py) and interfacing them with the PMs Pt, Ti and W, +we find the same characteristic sign changes in the emitted THz pulse as in previous magnetotransport +studies [11]. Consequently, we interpret our observations as signatures of ultrafast L2C and S2C. Remarkably, +the emitted THz field from Ni|W is strongly time-delayed and low-pass-filtered compared to that from Ni|Pt. +The bandwidth and amplitude of the underlying charge-current burst decreases with W thickness, whereas +its delay increases linearly. We assign this observation to long-distance ballistic 𝐿 transport in W, which has +a more than 10 times larger relaxation length than 𝑆 transport. Specifically, our data suggest a dominant +contribution to L2C through the inverse OREE (IOREE) at the W/SiO2. Interestingly, this effect is absent in +Ni|Ti and attributed to a dominant bulk L2C by the inverse OHE (IOHE). Our results may help establish an +ultrafast experimental and theoretical methodology to extract the propagation dynamics of 𝐿 currents. + + +FIGURE 1: Launching and detecting terahertz 𝑺 and 𝑳 currents. Upon ultrafast laser excitation of the FM, +the FM magnetization 𝐌 is quenched, leading to 𝑆 accumulation 𝜇�, 𝐿 accumulation 𝜇� and the injection of +spin and orbital currents 𝑗� and 𝑗�, respectively, into the PM. Various bulk and interfacial L2C and S2C +processes generate an ultrafast in-plane charge current 𝑗� that radiates a THz pulse with electric field 𝐸 vs +time 𝑡 directly behind the sample. +Conceptual background. Our approach is guided by the idea that 𝐿 currents obey the same phenomenology +as 𝑆 currents, whereas 𝐿 transport is expected to have comparatively different spatiotemporal dynamics on +ultrashort time and length scales [1, 9-11]. As schematically depicted in Fig. 1, a femtosecond optical pump +pulse excites a FM|PM stack and triggers ultrafast 𝑆 and 𝐿 currents with density 𝑗� and 𝑗�, respectively, from +FM to PM. S2C and L2C result in ultrafast in-plane charge currents acting as a sources of a THz +electromagnetic pulse [28]. The resulting THz electric-field amplitude 𝐸(𝑡) directly behind the sample is +proportional to the sheet charge current 𝐼�(𝑡), which reads + +Femtosecond +THz pulse +heating pulse +js +E +S2C +us +ee +UL +L2C +iL +M +jc +FM +PM +Z𝐸(𝑡) ∝ 𝐼�(𝑡) = +� +d𝑧 [θ���(𝑧)𝑗�(𝑧, 𝑡) + θ���(𝑧)𝑗�(𝑧, 𝑡)] +������� +� +. +(1) +Here, θ���(𝑧) and θ���(𝑧) describe the local efficiency of instantaneous L2C and S2C, respectively. They +include microscopic mechanisms like the inverse SHE (ISHE) or IOHE [27, 29], which occur in the bulk, or the +inverse SREE and IOREE, which require regions of broken inversion symmetry such as interfaces [30, 31]. +To understand the emergence of 𝑗� and 𝑗�, we note that sudden laser heating of the FM induces 𝑆 and 𝐿 +accumulations, 𝜇� and 𝜇� , respectively. The spin accumulation 𝜇� is proportional to the excess +magnetization, i.e., the difference between the instantaneous magnetization and the equilibrium +magnetization that would be attained at the instantaneous electron temperature [32-35]. Consequently, the +FM releases 𝑆 at a rate proportional to 𝜇�, by transferring 𝑆 to both the crystal lattice and the PM. +Recent studies on single-element FMs showed that the 𝑆- and 𝐿-type magnetizations exhibit very similar +ultrafast time evolution following laser excitation [36-38]. Therefore, we expect a very similar time evolution +of 𝜇� and 𝜇�, i.e., 𝜇�(𝑡) ∝ 𝜇�(𝑡), where their amplitudes depend on details of the electronic structure [14]. +Despite this common origin of 𝑆 and 𝐿 currents, the relation between 𝑗�(𝑧, 𝑡) and 𝑗�(𝑧, 𝑡) (Fig. 1) can be +highly nontrivial as 𝑆 and 𝐿 may propagate differently through the FM/NM interface and the NM bulk. +Eq. ( 1 ) does not account for contributions due to magnetic dipole radiation of the time-dependent +magnetization and of photocurrents even in magnetic order, because both components can be discriminated +experimentally [32, 39]. +Experiment details. We study thin film FM|PM samples, where the two FMs Py and Ni are chosen for their +high efficiency in generating 𝑆 and 𝐿 currents, respectively [11]. The PMs are chosen to have a strong ISHE +(Pt, W) and IOHE (W, Ti) response. The reported signs for the ISHE are opposite for Pt vs W with a vanishing +ISHE in Ti, but the expected IOHE signs are the same for all three PMs [40]. The studied FM|PM stacks have +thicknesses of a few nanometers deposited onto 500 μm thick glass substrates or 625 μm thick thermally +oxidized Si substrates (see Fig. S1 and Methods). The samples are characterized by optical and THz +transmission spectroscopy [41], yielding the pump absorptance, DC conductivity and Drude relaxation rate +(Fig. S2). +In our experiment (Fig. 1), ultrashort laser pulses (15 fs duration, 800 nm center wavelength, 80 MHz +repetition rate, 1.9 nJ pulse energy, 0.2 mJ/cm2 incident fluence) derived from a Ti:sapphire oscillator excite +the FM|PM samples. We record the emitted THz radiation by electrooptic sampling in a 1 mm or 10 µm thick +ZnTe(110) or a 250 μm thick GaP(110) electro-optic crystal [42]. The resulting THz emission signal 𝑆(𝐌, 𝑡) vs +time 𝑡 is proportional to the THz electric-field waveform 𝐸 (Fig. 1) convoluted with a setup-response function +[43]. The presented data is low-pass filtered by convolution with a Gaussian function with a full width at half +maximum of about 80 fs for better visibility unless noted otherwise. +All experiments are performed under ambient conditions unless stated otherwise. We apply an in-plane +magnetic field of about 10 mT to the sample and monitor the THz field component perpendicular to the +sample magnetization 𝐌. The component parallel to 𝐌 is found to be minor (Fig. S3). Measurements with +linearly and circularly polarized pump pulses reveal a negligible impact of the pump polarization on the THz +emission (Fig. S4). +To isolate magnetic signals, we reverse 𝐌 and focus on the odd-in- 𝐌 THz signal 𝑆(𝑡) = +[𝑆(+𝐌, t) − 𝑆(−𝐌, t)] 2 +⁄ . Even-in-𝐌 signal components are minor. As expected from a transport scenario, +further experiments, in which the samples are reversed, reveal a dominant structural-inversion-asymmetry +(SIA) character of the emitted THz signals compared to minor contributions unrelated to SIA, which most +likely arise from magnetic-dipole radiation due to ultrafast demagnetization (Fig. S5) [32]. + + + +FIGURE 2: Terahertz raw data. THz +emission signals 𝑆(𝑡) from FM|PM stacks +with a FM=Py and b FM=Ni capped with +PM=Pt, W or Ti. Note the rescaling of the +Pt-based sample signals. Film thicknesses +in nanometers are given as numerals in +parenthesis. As THz detector, a 1 mm +ZnTe(110) crystal was used. +Results +FM=Py. Figure 2a shows THz emission signals 𝑆 from Py|PM samples with PM=Pt, W, Ti, where the time-axis +origin is the same for all signals. All three waveforms have identical shapes. Minor differences in the shape +of 𝑆��|�� vs 𝑆��|�� are attributed to contributions unrelated to SIA (see above and Fig. S6). +The relative signal magnitudes as well as the opposite polarities for PM=Pt and W are consistent with +previous reports of ISHE-dominated THz emitters [28]. The polarity of the signal from Py|Ti is the same as +from Py|Pt and consistent with the calculations and measurements that found the same sign of the ISHE in +Pt and the IOHE in Ti [8, 27, 40]. However, the Py|Ti signal has a significantly smaller amplitude than the +Py|Pt signal even though Ti has a sizeable L2C efficiency. We ascribe this observation to a small amplitude of +the 𝐿 current injected into Ti, consistent with the small 𝐿 component of the Py magnetization [11]. +To summarize, for Py|PM, our THz signals are consistent with the notion that we predominantly observe +transport of 𝑆 and 𝐿 into the PM bulk and its conversion into a charge current through the ISHE and the IOHE. +A possible Rashba-type L2C or S2C, or skew-scattering at the FM/PM interface [44] may make an additional +yet relatively small contribution. +FM=Ni. When the FM=Py is replaced by Ni, the signal polarity remains the same for Pt and Ti, and the two +waveforms exhibit identical dynamics (Fig. 2b and Fig. S7). In stark contrast, however, the signal polarity for +Ni|W reverses, the waveform is less symmetric, and its maximum is time-shifted relative to Py|W. This +striking observation indicates that Py|W and Ni|W show competing THz-generation mechanisms, the +a +b + +X10-6 +3 +Ni(5)IPt(3) /3 +Ni(5)ITi(3) +2 +Ni(5)/W(3) +Terahertz signal +-2 +-3 +0 +1 +2 +Time (ps)3 +Py(5)/Pt(3) /3 +Py(5)/Ti(3) +2 +Py(5)/W(3) +Terahertz signal +-2 +.3 +0 +2 +1 +Time (ps)dominance of which depends sensitively on the FM material. To gain more insight into the different dynamics +in Ni|W, we next vary the W thickness. + + +FIGURE 3: Impact of W thickness in +Ni|W. THz emission signals for Ni|W +samples with varying W thickness +normalized to the absorbed pump- +pulse fraction in the Ni layer and to the +sample impedance (see Methods and +Table S1). Note the rescaling of the +reference signal from Ni|Pt. Film +thicknesses in nanometers are given as +numerals in parenthesis. A 250 µm +GaP(110) crystal was used as THz +detector. +Impact of W thickness. Figure 3 shows THz emission signals from Ni|W(𝑑�) for various 𝑑� and from a Ni|Pt +reference sample. Consistent with Fig. 2b, we see a clear trend with increasing W thickness relative to Ni|Pt: +The THz signal amplitude has a reversed sign, reduces with increasing 𝑑� and undergoes a significant +reshaping from asymmetric (Ni|Pt) to more symmetric (Ni|W) around the signal maximum. Interestingly, +𝑑� = 2 nm is already sufficient to induce a shift of the maximum of the THz signal by about 100 fs. +We emphasize that the changes in THz-signal dynamics solely originate from changing the PM thickness. +Therefore, the FM is not primarily responsible for the signal-dynamics changes and, thus, considered as an +PM-independent 𝑆 and 𝐿 injector in the following. + +FIGURE 4: Ultrafast charge currents in Ni|W. a Charge sheet currents in Ni|W for various W thicknesses 𝑑� +as extracted from the data of Fig. 3. The feature at 0.9 ps is a remainder of a THz-field reflection echo in the +10 µm ZnTe electro-optic detection crystal (see Methods). Film thicknesses in nanometers are given as +numerals in parenthesis. Note the rescaling of the Pt-based sample signal. The apparent signal delays and +amplitudes are highlighted by a circular marker. b Extracted time delay with a straight line as a guide to the +eye, c relative amplitude at the delay marked in panel a, d temporal width at half maximum, and e integrated +charge current between 0.2 to 0.9 ps vs 𝑑� from the data in panel a. Error bars are estimated for panels c +a +b +c +d +e + +X10-9 +5 +Ni(5)/Pt(3) /6 +Ni(5)/W(2) +4 +Norm. terahertz signal +Ni(5)/W(3) +Ni(5)/W(5) +3 +Ni(5)/W(10) +Ni(5)/W(15) +2 +Ni(5)/W(20) +0 +7 +-2 +-3 +-0.5 +0 +0.5 +Time (ps)Ni(5)/Pt(3) /6 +14 +Ni(5)/W(2) +12 +Ni(5)/W(3) +Ni(5)/W(5) +2 +per abs. fluence in Ni (J/m +10 +Ni(5)/W(10) +Ni(5)/W(15) +8 +Ni(5)IW(20) +delay +6 +4 +2 +0 +-2 +-4 +-6 +0 +0.5 +Time (ps)80 +Ampl. (norm. +Delay (fs) +60 +40 +0.5 +20 +0 +0 +0 +10 +20 +10 +20 +0 +340 +320 +Area (norm.) +(fs) +300 +280 +Width +260 +0.5 +240 +220 +0 +0 +10 +20 +10 +20 +0 +d... (nm) +d... (nm)and e from the signal-to-noise ratio in panel a, for panels b and d as 10% of the delay and width, respectively, +and in all panels b-e as ± 1 nm for 𝑑�. +Current dynamics in Ni|W. To obtain a sample-intrinsic measurement of the L2C dynamics, we extract the +sheet charge current 𝐼�(𝑡) flowing in Ni|W (Eq. (1)) normalized to the absorbed laser fluence in the Ni layer +(see Methods). This procedure eliminates any impact of sample exchange on pump-pulse absorption +efficiency, sample impedance or setup response function (see Methods). +Figure 4a presents 𝐼�(𝑡) in Ni|W with a resolution of 50 fs for various W thicknesses 𝑑�. The 𝐼�(𝑡) traces +have striking features. (i) They have opposite polarity relative to Py|W. (ii) Their maximum shifts by delays +Δ𝑡��� ∝ 𝑑� at a rate Δ𝑡��� 𝑑� +⁄ +≈ 4 fs/nm (Fig. 4b), implying a velocity of 0.25 nm/fs. (iii) The 𝐼� peak +value decreases approximately linearly with 𝑑� to about 50% after 20 nm (Fig. 4c), indicating attenuation +and dispersion upon propagation. (iv) The 𝐼� width increases linearly at a rate of ≈ 8 fs/nm (Fig. 4d). (v) The +time-integrated current ∫ d𝑡 𝐼�(𝑡) is only weakly dependent on 𝑑� with a decreasing trend, thereby +indicating an extremely large relaxation length >20 nm (Fig. 4e). +Features (i) and (iii) imply that 𝐼�(𝑡) cannot arise from 𝑆 transport. Otherwise, an opposite signal polarity +would result because S2C in W is dominated by the ISHE [28]. In addition, 𝑆 currents in W relax over distances +much smaller than 20 nm [41]. Our data, therefore, strongly indicate that 𝐿 transport plus L2C is the +dominant origin of the THz charge current in Ni|W. Second, features (ii) and (iv) are a hallmark of a signal +arising from ballistic transport of a pulse that is detected in an arrival layer. In this picture, the increase of +the 𝐼�(𝑡) width with 𝑑� arises from velocity dispersion along the 𝑧-direction of the particles that make up +the pulse (Fig. 5a). Feature (v) implies a minor L2C in the W bulk because it would otherwise result in an +integrated charge current ∫ d𝑡 𝐼�(𝑡) that increases monotonically with 𝑑�. +Model: 𝑳 current and IOREE in Ni|W. The preceding discussion suggests the following transport scenario in +Ni|W. Upon excitation of the Ni layer, a transient 𝑆 and 𝐿 accumulation is induced (Fig. 1). Their dynamics +are expected to be very similar (see above) [36-38] and monitored well by the ISHE charge current in Ni|Pt +(Fig. 4a). Finally, L2C is dominated in regions close to the W/SiO2 interface (Fig. 5a). Such interfacial L2C can +be very efficient, as argued in previous works [7, 24, 45-49], which, however, lacked the required +femtosecond resolution. +This scenario can explain all charge-current features (i)-(v) (Fig. 4) and is consistent the above experimental +findings. As the 𝑗� pulse propagates predominantly ballistically, its arrival in the W/SiO2 L2C region is delayed +by a time Δ𝑡��� ∝ 𝑑�. The velocity of the 𝑗�-pulse peak (∼ 0.1 nm/fs) is consistent with that of 𝐿-carrying +d-band states of W [50]. During propagation through PM=W, the 𝑗� pulse disperses due to different electron +velocities along the surface normal (Fig. 5a) and attenuation with a typical relaxation length >20 nm. +To quantitatively model the charge-current dynamics in Ni|W (Fig. 3), we assume ballistic 𝐿 transport with a +characteristic decay length 𝜆� in W. The 𝑗� arriving at the W/SiO2 interface is obtained by summing over all +Fermi-surface states with positive group velocity along the 𝑧-axis (see Fig. 5a and Methods). The resulting 𝑗� +at the W/SiO2 interface induced by a fictitious 𝛿(𝑡)-like 𝐿 accumulation in Ni reads +𝑟(𝑡) ∝ 𝑑� +𝑡� Θ(𝑣�𝑡 − 𝑑�)e���� �� +⁄ +, +(2) +where Θ is the Heaviside step function, and 𝑣� is the Fermi velocity of the 𝐿-polarized electrons in W. We +convolute 𝑟(𝑡) with 𝜇�(𝑡) ∝ 𝜇�(𝑡), which is given by the charge current measured in Ni|Pt (Fig. 4a). Our +modeled 𝐼�(𝑡) curves (Fig. 5c) reproduce the measured charge currents in Ni|W (Fig. 4a) semiquantitatively +for the choice 𝜆� = 80 nm and 𝑣� = 0.14 nm/fs. These values are in good agreement with the estimates +obtained above (Figs. 4b-e). + + + + +FIGURE 5: Simulated ultrafast inverse +orbital Rashba-Edelstein effect in W. +a Schematic of the suggested scenario for +𝐿 transport and L2C by the IOREE in Ni|W +showing the different wave vector +contributions of the 𝐿 currents inside the +W layer driven by the magnetization +quenching in the Ni layer. Upon reaching +the W back surface, the orbital currents 𝐣� +are converted into a transverse charge +current 𝐣� by the inverse orbital Rashba- +Edelstein +effect +(IOREE). +In +the +experiment, many of the point-like +sources +of +orbital +currents +are +superimposed +along +the +FM/PM +interface. +b Qualitative +response +functions 𝑟(𝑡) to a fictitious delta-like 𝑗� +pulse injected at the Ni/W interface for +different W layer thicknesses 𝑑�, where +𝑑�� < 𝑑�� < 𝑑�� . c Simulated IOREE +charge +currents 𝐼�(𝑡) obtained +by +convoluting 𝑟(𝑡) [Eq. (2), panel b] with +the 𝐼�(𝑡) of the Ni|Pt reference sample. +Inputs for the simulation are a ballistic 𝐿 +velocity of 0.14 fs/nm, an 𝐿 decay length +of 𝜆� = 80 nm and a global scaling factor. +To summarize, the THz charge currents in Ni|W (Fig. 4) can be considered as signatures of 𝐿 currents injected +into W. The charge-current generation [see Eq. (1)] is dominated by an extremely long-range 𝑗� and L2C at +the W/SiO2 interface, i.e., by θ��� at 𝑧 = 𝑑�� + 𝑑��. Such long-range 𝐿 transport is a unique feature of +orbitronic materials, and first indications for it were found previously in Ti [10, 11]. Note that within our +interpretation, the sign of 𝑆��|� agrees coincidentally with the calculated sign of θ��� for the IOHE in W [40]. +Discussion +Our interpretation neglects other possible contributions to the THz charge current. First, the inverse Faraday +effect as a source of 𝑆 and 𝐿 currents can be ruled out by the pump-polarization independence (see Fig. S4). +Second, for the 𝑆 channel, a dominant Seebeck-type contribution due to an electronic temperature +difference Δ𝑇��/�� across the Ni/PM interface is neglected as found in previous studies [32]. For the 𝐿 +channel, we estimate Δ𝑇��/�� right after pump pulse absorption (see Methods) and find Δ𝑇��/�� ∼ +400 K +and Δ𝑇��/�� ∼ −100 K in Ni|Ti and Ni|W. The observed THz-emission signals, in contrast, show the same +sign from all three samples (Fig. 2b). Therefore, interfacial electronic temperature differences are a minor +a +b +c + +14 +Ni(5)/Pt(3) /6 +12 +Ni(5)/W(2) +Ni(5)/W(3) +10 +Ni(5)/W(5) +Ni(5)/W(10) +8 +Ni(5)/W(15) +Ni(5)/W(20) +6 +4 +2 +per abs. +0 +-2 +4 +-6 +0 +0.5 +1 +Time (ps)W +Ni +eee +μL +IOREE +GG +jL +dw +Z +r(t) +r (0) +0 +t +t(dw1) t(dw2) t(dw3driving force. Additional pump-propagation simulations show that, even for the thickest samples, pump- +intensity gradients in the FM and PM bulk are relatively small (Fig. S8). +Third, regarding transport in W, we consider dominant angular-momentum transport by magnons unlikely +because W is not magnetically ordered. An outstandingly long propagation of 𝑆 transport is ruled out, too, +because the Drude scattering times for all studied samples are substantially shorter (<50 fs, Figs. S2) than the +peak delays of 𝐼�(𝑡) (Fig. 4a). +Fourth, even though our data imply a dominant IOREE contribution to charge-current generation (see above), +the positive shoulder-like feature at around 0.1 ps for 𝑑� ≤ 3 nm in Fig. 4a may indicate a small contribution +of bulk L2C, i.e., the IOHE. A 𝐿-to-𝑆 conversion plus ISHE in the PM [23] might contribute but is considered +negligible here given the good agreement of our experimental data (Fig. 4) and the IOREE scenario (Fig. 5). +The dominance of an 𝐿-type angular momentum current in Ni|W highlights the role of Ni as an 𝐿 source and +indicates that the Ni/W interface may transmit 𝐿 currents more efficiently than 𝑆 currents. +We finally turn to other interesting aspects of our study. A more detailed comparison of Fig. 2a and 2b reveals +further changes in amplitude between Ni- and Py-based samples. The pronounced amplitude changes for +PM=W or Pt when changing FM=Py to Ni are related to the intricate interplay of all parameters in Eq. (1) in +addition to changes in the relative amplitudes of 𝜇� and 𝜇� and interface transmission coefficients for 𝑗� and +𝑗�. Therefore, further experiments for a robust separation of 𝑆 and 𝐿 transport are needed. +We further find that the THz signals from the Ni-based samples increase linearly with pump fluence. Slight +sublinearities at the highest fluences do not alter the THz emission dynamics (Fig. S9). We emphasize that +samples deposited on Si rather than glass substrates show very similar THz emission characteristics (Fig. S1). +These observations demonstrate the robustness of the observed effects. +When adding a Cu layer on top of the Ni|W sample, we find almost no change in the THz emission signal +(Fig. S10). Future studies are needed to elaborate the exact character of the IOREE for different interfaces. +Interestingly, a Cu intermediate layer in Ni|Cu|W slightly modifies the amplitude and dynamics of the THz +signal, suggesting that Cu does not block 𝐿 transport strongly. +Regarding earlier reports of different THz emission dynamics in Fe|Au and Fe|Ru samples [29], we note that +a possible IOREE in Ru can, in hindsight, not be excluded. A dominant IOREE might also explain the seemingly +strong dependence of the Fe|Ru THz emission dynamics on the exact growth details [29, 51, 52]. +In conclusion, we observe THz-emission signals from optically excited Ni|W stacks that are consistent with +an ultrafast injection of 𝐿 currents into W and long-distance ballistic transport through W. Remarkably, we +find strong indications for the occurrence of the IOREE. This result can be considered as time-domain +evidence of the long-range nature of orbital currents and IOREE in typical metals such as W. +Our study highlights the power of broadband THz emission spectroscopy in disentangling of spin and orbital +transport and Hall- and Rashba-Edelstein-like angular-momentum conversion processes through ultrafast +time-of-flight experiments. Future studies may aim at exploiting L2C physics in materials with nontrivial +topology, in which the OHE is predicted to be drastically enhanced close to Weyl points [53]. Our study opens +the field of THz orbitronics whose distinct dynamical features allow for tailoring ultrafast spin and orbital +currents on femtosecond time scales by a targeted material, thickness and interface engineering of +multilayers. + + + +Methods +Current extraction. To extract the in-plane sheet current flowing inside the sample from the measured THz +signal 𝑆, we first measure our setup response function ℎ by having a reference electro-optic emitter (50 μm +GaP on a 500 μm glass substrate) at the same position as the sample, which yields a reference THz signal 𝑆���. +By calculating the emitted THz electric field from that reference emitter 𝐸���, ℎ is determined by solving the +convolution 𝑆��� = (ℎ ∗ 𝐸���)(𝑡) for ℎ [43]. Further measured inputs for this calculation are the excitation +spot size with a full width at half maximum of 22 μm, the excitation pulse energy of 1.9 nJ and a transform +limited pump pulse with a spectrum centered at 800 nm and 110 nm full width at half maximum. We perform +the deconvolution directly in the time domain by recasting it as a matrix equation [54]. +Next, the electric field 𝐸 directly behind the sample is obtained from the recorded THz signal 𝑆 with the help +of the derived function ℎ by solving again the similar equation 𝑆 = (ℎ ∗ 𝐸)(𝑡) for 𝐸. Finally, the sheet charge +current (see Table S1) as shown in Fig. 3 is derived from a generalized Ohm’s law [28] that reads 𝐸(𝜔) = +𝑒𝑍(𝜔)𝐼�(𝜔) , where −𝑒 is the electron charge and the sample impedance is given by 𝑍(𝜔) = +𝑍� [1 + 𝑛��� + 𝑍�𝑑𝜎(𝜔)] +⁄ + with the free space impedance 𝑍�, the metal-stack thickness 𝑑 and the measured +mean sample conductivity 𝜎 (see Table S1) that we assume to be frequency independent due to the large +Drude scattering rate (see Fig. S2). To enable comparison of THz currents from different samples, we +normalize 𝐼� by the absorbed fluence in the FM layer. The data shown in Fig. 3 was obtained in a dry-air +atmosphere. +Sample preparation. The FM|PM samples (FM = Ni and Py, PM = Pt, Ti, Cu and W) were fabricated on glass +substrates of 500 μm thickness or thermally oxidized Si substrates of 625 μm thickness by radio frequency +(RF) magnetron sputtering under 6N-purity-Ar atmosphere. The sample structure and thickness are described +in Table S1. For the sputtering, the base pressure in the chamber was better than 5.0 × 10-7 Pa. To avoid +oxidation, 4-nm-thick SiO2 was sputtered on the surface of the films. All sputtering processes were performed +at room temperature. +Estimate of electronic temperatures. We calculate the electronic temperatures increase upon pump-pulse +absorption by +Δ𝑇� = �𝑇� +� + 2𝑓 +𝑑𝛾 − 𝑇�. +(3) +Here, 𝑇� = 300 K is the ambient temperature, 𝑓 is the absorbed fluence in the respective layer (see +Table S1), 𝑑 is the layer thickness and 𝛾 is the specific electronic heat capacity that is 300 J/m3 K2 for W, 320 +J/m3 K2 for Ni, 330 J/m3 K2 for Ti and 90 J/m3 K2 for Pt [55]. To obtain the absorbed fluences in each layer, we +note that the pump electric field is almost constant throughout the sample (see Fig S8). Therefore, local pump +absorption scales solely with the imaginary part of the dielectric function at a wavelength of 800 nm, which +equals 22.07 for Ni, 9.31 for Pt, 19.41 for Ti and 19.71 for W [56]. Consequently, the absorbed fluence is +determined by +𝑓��/�� = 𝑓��� +𝑑��/�� Im𝜀��/�� +𝑑�� Im𝜀�� + 𝑑�� Im𝜀�� + +(4) +with the total absorbed fluence 𝑓��� that is obtained from the absorbed pump power (see Table S1) and the +beam size on the sample (see above). +Model of orbital transport. To model the ballistic current in the PM, we assume that a 𝛿(𝑡)-like transient +spin accumulation in the FM causes a change Δ𝑛𝐤� in the occupation of each electronic wavepacket in the +PM with mean wavevector 𝐤 right behind the FM/PM interface at 𝑧 = 0� (Fig. 5a). Subsequently, this +occupation change propagates into the PM bulk according to Δ𝑛𝐤(𝑧, 𝑡) = Δ𝑛𝐤�𝛿(𝑧 − 𝑣𝐤𝑡), where 𝑣𝐤 is the +𝑧 component of the group velocity of the electronic wavepacket. Note that we here restrict ourselves to 𝐤 + +with nonnegative 𝑣𝐤. The occupation change Δ𝑛𝐤(𝑧, 𝑡) is accompanied by a particle current density Δ𝑗𝐤 = +𝑣𝐤Δ𝑛𝐤. The total pump-induced current density flowing into the depth of the PM layer is given by the sum +Δ𝑗(𝑧, 𝑡) = +� +Δ𝑛𝐤�𝑣𝐤𝛿(𝑧 − 𝑣𝐤𝑡) +𝐤, �𝐤�� +. +(5) +Because Δ𝑛𝐤� predominantly affects states close to the Fermi energy, the summation of Eq. ( 5 ) is +approximately proportional to an integration over the Fermi-surface parts with 𝑣𝐤 ≥ 0. One obtains +Δ𝑗(𝑧, 𝑡) = � d𝑣 𝑤(𝑣)𝑣 +� +� +𝛿(𝑧 − 𝑣𝑡), +(6) +where 𝑤(𝑣) is the weight of the group velocity 𝑣 along the 𝑧 axis. We assume that the Fermi surface is a +sphere with isotropic occupation change Δ𝑛𝐤� and radius 𝑘�. Consequently, the integrand Δ𝑛𝐤�𝑣𝐤 d�𝐤 +becomes Δ𝑛𝐤�𝑣� cos 𝜃 𝑘�d𝑘d𝜑d cos𝜃 ∝ 𝑣 d𝑣d𝑘d𝜑, where 𝑣� is the Fermi velocity. In other words, all +velocities from 0 to the Fermi velocity 𝑣� have equal weight, and we find +Δ𝑗(𝑧, 𝑡) ∝ � +d𝑣 𝑣 +�� +� +𝛿(𝑧 − 𝑣𝑡) = 𝑧 +𝑡� Θ(𝑣�𝑡 − 𝑧), +(7) +for 𝑧 > 0. Finally, we phenomenologically account for relaxation of the ballistic current with time constant 𝜏 +by multiplying Δ𝑗(𝑧, 𝑡) with e��/�, which directly takes us to Eq. (2). + +Acknowledgements +We thank G. Sala for fruitful discussions. TSS, RR and TK acknowledge funding by the German Research +Foundation (DFG) through the collaborative research center SFB TRR 227 “Ultrafast spin dynamics” (project +ID 328545488, projects A05 and B02) and financial support from the Horizon 2020 Framework Programme +of the European Commission under FET-Open Grant No. 863155 (s-Nebula). FF and YM acknowledge DFG +collaborative research center SFB TRR 173/2 “Spin+X”(project ID 268565370, project A11). 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Applied Optics, 1983. +22: p. 1099. +[57] Zak, J., E. Moog, C. Liu, and S. Bader Universal approach to magneto-optics. Journal of Magnetism and +Magnetic Materials, 1990. 89: p. 107. + + + + +Supplementary Materials +First, we summarize briefly the content of the Supplementary Materials before showing the +corresponding data: + + +Samples on Si show qualitatively the same THz emission waveforms for Ni with Pt, W and Ti. +Most importantly, the strong change in W dynamics is also observed on Si (Fig. S1). However, the +THz waveforms of Si vs glass differ in the details, which might be related to slightly changed +transport times. + +Drude scattering times are estimated to be <50 fs for all studied samples (Figs. S2). None of the +samples showed any indication of a drastically different Drude scattering time compared to all +other samples. + +Emitted THz signals are found to be linearly polarized and perpendicular to the sample +magnetization (Fig. S3). + +Pump-polarization dependent studies (pump helicity and linear polarization direction) show a +minor impact on the measured THz emission signal (Figs. S4). + +We perform THz emission measurements upon reversing the sample. Only the pure Ni film shows +a dominant contribution even in sample rotation, which we ascribe to SIA or magnetic dipole +radiation (Fig. S5) [32, 39]. + +For all Py-based bilayer samples, we find almost identical THz emission waveform shapes even +for PM thicknesses of 20 nm (Fig. S6). + +For Ni|Ti samples, we find almost identical THz emission dynamics to Ni|Pt (Fig. S7). + +Currents driven by pump light gradients in thick films of Ni|W and Ni|Ti can be neglected (Fig. +S8). + +All fluence dependencies are to a good approximation linear (Fig S9) with minor sublinearities +overserved for Ni|Ti and Ni|W samples. Related to that, only minor changes in the THz waveform +dynamics can be observed for different pumping fluences (Fig S9) + +Cu has only minor impact on the emitted THz waveforms (Fig. S10), either as a spacer layer or as +a capping layer as confirmed by comparison to the same sample without Cu. + +All data in the Supplementary Materials was measured with a 1 mm ZnTe(110) detection crystal. + + + + + +FIGURE S1: Si vs glass substrate. a Terahertz-emission waveforms from Ni|PM stacks on Si substrates. THz +waveforms for Si based samples are multiplied by -1 to account for the reversed sample orientation due +to the intransparency of the Si substrate for the pump pulse. a Terahertz-emission waveforms from Ni|PM +stacks on glass substrates. Film thicknesses in nanometers are given as numerals in parenthesis. Note the +rescaling of the Ni|Pt sample THz waveforms. + + + +b +a + +Glass substrate +X10-7 +5 +4 +3 +Terahertz signal +2 +-2 +-3 +-1.5 +-1 +-0.5 +0 +0.5 +Time +e (ps)Si substrate +X10-7 +2.5 +Ni(5)/Pt(3)/4 +2 +Ni(5)ITi(3) +Ni(5)/Ti(20) +1.5 +Ni(5)IW(3) +Terahertz signal +Ni(5)/W(20) +1 +0.5 +O +-0.5 +-1 +-1.5 +-2 +-0.5 +0 +0.5 +1 +1 +Time +(ps)FIGURE S2: Terahertz conductivities for samples on glass. Mean complex-valued terahertz conductivities +obtained from terahertz transmission measurements for a Ni, b Ni|Ti, c Ni|Pt and d Ni|W samples. For the +extraction, a thin film formula is applied [41] and a terahertz refractive index of 2.1 for glass is assumed. +Film thicknesses in nanometers are given as numerals in parenthesis. + + + +c +d +a +b + +Ni(5) +X106 +7 +6 +real +imag +5 +Conductivity (S/m) +4 +3 +2 +1 +0 +-1 +1.5 +2 +1 +2.5 +3 +3.5 +Freguency (THz)Ni(5)[Pt(3) +X106 +7 +real +6 +imag +5 +Conductivity (S/m) +4 +3 +2 +1 +0 +-1 +1 +1.5 +2 +2.5 +3 +3.5 +Freguency (THz)Ni(5)/W(3) +X106 +7 +real +6 +imag +5 +Conductivity (S/m) +4 +3 +2 +1 +0 +-1 +1 +1.5 +2 +2.5 +3 +3.5 +Freguency (THz)Ni(5)/Ti(3) +X106 +7 +real +6 +imag +5 +Conductivity (S/m) +4 +3 +2 +1 +0 +-1 +1 +1.5 +2 +2.5 +3 +3.5 +Freguency (THz) +FIGURE S3: Polarization state of the THz emission signal. Samples are magnetized along the s-direction +and pump pulses are polarized along the p-direction. Film thicknesses in nanometers are given as numerals +in parenthesis. + +-2 +-1 +0 +1 +2 +Time (ps) +-2 +0 +2 +4 +6 +8 +10 +Terahertz signal +10-7 +Ni(5)|Pt(3),p-pol +Ni(5)|Ti(3),p-pol +Ni(5)|W(3),p-pol +Ni(5)|Pt(3),s-pol +Ni(5)|Ti(3),s-pol +Ni(5)|W(3),s-pol + + +FIGURE S4: Impact of pump polarization. a and b: Circular pump polarization. Terahertz emission signals +even and odd in change of pump helicity for terahertz emission polarized along the p-direction (a) and s- +direction (b). c and d: Linear pump polarization. Terahertz emission signals even and odd in change of +pump polarization from s to p polarized for terahertz emission polarized along the p-direction (c) and s- +direction (d). Samples are magnetized along the s-direction. Film thicknesses in nanometers are given as +numerals in parenthesis. + +-2 +-1 +0 +1 +2 +Time (ps) +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +Terahertz signal +10-7 +helicity,THz p-pol +Ni(5)|Pt(3),even +Ni(5)|Ti(3),even +Ni(5)|W(3),even +Ni(5)|Pt(3),odd +Ni(5)|Ti(3),odd +Ni(5)|W(3),odd +-2 +-1 +0 +1 +2 +Time (ps) +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +Terahertz signal +10-7 +helicity,THzs-pol +-2 +-1 +0 +1 +2 +Time (ps) +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +Terahertz signal +10-7s vs p pump,THz p-pol +Ni(5)|Pt(3),even +Ni(5)|Ti(3),even +Ni(5)|W(3),even +Ni(5)|Pt(3),odd +Ni(5)|Ti(3),odd +Ni(5)|W(3),odd +-2 +-1 +0 +1 +2 +Time (ps) +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +Terahertz signal +10-7 s vs p pump,THz s-pol +a +b +c +d + + +FIGURE S5: Front-side vs back-side pump geometry. a Samples pumped from the front side. b Samples +pumped from the back side. The back-side pumping is defined as the direction where the pump pulse first +traverses the substrate before exciting the sample and is the standard direction used for all measurements +throughout this work. Film thicknesses in nanometers are given as numerals in parenthesis. + + +FIGURE S6: Terahertz emission signals for Py based samples as shown in Fig. 2a in addition to terahertz +emission signals from thicker Ti and W layers on Py. Film thicknesses in nanometers are given as numerals +in parenthesis. + + +b +a +b +a + +front-side +back-side +X10-6 +X10-6 +Ni(5) +6 +6 +Ni(5)/Pt(3)/5 +5 +5 +Ni(5)/Ti(3) +4 +4 +Ni(5)IPt(3)/5 +Terahertz signal +3 +Y +erahertz +2 +Ni(5)/W(3) +Ni(5)IPt(3)/5 +0 +0 +-2 +.1 +0 +2 +-2 +-1 +0 +1 +2 +Time (ps) +Time (ps)X10-6 +3 +Py(5)/Pt(3) /3 +Py(5)/Ti(3) +2 +Py(5)/W(3) +Py(5)ITi(20) +Terahertz signal +Py(5)IW(20) +0 +-2 +-3 +0 +2 +1 +Time (ps)Py(5)IPt(3) +Py(5)/Ti(3) +Norm. terahertz signal +Py(5)/W(3) +0.5 +-0.5 +0 +2 +1 +Time +(ps) +FIGURE S7: Ni|Pt vs Ni|Ti. Film thicknesses in nanometers are given as numerals in parenthesis. Note the +rescaling of the Ni|Pt sample waveform. + + +FIGURE S8: Calculated pump-light gradient in Ni for Ni(5)|Ti(20) and Ni(5)|W(20) samples, which are the +thickest samples measured. However, even in these thickest samples, the pump-light gradient is minor. +The calculation is based on a transfer matrix formalism [57]. Film thicknesses in nanometers are given as +numerals in parenthesis. + +0 +1 +2 +3 +4 +Time (ps) +-3 +-2 +-1 +0 +1 +2 +3 +4 +Terahertz signal +10-7 +Ni(5)|Pt(3)/5 +Ni(5)|Ti(3) +Ni(5)|Ti(20) +0 +2 +4 +Thickness (nm) +0 +0.05 +0.1 +0.15 +0.2 +0.25 +Rel. pump light intensity +Ni(5)|Ti(20) +Ni(5)|W(20) + + +FIGURE S9: Pump fluence dependencies. a Fluence dependencies of Ni capped with Pt, W or Ti. The data +was contracted by taking the root mean square (RMS) of the time-domain traces. b-f Normalized THz +emission signals for different pump fluences. Film thicknesses in nanometers are given as numerals in +parenthesis. + + +b +a +d +c +f +e + +X10-6 +10 +RMS of terahertz signal +Ni(5)IPt(3)/3 +Ni(5)ITi(3) +8 +Ni(5)IW(3) +6 +4 +2 +0 +0.05 +0.1 +Incident fluence +(mJ/cm²Ni(5)[Pt(3) +0.25 +0.5 +0.75 +Norm. terahertz signal +0.5 +1 +-0.5 +-0.5 +0 +0.5 +Time +(psNi(5)/Ti(3) +Norm. terahertz signal +0.5 +0 +-0.5 +-1.5 +-0.5 +0 +0.5 +7 +Time +(psNi(5)/Ti(20) +0.5 +terahertz signal +-0.5 +Norm. t +-1.5 +-0.5 +0 +0.5 +Time +e (ps)Ni(5)/W(3) +0.5 +Norm. terahertz signal +-0.5 +-1.5 +-0.5 +0 +0.5 +Time +(psNi(5)/W(20) +0.5 +Norm. terahertz signal +-0.5 +-1.5 +-0.5 +0 +0.5 +Time +e (ps)FIGURE S10: Impact of cupper inter- and capping layers. a Reference samples without Cu b Samples with +Cu intermediate layer c Samples with Cu capping layer. Film thicknesses in nanometers are given as +numerals in parenthesis. + + + +a +b +c + +X10-6 +2 +Ni(5)/W(3)/Cu(2 +Ni(5)IPt(3)ICu(2) +1.5 +Ni(5)/Ti(3)ICu(2) +Ni(5)ICu(2) +Terahertz signal +0.5 +0 +-0.5 +-2 +0 +2 +Time (ps)X10-6 +1.5 +Ni(5)/Cu(2)W(3) +Ni(5)ICu(2)IPt(3) +Ni(5)ICu(2)/Ti(3) +Ni(5)ICu(2) +Terahertz signal +0.5 +0 +-0.5 +-2 +0 +2 +Time (ps)X10-6 +Ni(5)/W(3) +Ni(5)IPt(3) +3 +Ni(5)ITi(3) +Terahertz signal +2 +-2 +-2 +-1 +0 +2 +Time (ps) +Sample +Absorptance +Absorbed fluence +in the FM layer +(mJ/cm2) +Absorbed +fluence in the +PM layer +(mJ/cm2) +Conductivity (1e6 +S/m) +Glass| Ti(50) +- +- +- +1.6 +Glass| Ni(5)|W(20) +0.52 +0.06 +0.20 +5.1 +Glass| Ni(5)|Pt(3) +0.63 +0.25 +0.06 +3.6 +Glass| Ni(5)|Ti(3) +0.58 +0.19 +0.10 +2.2 +Glass| Ni(5)|W(3) +0.58 +0.19 +0.10 +2.1 +Glass| Ni(5)|Ti(20) +0.51 +0.06 +0.20 +1.6 +Glass| Ni(5) +0.51 +0.25 +- +1.7 +Glass| Py(5)|W(3) +- +- +- +2.2 +Glass| Py(5)|Ti(3) +- +- +- +1.5 +Glass| Py(5)|Pt(3) +- +- +- +2.5 +Glass| Py(5)|W(20) +- +- +- +5.3 +Glass| Py(5) +- +- +- +2.4 +Glass| Py(5)|Ti(20) +- +- +- +1.2 +Glass| Ni(5)|Ti(3)|Cu(2) +0.53 +- +- +3.1 +Glass| Ni(5)|Pt(3)|Cu(2) +0.54 +- +- +4.2 +Glass| Ni(5)|W(3)|Cu(2) +0.58 +- +- +4.0 +Glass| Ni(5)|Cu(2) +0.52 +- +- +4.5 +Glass| Ni(5)|Cu(2)|Ti(3) +0.56 +- +- +3.4 +Glass| Ni(5)|Cu(2)|Pt(3) +0.54 +- +- +3.7 +Glass| Ni(5)|Cu(2)|W(3) +0.57 +- +- +3.4 +Glass| Ni(5)|W(15) +0.54 +0.07 +0.19 +4.7 +Glass| Ni(5)|W(10) +0.57 +0.10 +0.18 +4.2 +Glass| Ni(5)|W(5) +0.63 +0.16 +0.15 +3.6 +Glass| Ni(5)|W(2) +0.60 +0.22 +0.08 +2.9 +Si| Ni(5)|W(3) +- +- +- +2.9 +Si| Ni(5)|Ti(20) +- +- +- +1.6 +Si| Ti(50) +- +- +- +1.5 +Si| Ni(5)|Pt(3) +- +- +- +3.4 +Si| Ni(5)|W(20) +- +- +- +4.3 +Si| Ni(5) +- +- +- +3.3 +Si| Ni(5)|Ti(3) +- +- +- +2.3 +Table S1. Optical properties of all studied samples. To obtain the absorbed fluence in the FM and PM layer, we +assume imaginary parts of the dielectric constants at 800 nm of 22.07 for Ni, 9.31 for Pt, 19.41 for Ti and 19.71 for W +[56]. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Germany 2Fritz Haber Institut der Max-Planck-Gesellschaft,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 14195 Berlin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Germany 3Forschungszentrum Jülich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 52425 Jülich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Germany 4Department of Applied Physics and Physico-Informatics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Keio University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Yokohama 223-8522,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Japan 5Johannes Gutenberg-Universität Mainz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 55099 Mainz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Germany 6Keio Institute of Pure and Applied Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Keio University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Yokohama 223-8522,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Japan 7Center for Spintronics Research Network,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Keio University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Yokohama 223-8522,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Japan Abstract The emerging field of orbitronics exploits the electron orbital momentum 𝐿,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' which may allow magnetic- information transfer with significantly higher density over longer distances in more materials than possible with spin-polarized electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' However, direct experimental observation of 𝐿 currents, their extended propagation lengths and their conversion into charge currents has remained challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Here, we optically trigger ultrafast angular-momentum transport in Ni|W|SiO2 thin-film stacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' The resulting terahertz charge- current bursts exhibit a marked delay and width that grow linearly with W thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' We consistently ascribe these observations to a ballistic 𝐿 current from Ni through W with giant decay length (∼ 80 nm) and slow velocity (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='1 nm/fs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' At the W/SiO2 interface, the 𝐿 flow is converted into a charge current by the inverse orbital Rashba-Edelstein effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Our findings establish orbitronic materials with long-distance ballistic 𝐿 transport as possible candidates for future ultrafast devices and an approach to discriminate Hall- and Rashba-Edelstein-like conversion processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Introduction Spintronics research aims at utilizing the flow of spin angular momentum carried by electrons to transport information and eventually manipulate magnetic order [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Actually, electrons have two distinct channels of angular momentum: the electron spin 𝑆 and orbital angular momentum 𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' While 𝑆 is successfully exploited in the field of spintronics to transport information by spin currents and to convert the latter into detectable charge currents by spin-to-charge conversion (S2C) [2], 𝐿 has only recently gained attention in the field of orbitronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' To make this fascinating concept compatible and competitive with conventional electronics [3, 4], the speed of spin-orbitronic functionalities needs to be scalable to terahertz (THz) rates [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' A first key advantage of 𝐿 over 𝑆 is that it can assume arbitrarily high values for one electron, which is interesting for efficient manipulation of future orbitronic devices [1, 6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Second, 𝐿-to-charge conversion (L2C) does not rely on spin-orbit interaction (SOI), which opens the orbitronic workbench to abundant light metals [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Third, 𝐿-currents are predicted to propagate over increased lengths reaching almost 100 nm [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Finally, 𝐿-induced torques should have a starkly different behavior compared to 𝑆-induced torques [10-14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Recent studies provided strong indications of 𝐿 transport and charge-to-𝐿-current conversion by the orbital Hall effect (OHE) in a thin layer of a paramagnetic material (PM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' The 𝑆 or 𝐿 accumulation resulting from an in-plane charge current was interrogated by magnetooptic imaging [8] or by measuring the torque it exerted on the magnetization of an adjacent thin-film ferromagnetic material (FM) [1, 9-24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' The FM was chosen to be either susceptible to 𝑆 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=', Ni81Fe19, CoFeB) or 𝐿 accumulation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=', Ni).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Unfortunately, it remains experimentally challenging to measure 𝐿 curents by L2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' First, it is difficult to distinguish L2C by the OHE from L2C by an orbital Rashba-Edelstein effect (OREE) because both phenomena obey identical macroscopic symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Second and for the same reason, OHE and OREE are difficult to separate from their S2C counterparts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=', from the spin Hall effect (SHE) and the spin-based Rashba- Edelstein effect (SREE) [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Previous work, however, indicates different spatial propagation and relaxation dynamics of 𝑆 and 𝐿 currents [9-11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Therefore, an experimental approach such as THz emission spectroscopy [26, 27], which monitors currents with femtosecond resolution, is perfectly suited to access the possibly different ultrafast 𝐿/𝑆 propagation and conversion dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Here, we study ultrafast signatures of 𝑆 and 𝐿 transport from a FM into a PM that is launched by exciting FM|PM stacks with a femtosecond laser pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' L2C and S2C in the PM is measured by monitoring the emitted THz pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Upon changing the FM from Ni to Ni81Fe19 (Py) and interfacing them with the PMs Pt, Ti and W, we find the same characteristic sign changes in the emitted THz pulse as in previous magnetotransport studies [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Consequently, we interpret our observations as signatures of ultrafast L2C and S2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Remarkably, the emitted THz field from Ni|W is strongly time-delayed and low-pass-filtered compared to that from Ni|Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' The bandwidth and amplitude of the underlying charge-current burst decreases with W thickness, whereas its delay increases linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' We assign this observation to long-distance ballistic 𝐿 transport in W, which has a more than 10 times larger relaxation length than 𝑆 transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Specifically, our data suggest a dominant contribution to L2C through the inverse OREE (IOREE) at the W/SiO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Interestingly, this effect is absent in Ni|Ti and attributed to a dominant bulk L2C by the inverse OHE (IOHE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Our results may help establish an ultrafast experimental and theoretical methodology to extract the propagation dynamics of 𝐿 currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' FIGURE 1: Launching and detecting terahertz 𝑺 and 𝑳 currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Upon ultrafast laser excitation of the FM, the FM magnetization 𝐌 is quenched, leading to 𝑆 accumulation 𝜇�, 𝐿 accumulation 𝜇� and the injection of spin and orbital currents 𝑗� and 𝑗�, respectively, into the PM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Various bulk and interfacial L2C and S2C processes generate an ultrafast in-plane charge current 𝑗� that radiates a THz pulse with electric field 𝐸 vs time 𝑡 directly behind the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Conceptual background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Our approach is guided by the idea that 𝐿 currents obey the same phenomenology as 𝑆 currents, whereas 𝐿 transport is expected to have comparatively different spatiotemporal dynamics on ultrashort time and length scales [1, 9-11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' As schematically depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 1, a femtosecond optical pump pulse excites a FM|PM stack and triggers ultrafast 𝑆 and 𝐿 currents with density 𝑗� and 𝑗�, respectively, from FM to PM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' S2C and L2C result in ultrafast in-plane charge currents acting as a sources of a THz electromagnetic pulse [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' The resulting THz electric-field amplitude 𝐸(𝑡) directly behind the sample is proportional to the sheet charge current 𝐼�(𝑡), which reads Femtosecond THz pulse heating pulse js E S2C us ee UL L2C iL M jc FM PM Z𝐸(𝑡) ∝ 𝐼�(𝑡) = � d𝑧 [θ���(𝑧)𝑗�(𝑧, 𝑡) + θ���(𝑧)𝑗�(𝑧, 𝑡)] ������� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' (1) Here, θ���(𝑧) and θ���(𝑧) describe the local efficiency of instantaneous L2C and S2C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' They include microscopic mechanisms like the inverse SHE (ISHE) or IOHE [27, 29], which occur in the bulk, or the inverse SREE and IOREE, which require regions of broken inversion symmetry such as interfaces [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' To understand the emergence of 𝑗� and 𝑗�, we note that sudden laser heating of the FM induces 𝑆 and 𝐿 accumulations, 𝜇� and 𝜇� , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' The spin accumulation 𝜇� is proportional to the excess magnetization, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=', the difference between the instantaneous magnetization and the equilibrium magnetization that would be attained at the instantaneous electron temperature [32-35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Consequently, the FM releases 𝑆 at a rate proportional to 𝜇�, by transferring 𝑆 to both the crystal lattice and the PM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Recent studies on single-element FMs showed that the 𝑆- and 𝐿-type magnetizations exhibit very similar ultrafast time evolution following laser excitation [36-38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Therefore, we expect a very similar time evolution of 𝜇� and 𝜇�, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=', 𝜇�(𝑡) ∝ 𝜇�(𝑡), where their amplitudes depend on details of the electronic structure [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Despite this common origin of 𝑆 and 𝐿 currents, the relation between 𝑗�(𝑧, 𝑡) and 𝑗�(𝑧, 𝑡) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 1) can be highly nontrivial as 𝑆 and 𝐿 may propagate differently through the FM/NM interface and the NM bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' ( 1 ) does not account for contributions due to magnetic dipole radiation of the time-dependent magnetization and of photocurrents even in magnetic order, because both components can be discriminated experimentally [32, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Experiment details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' We study thin film FM|PM samples, where the two FMs Py and Ni are chosen for their high efficiency in generating 𝑆 and 𝐿 currents, respectively [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' The PMs are chosen to have a strong ISHE (Pt, W) and IOHE (W, Ti) response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' The reported signs for the ISHE are opposite for Pt vs W with a vanishing ISHE in Ti, but the expected IOHE signs are the same for all three PMs [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' The studied FM|PM stacks have thicknesses of a few nanometers deposited onto 500 μm thick glass substrates or 625 μm thick thermally oxidized Si substrates (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' S1 and Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' The samples are characterized by optical and THz transmission spectroscopy [41], yielding the pump absorptance, DC conductivity and Drude relaxation rate (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' In our experiment (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 1), ultrashort laser pulses (15 fs duration, 800 nm center wavelength, 80 MHz repetition rate, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='9 nJ pulse energy, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='2 mJ/cm2 incident fluence) derived from a Ti:sapphire oscillator excite the FM|PM samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' We record the emitted THz radiation by electrooptic sampling in a 1 mm or 10 µm thick ZnTe(110) or a 250 μm thick GaP(110) electro-optic crystal [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' The resulting THz emission signal 𝑆(𝐌, 𝑡) vs time 𝑡 is proportional to the THz electric-field waveform 𝐸 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 1) convoluted with a setup-response function [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' The presented data is low-pass filtered by convolution with a Gaussian function with a full width at half maximum of about 80 fs for better visibility unless noted otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' All experiments are performed under ambient conditions unless stated otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' We apply an in-plane magnetic field of about 10 mT to the sample and monitor the THz field component perpendicular to the sample magnetization 𝐌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' The component parallel to 𝐌 is found to be minor (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Measurements with linearly and circularly polarized pump pulses reveal a negligible impact of the pump polarization on the THz emission (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' To isolate magnetic signals, we reverse 𝐌 and focus on the odd-in- 𝐌 THz signal 𝑆(𝑡) = [𝑆(+𝐌, t) − 𝑆(−𝐌, t)] 2 ⁄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Even-in-𝐌 signal components are minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' As expected from a transport scenario, further experiments, in which the samples are reversed, reveal a dominant structural-inversion-asymmetry (SIA) character of the emitted THz signals compared to minor contributions unrelated to SIA, which most likely arise from magnetic-dipole radiation due to ultrafast demagnetization (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' S5) [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' FIGURE 2: Terahertz raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' THz emission signals 𝑆(𝑡) from FM|PM stacks with a FM=Py and b FM=Ni capped with PM=Pt, W or Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Note the rescaling of the Pt-based sample signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Film thicknesses in nanometers are given as numerals in parenthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' As THz detector, a 1 mm ZnTe(110) crystal was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Results FM=Py.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Figure 2a shows THz emission signals 𝑆 from Py|PM samples with PM=Pt, W, Ti, where the time-axis origin is the same for all signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' All three waveforms have identical shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Minor differences in the shape of 𝑆��|�� vs 𝑆��|�� are attributed to contributions unrelated to SIA (see above and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' The relative signal magnitudes as well as the opposite polarities for PM=Pt and W are consistent with previous reports of ISHE-dominated THz emitters [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' The polarity of the signal from Py|Ti is the same as from Py|Pt and consistent with the calculations and measurements that found the same sign of the ISHE in Pt and the IOHE in Ti [8, 27, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' However, the Py|Ti signal has a significantly smaller amplitude than the Py|Pt signal even though Ti has a sizeable L2C efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' We ascribe this observation to a small amplitude of the 𝐿 current injected into Ti, consistent with the small 𝐿 component of the Py magnetization [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' To summarize, for Py|PM, our THz signals are consistent with the notion that we predominantly observe transport of 𝑆 and 𝐿 into the PM bulk and its conversion into a charge current through the ISHE and the IOHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' A possible Rashba-type L2C or S2C, or skew-scattering at the FM/PM interface [44] may make an additional yet relatively small contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' FM=Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' When the FM=Py is replaced by Ni, the signal polarity remains the same for Pt and Ti, and the two waveforms exhibit identical dynamics (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 2b and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' S7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' In stark contrast, however, the signal polarity for Ni|W reverses, the waveform is less symmetric, and its maximum is time-shifted relative to Py|W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' This striking observation indicates that Py|W and Ni|W show competing THz-generation mechanisms, the a b X10-6 3 Ni(5)IPt(3) /3 Ni(5)ITi(3) 2 Ni(5)/W(3) Terahertz signal 2 3 0 1 2 Time (ps)3 Py(5)/Pt(3) /3 Py(5)/Ti(3) 2 Py(5)/W(3) Terahertz signal 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='3 0 2 1 Time (ps)dominance of which depends sensitively on the FM material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' To gain more insight into the different dynamics in Ni|W, we next vary the W thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' FIGURE 3: Impact of W thickness in Ni|W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' THz emission signals for Ni|W samples with varying W thickness normalized to the absorbed pump- pulse fraction in the Ni layer and to the sample impedance (see Methods and Table S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Note the rescaling of the reference signal from Ni|Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Film thicknesses in nanometers are given as numerals in parenthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' A 250 µm GaP(110) crystal was used as THz detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Impact of W thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Figure 3 shows THz emission signals from Ni|W(𝑑�) for various 𝑑� and from a Ni|Pt reference sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Consistent with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 2b, we see a clear trend with increasing W thickness relative to Ni|Pt: The THz signal amplitude has a reversed sign, reduces with increasing 𝑑� and undergoes a significant reshaping from asymmetric (Ni|Pt) to more symmetric (Ni|W) around the signal maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Interestingly, 𝑑� = 2 nm is already sufficient to induce a shift of the maximum of the THz signal by about 100 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' We emphasize that the changes in THz-signal dynamics solely originate from changing the PM thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Therefore, the FM is not primarily responsible for the signal-dynamics changes and, thus, considered as an PM-independent 𝑆 and 𝐿 injector in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' FIGURE 4: Ultrafast charge currents in Ni|W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' a Charge sheet currents in Ni|W for various W thicknesses 𝑑� as extracted from the data of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' The feature at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='9 ps is a remainder of a THz-field reflection echo in the 10 µm ZnTe electro-optic detection crystal (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Film thicknesses in nanometers are given as numerals in parenthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Note the rescaling of the Pt-based sample signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' The apparent signal delays and amplitudes are highlighted by a circular marker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' b Extracted time delay with a straight line as a guide to the eye, c relative amplitude at the delay marked in panel a, d temporal width at half maximum, and e integrated charge current between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='2 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='9 ps vs 𝑑� from the data in panel a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Error bars are estimated for panels c a b c d e X10-9 5 Ni(5)/Pt(3) /6 Ni(5)/W(2) 4 Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' terahertz signal Ni(5)/W(3) Ni(5)/W(5) 3 Ni(5)/W(10) Ni(5)/W(15) 2 Ni(5)/W(20) 0 7 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 Time (ps)Ni(5)/Pt(3) /6 14 Ni(5)/W(2) 12 Ni(5)/W(3) Ni(5)/W(5) 2 per abs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' fluence in Ni (J/m 10 Ni(5)/W(10) Ni(5)/W(15) 8 Ni(5)IW(20) delay 6 4 2 0 2 4 6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 Time (ps)80 Ampl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' (norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Delay (fs) 60 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 20 0 0 0 10 20 10 20 0 340 320 Area (norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=') (fs) 300 280 Width 260 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 240 220 0 0 10 20 10 20 0 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' (nm) d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' (nm)and e from the signal-to-noise ratio in panel a, for panels b and d as 10% of the delay and width, respectively, and in all panels b-e as ± 1 nm for 𝑑�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Current dynamics in Ni|W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' To obtain a sample-intrinsic measurement of the L2C dynamics, we extract the sheet charge current 𝐼�(𝑡) flowing in Ni|W (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' (1)) normalized to the absorbed laser fluence in the Ni layer (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' This procedure eliminates any impact of sample exchange on pump-pulse absorption efficiency, sample impedance or setup response function (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Figure 4a presents 𝐼�(𝑡) in Ni|W with a resolution of 50 fs for various W thicknesses 𝑑�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' The 𝐼�(𝑡) traces have striking features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' (i) They have opposite polarity relative to Py|W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' (ii) Their maximum shifts by delays Δ𝑡��� ∝ 𝑑� at a rate Δ𝑡��� 𝑑� ⁄ ≈ 4 fs/nm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 4b), implying a velocity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='25 nm/fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' (iii) The 𝐼� peak value decreases approximately linearly with 𝑑� to about 50% after 20 nm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 4c), indicating attenuation and dispersion upon propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' (iv) The 𝐼� width increases linearly at a rate of ≈ 8 fs/nm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 4d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' (v) The time-integrated current ∫ d𝑡 𝐼�(𝑡) is only weakly dependent on 𝑑� with a decreasing trend, thereby indicating an extremely large relaxation length >20 nm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 4e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Features (i) and (iii) imply that 𝐼�(𝑡) cannot arise from 𝑆 transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Otherwise, an opposite signal polarity would result because S2C in W is dominated by the ISHE [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' In addition, 𝑆 currents in W relax over distances much smaller than 20 nm [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Our data, therefore, strongly indicate that 𝐿 transport plus L2C is the dominant origin of the THz charge current in Ni|W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Second, features (ii) and (iv) are a hallmark of a signal arising from ballistic transport of a pulse that is detected in an arrival layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' In this picture, the increase of the 𝐼�(𝑡) width with 𝑑� arises from velocity dispersion along the 𝑧-direction of the particles that make up the pulse (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Feature (v) implies a minor L2C in the W bulk because it would otherwise result in an integrated charge current ∫ d𝑡 𝐼�(𝑡) that increases monotonically with 𝑑�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Model: 𝑳 current and IOREE in Ni|W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' The preceding discussion suggests the following transport scenario in Ni|W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Upon excitation of the Ni layer, a transient 𝑆 and 𝐿 accumulation is induced (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Their dynamics are expected to be very similar (see above) [36-38] and monitored well by the ISHE charge current in Ni|Pt (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Finally, L2C is dominated in regions close to the W/SiO2 interface (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Such interfacial L2C can be very efficient, as argued in previous works [7, 24, 45-49], which, however, lacked the required femtosecond resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' This scenario can explain all charge-current features (i)-(v) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 4) and is consistent the above experimental findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' As the 𝑗� pulse propagates predominantly ballistically, its arrival in the W/SiO2 L2C region is delayed by a time Δ𝑡��� ∝ 𝑑�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' The velocity of the 𝑗�-pulse peak (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='1 nm/fs) is consistent with that of 𝐿-carrying d-band states of W [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' During propagation through PM=W, the 𝑗� pulse disperses due to different electron velocities along the surface normal (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 5a) and attenuation with a typical relaxation length >20 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' To quantitatively model the charge-current dynamics in Ni|W (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 3), we assume ballistic 𝐿 transport with a characteristic decay length 𝜆� in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' The 𝑗� arriving at the W/SiO2 interface is obtained by summing over all Fermi-surface states with positive group velocity along the 𝑧-axis (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 5a and Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' The resulting 𝑗� at the W/SiO2 interface induced by a fictitious 𝛿(𝑡)-like 𝐿 accumulation in Ni reads 𝑟(𝑡) ∝ 𝑑� 𝑡� Θ(𝑣�𝑡 − 𝑑�)e���� �� ⁄ , (2) where Θ is the Heaviside step function, and 𝑣� is the Fermi velocity of the 𝐿-polarized electrons in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' We convolute 𝑟(𝑡) with 𝜇�(𝑡) ∝ 𝜇�(𝑡), which is given by the charge current measured in Ni|Pt (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Our modeled 𝐼�(𝑡) curves (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 5c) reproduce the measured charge currents in Ni|W (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 4a) semiquantitatively for the choice 𝜆� = 80 nm and 𝑣� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='14 nm/fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' These values are in good agreement with the estimates obtained above (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 4b-e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' FIGURE 5: Simulated ultrafast inverse orbital Rashba-Edelstein effect in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' a Schematic of the suggested scenario for 𝐿 transport and L2C by the IOREE in Ni|W showing the different wave vector contributions of the 𝐿 currents inside the W layer driven by the magnetization quenching in the Ni layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Upon reaching the W back surface, the orbital currents 𝐣� are converted into a transverse charge current 𝐣� by the inverse orbital Rashba- Edelstein effect (IOREE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' In the experiment, many of the point-like sources of orbital currents are superimposed along the FM/PM interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' b Qualitative response functions 𝑟(𝑡) to a fictitious delta-like 𝑗� pulse injected at the Ni/W interface for different W layer thicknesses 𝑑�, where 𝑑�� < 𝑑�� < 𝑑�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' c Simulated IOREE charge currents 𝐼�(𝑡) obtained by convoluting 𝑟(𝑡) [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' (2), panel b] with the 𝐼�(𝑡) of the Ni|Pt reference sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Inputs for the simulation are a ballistic 𝐿 velocity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='14 fs/nm, an 𝐿 decay length of 𝜆� = 80 nm and a global scaling factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' To summarize, the THz charge currents in Ni|W (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 4) can be considered as signatures of 𝐿 currents injected into W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' The charge-current generation [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' (1)] is dominated by an extremely long-range 𝑗� and L2C at the W/SiO2 interface, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=', by θ��� at 𝑧 = 𝑑�� + 𝑑��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Such long-range 𝐿 transport is a unique feature of orbitronic materials, and first indications for it were found previously in Ti [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Note that within our interpretation, the sign of 𝑆��|� agrees coincidentally with the calculated sign of θ��� for the IOHE in W [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Discussion Our interpretation neglects other possible contributions to the THz charge current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' First, the inverse Faraday effect as a source of 𝑆 and 𝐿 currents can be ruled out by the pump-polarization independence (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Second, for the 𝑆 channel, a dominant Seebeck-type contribution due to an electronic temperature difference Δ𝑇��/�� across the Ni/PM interface is neglected as found in previous studies [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' For the 𝐿 channel, we estimate Δ𝑇��/�� right after pump pulse absorption (see Methods) and find Δ𝑇��/�� ∼ +400 K and Δ𝑇��/�� ∼ −100 K in Ni|Ti and Ni|W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' The observed THz-emission signals, in contrast, show the same sign from all three samples (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Therefore, interfacial electronic temperature differences are a minor a b c 14 Ni(5)/Pt(3) /6 12 Ni(5)/W(2) Ni(5)/W(3) 10 Ni(5)/W(5) Ni(5)/W(10) 8 Ni(5)/W(15) Ni(5)/W(20) 6 4 2 per abs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 0 2 4 6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 1 Time (ps)W Ni eee μL IOREE GG jL dw Z r(t) r (0) 0 t t(dw1) t(dw2) t(dw3driving force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Additional pump-propagation simulations show that, even for the thickest samples, pump- intensity gradients in the FM and PM bulk are relatively small (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' S8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Third, regarding transport in W, we consider dominant angular-momentum transport by magnons unlikely because W is not magnetically ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' An outstandingly long propagation of 𝑆 transport is ruled out, too, because the Drude scattering times for all studied samples are substantially shorter (<50 fs, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' S2) than the peak delays of 𝐼�(𝑡) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Fourth, even though our data imply a dominant IOREE contribution to charge-current generation (see above), the positive shoulder-like feature at around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='1 ps for 𝑑� ≤ 3 nm in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 4a may indicate a small contribution of bulk L2C, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=', the IOHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' A 𝐿-to-𝑆 conversion plus ISHE in the PM [23] might contribute but is considered negligible here given the good agreement of our experimental data (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 4) and the IOREE scenario (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' The dominance of an 𝐿-type angular momentum current in Ni|W highlights the role of Ni as an 𝐿 source and indicates that the Ni/W interface may transmit 𝐿 currents more efficiently than 𝑆 currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' We finally turn to other interesting aspects of our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' A more detailed comparison of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 2a and 2b reveals further changes in amplitude between Ni- and Py-based samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' The pronounced amplitude changes for PM=W or Pt when changing FM=Py to Ni are related to the intricate interplay of all parameters in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' (1) in addition to changes in the relative amplitudes of 𝜇� and 𝜇� and interface transmission coefficients for 𝑗� and 𝑗�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Therefore, further experiments for a robust separation of 𝑆 and 𝐿 transport are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' We further find that the THz signals from the Ni-based samples increase linearly with pump fluence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Slight sublinearities at the highest fluences do not alter the THz emission dynamics (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' S9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' We emphasize that samples deposited on Si rather than glass substrates show very similar THz emission characteristics (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' These observations demonstrate the robustness of the observed effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' When adding a Cu layer on top of the Ni|W sample, we find almost no change in the THz emission signal (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' S10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Future studies are needed to elaborate the exact character of the IOREE for different interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Interestingly, a Cu intermediate layer in Ni|Cu|W slightly modifies the amplitude and dynamics of the THz signal, suggesting that Cu does not block 𝐿 transport strongly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Regarding earlier reports of different THz emission dynamics in Fe|Au and Fe|Ru samples [29], we note that a possible IOREE in Ru can, in hindsight, not be excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' A dominant IOREE might also explain the seemingly strong dependence of the Fe|Ru THz emission dynamics on the exact growth details [29, 51, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' In conclusion, we observe THz-emission signals from optically excited Ni|W stacks that are consistent with an ultrafast injection of 𝐿 currents into W and long-distance ballistic transport through W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Remarkably, we find strong indications for the occurrence of the IOREE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' This result can be considered as time-domain evidence of the long-range nature of orbital currents and IOREE in typical metals such as W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Our study highlights the power of broadband THz emission spectroscopy in disentangling of spin and orbital transport and Hall- and Rashba-Edelstein-like angular-momentum conversion processes through ultrafast time-of-flight experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Future studies may aim at exploiting L2C physics in materials with nontrivial topology, in which the OHE is predicted to be drastically enhanced close to Weyl points [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Our study opens the field of THz orbitronics whose distinct dynamical features allow for tailoring ultrafast spin and orbital currents on femtosecond time scales by a targeted material, thickness and interface engineering of multilayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Methods Current extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' To extract the in-plane sheet current flowing inside the sample from the measured THz signal 𝑆, we first measure our setup response function ℎ by having a reference electro-optic emitter (50 μm GaP on a 500 μm glass substrate) at the same position as the sample, which yields a reference THz signal 𝑆���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' By calculating the emitted THz electric field from that reference emitter 𝐸���, ℎ is determined by solving the convolution 𝑆��� = (ℎ ∗ 𝐸���)(𝑡) for ℎ [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Further measured inputs for this calculation are the excitation spot size with a full width at half maximum of 22 μm, the excitation pulse energy of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='9 nJ and a transform limited pump pulse with a spectrum centered at 800 nm and 110 nm full width at half maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' We perform the deconvolution directly in the time domain by recasting it as a matrix equation [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Next, the electric field 𝐸 directly behind the sample is obtained from the recorded THz signal 𝑆 with the help of the derived function ℎ by solving again the similar equation 𝑆 = (ℎ ∗ 𝐸)(𝑡) for 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Finally, the sheet charge current (see Table S1) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 3 is derived from a generalized Ohm’s law [28] that reads 𝐸(𝜔) = 𝑒𝑍(𝜔)𝐼�(𝜔) , where −𝑒 is the electron charge and the sample impedance is given by 𝑍(𝜔) = 𝑍� [1 + 𝑛��� + 𝑍�𝑑𝜎(𝜔)] ⁄ with the free space impedance 𝑍�, the metal-stack thickness 𝑑 and the measured mean sample conductivity 𝜎 (see Table S1) that we assume to be frequency independent due to the large Drude scattering rate (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' To enable comparison of THz currents from different samples, we normalize 𝐼� by the absorbed fluence in the FM layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' The data shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 3 was obtained in a dry-air atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Sample preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' The FM|PM samples (FM = Ni and Py, PM = Pt, Ti, Cu and W) were fabricated on glass substrates of 500 μm thickness or thermally oxidized Si substrates of 625 μm thickness by radio frequency (RF) magnetron sputtering under 6N-purity-Ar atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' The sample structure and thickness are described in Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' For the sputtering, the base pressure in the chamber was better than 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='0 × 10-7 Pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' To avoid oxidation, 4-nm-thick SiO2 was sputtered on the surface of the films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' All sputtering processes were performed at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Estimate of electronic temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' We calculate the electronic temperatures increase upon pump-pulse absorption by Δ𝑇� = �𝑇� � + 2𝑓 𝑑𝛾 − 𝑇�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' (3) Here, 𝑇� = 300 K is the ambient temperature, 𝑓 is the absorbed fluence in the respective layer (see Table S1), 𝑑 is the layer thickness and 𝛾 is the specific electronic heat capacity that is 300 J/m3 K2 for W, 320 J/m3 K2 for Ni, 330 J/m3 K2 for Ti and 90 J/m3 K2 for Pt [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' To obtain the absorbed fluences in each layer, we note that the pump electric field is almost constant throughout the sample (see Fig S8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Therefore, local pump absorption scales solely with the imaginary part of the dielectric function at a wavelength of 800 nm, which equals 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='07 for Ni, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='31 for Pt, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='41 for Ti and 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='71 for W [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Consequently, the absorbed fluence is determined by 𝑓��/�� = 𝑓��� 𝑑��/�� Im𝜀��/�� 𝑑�� Im𝜀�� + 𝑑�� Im𝜀�� (4) with the total absorbed fluence 𝑓��� that is obtained from the absorbed pump power (see Table S1) and the beam size on the sample (see above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Model of orbital transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' To model the ballistic current in the PM, we assume that a 𝛿(𝑡)-like transient spin accumulation in the FM causes a change Δ𝑛𝐤� in the occupation of each electronic wavepacket in the PM with mean wavevector 𝐤 right behind the FM/PM interface at 𝑧 = 0� (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Subsequently, this occupation change propagates into the PM bulk according to Δ𝑛𝐤(𝑧, 𝑡) = Δ𝑛𝐤�𝛿(𝑧 − 𝑣𝐤𝑡), where 𝑣𝐤 is the 𝑧 component of the group velocity of the electronic wavepacket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Note that we here restrict ourselves to 𝐤 with nonnegative 𝑣𝐤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' The occupation change Δ𝑛𝐤(𝑧, 𝑡) is accompanied by a particle current density Δ𝑗𝐤 = 𝑣𝐤Δ𝑛𝐤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' The total pump-induced current density flowing into the depth of the PM layer is given by the sum Δ𝑗(𝑧, 𝑡) = � Δ𝑛𝐤�𝑣𝐤𝛿(𝑧 − 𝑣𝐤𝑡) 𝐤, �𝐤�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' (5) Because Δ𝑛𝐤� predominantly affects states close to the Fermi energy, the summation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' ( 5 ) is approximately proportional to an integration over the Fermi-surface parts with 𝑣𝐤 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' One obtains Δ𝑗(𝑧, 𝑡) = � d𝑣 𝑤(𝑣)𝑣 � � 𝛿(𝑧 − 𝑣𝑡), (6) where 𝑤(𝑣) is the weight of the group velocity 𝑣 along the 𝑧 axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' We assume that the Fermi surface is a sphere with isotropic occupation change Δ𝑛𝐤� and radius 𝑘�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Consequently, the integrand Δ𝑛𝐤�𝑣𝐤 d�𝐤 becomes Δ𝑛𝐤�𝑣� cos 𝜃 𝑘�d𝑘d𝜑d cos𝜃 ∝ 𝑣 d𝑣d𝑘d𝜑, where 𝑣� is the Fermi velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' In other words, all velocities from 0 to the Fermi velocity 𝑣� have equal weight, and we find Δ𝑗(𝑧, 𝑡) ∝ � d𝑣 𝑣 �� � 𝛿(𝑧 − 𝑣𝑡) = 𝑧 𝑡� Θ(𝑣�𝑡 − 𝑧), (7) for 𝑧 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Finally, we phenomenologically account for relaxation of the ballistic current with time constant 𝜏 by multiplying Δ𝑗(𝑧, 𝑡) with e��/�, which directly takes us to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Acknowledgements We thank G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Sala for fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' TSS, RR and TK acknowledge funding by the German Research Foundation (DFG) through the collaborative research center SFB TRR 227 “Ultrafast spin dynamics” (project ID 328545488, projects A05 and B02) and financial support from the Horizon 2020 Framework Programme of the European Commission under FET-Open Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 863155 (s-Nebula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' FF and YM acknowledge DFG collaborative research center SFB TRR 173/2 “Spin+X”(project ID 268565370, project A11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' KA and HH acknowledge funding by JSPS (Grant Number 22H04964 and 20J20663) and Spintronics Research Network of Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' References [1] Go, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=', D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Jo, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Lee, M.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Alexander, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Ward Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Applied Optics, 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 22: p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 1099.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' [57] Zak, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=', E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Moog, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Liu, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Bader Universal approach to magneto-optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Journal of Magnetism and Magnetic Materials, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 89: p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Supplementary Materials First, we summarize briefly the content of the Supplementary Materials before showing the corresponding data: Samples on Si show qualitatively the same THz emission waveforms for Ni with Pt, W and Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Most importantly, the strong change in W dynamics is also observed on Si (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' However, the THz waveforms of Si vs glass differ in the details, which might be related to slightly changed transport times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Drude scattering times are estimated to be <50 fs for all studied samples (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' None of the samples showed any indication of a drastically different Drude scattering time compared to all other samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Emitted THz signals are found to be linearly polarized and perpendicular to the sample magnetization (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Pump-polarization dependent studies (pump helicity and linear polarization direction) show a minor impact on the measured THz emission signal (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' We perform THz emission measurements upon reversing the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Only the pure Ni film shows a dominant contribution even in sample rotation, which we ascribe to SIA or magnetic dipole radiation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' S5) [32, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' For all Py-based bilayer samples, we find almost identical THz emission waveform shapes even for PM thicknesses of 20 nm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' For Ni|Ti samples, we find almost identical THz emission dynamics to Ni|Pt (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' S7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Currents driven by pump light gradients in thick films of Ni|W and Ni|Ti can be neglected (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' S8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' All fluence dependencies are to a good approximation linear (Fig S9) with minor sublinearities overserved for Ni|Ti and Ni|W samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Related to that, only minor changes in the THz waveform dynamics can be observed for different pumping fluences (Fig S9) Cu has only minor impact on the emitted THz waveforms (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' S10), either as a spacer layer or as a capping layer as confirmed by comparison to the same sample without Cu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' All data in the Supplementary Materials was measured with a 1 mm ZnTe(110) detection crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' FIGURE S1: Si vs glass substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' a Terahertz-emission waveforms from Ni|PM stacks on Si substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' THz waveforms for Si based samples are multiplied by -1 to account for the reversed sample orientation due to the intransparency of the Si substrate for the pump pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' a Terahertz-emission waveforms from Ni|PM stacks on glass substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Film thicknesses in nanometers are given as numerals in parenthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Note the rescaling of the Ni|Pt sample THz waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' b a Glass substrate X10-7 5 4 3 Terahertz signal 2 2 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 Time e (ps)Si substrate X10-7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 Ni(5)/Pt(3)/4 2 Ni(5)ITi(3) Ni(5)/Ti(20) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 Ni(5)IW(3) Terahertz signal Ni(5)/W(20) 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 1 1 Time (ps)FIGURE S2: Terahertz conductivities for samples on glass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Mean complex-valued terahertz conductivities obtained from terahertz transmission measurements for a Ni, b Ni|Ti, c Ni|Pt and d Ni|W samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' For the extraction, a thin film formula is applied [41] and a terahertz refractive index of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='1 for glass is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Film thicknesses in nanometers are given as numerals in parenthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' c d a b Ni(5) X106 7 6 real imag 5 Conductivity (S/m) 4 3 2 1 0 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 2 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 Freguency (THz)Ni(5)[Pt(3) X106 7 real 6 imag 5 Conductivity (S/m) 4 3 2 1 0 1 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 Freguency (THz)Ni(5)/W(3) X106 7 real 6 imag 5 Conductivity (S/m) 4 3 2 1 0 1 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 Freguency (THz)Ni(5)/Ti(3) X106 7 real 6 imag 5 Conductivity (S/m) 4 3 2 1 0 1 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 Freguency (THz) FIGURE S3: Polarization state of the THz emission signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Samples are magnetized along the s-direction and pump pulses are polarized along the p-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Film thicknesses in nanometers are given as numerals in parenthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 2 1 0 1 2 Time (ps) 2 0 2 4 6 8 10 Terahertz signal 10-7 Ni(5)|Pt(3),p-pol Ni(5)|Ti(3),p-pol Ni(5)|W(3),p-pol Ni(5)|Pt(3),s-pol Ni(5)|Ti(3),s-pol Ni(5)|W(3),s-pol FIGURE S4: Impact of pump polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' a and b: Circular pump polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Terahertz emission signals even and odd in change of pump helicity for terahertz emission polarized along the p-direction (a) and s- direction (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' c and d: Linear pump polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Terahertz emission signals even and odd in change of pump polarization from s to p polarized for terahertz emission polarized along the p-direction (c) and s- direction (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Samples are magnetized along the s-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Film thicknesses in nanometers are given as numerals in parenthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 2 1 0 1 2 Time (ps) 0 2 4 6 8 10 12 14 16 18 Terahertz signal 10-7 helicity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='THz p-pol Ni(5)|Pt(3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='even Ni(5)|Ti(3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='even Ni(5)|W(3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='even Ni(5)|Pt(3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='odd Ni(5)|Ti(3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='odd Ni(5)|W(3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='odd 2 1 0 1 2 Time (ps) 0 2 4 6 8 10 12 14 16 18 Terahertz signal 10-7 helicity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='THzs-pol 2 1 0 1 2 Time (ps) 0 2 4 6 8 10 12 14 16 18 Terahertz signal 10-7s vs p pump,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='THz p-pol Ni(5)|Pt(3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='even Ni(5)|Ti(3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='even Ni(5)|W(3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='even Ni(5)|Pt(3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='odd Ni(5)|Ti(3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='odd Ni(5)|W(3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='odd 2 1 0 1 2 Time (ps) 0 2 4 6 8 10 12 14 16 18 Terahertz signal 10-7 s vs p pump,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='THz s-pol a b c d FIGURE S5: Front-side vs back-side pump geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' a Samples pumped from the front side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' b Samples pumped from the back side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' The back-side pumping is defined as the direction where the pump pulse first traverses the substrate before exciting the sample and is the standard direction used for all measurements throughout this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Film thicknesses in nanometers are given as numerals in parenthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' FIGURE S6: Terahertz emission signals for Py based samples as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 2a in addition to terahertz emission signals from thicker Ti and W layers on Py.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Film thicknesses in nanometers are given as numerals in parenthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' b a b a front-side back-side X10-6 X10-6 Ni(5) 6 6 Ni(5)/Pt(3)/5 5 5 Ni(5)/Ti(3) 4 4 Ni(5)IPt(3)/5 Terahertz signal 3 Y erahertz 2 Ni(5)/W(3) Ni(5)IPt(3)/5 0 0 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='1 0 2 2 1 0 1 2 Time (ps) Time (ps)X10-6 3 Py(5)/Pt(3) /3 Py(5)/Ti(3) 2 Py(5)/W(3) Py(5)ITi(20) Terahertz signal Py(5)IW(20) 0 2 3 0 2 1 Time (ps)Py(5)IPt(3) Py(5)/Ti(3) Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' terahertz signal Py(5)/W(3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 0 2 1 Time (ps) FIGURE S7: Ni|Pt vs Ni|Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Film thicknesses in nanometers are given as numerals in parenthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Note the rescaling of the Ni|Pt sample waveform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' FIGURE S8: Calculated pump-light gradient in Ni for Ni(5)|Ti(20) and Ni(5)|W(20) samples, which are the thickest samples measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' However, even in these thickest samples, the pump-light gradient is minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' The calculation is based on a transfer matrix formalism [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Film thicknesses in nanometers are given as numerals in parenthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' 0 1 2 3 4 Time (ps) 3 2 1 0 1 2 3 4 Terahertz signal 10-7 Ni(5)|Pt(3)/5 Ni(5)|Ti(3) Ni(5)|Ti(20) 0 2 4 Thickness (nm) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='25 Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' pump light intensity Ni(5)|Ti(20) Ni(5)|W(20) FIGURE S9: Pump fluence dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' a Fluence dependencies of Ni capped with Pt, W or Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' The data was contracted by taking the root mean square (RMS) of the time-domain traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' b-f Normalized THz emission signals for different pump fluences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Film thicknesses in nanometers are given as numerals in parenthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' b a d c f e X10-6 10 RMS of terahertz signal Ni(5)IPt(3)/3 Ni(5)ITi(3) 8 Ni(5)IW(3) 6 4 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='1 Incident fluence (mJ/cm²Ni(5)[Pt(3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='75 Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' terahertz signal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 Time (psNi(5)/Ti(3) Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' terahertz signal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 7 Time (psNi(5)/Ti(20) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 terahertz signal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' t 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 Time e (ps)Ni(5)/W(3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' terahertz signal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 Time (psNi(5)/W(20) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' terahertz signal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 Time e (ps)FIGURE S10: Impact of cupper inter- and capping layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' a Reference samples without Cu b Samples with Cu intermediate layer c Samples with Cu capping layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Film thicknesses in nanometers are given as numerals in parenthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' a b c X10-6 2 Ni(5)/W(3)/Cu(2 Ni(5)IPt(3)ICu(2) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 Ni(5)/Ti(3)ICu(2) Ni(5)ICu(2) Terahertz signal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 2 0 2 Time (ps)X10-6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 Ni(5)/Cu(2)W(3) Ni(5)ICu(2)IPt(3) Ni(5)ICu(2)/Ti(3) Ni(5)ICu(2) Terahertz signal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 2 0 2 Time (ps)X10-6 Ni(5)/W(3) Ni(5)IPt(3) 3 Ni(5)ITi(3) Terahertz signal 2 2 2 1 0 2 Time (ps) Sample Absorptance Absorbed fluence in the FM layer (mJ/cm2) Absorbed fluence in the PM layer (mJ/cm2) Conductivity (1e6 S/m) Glass| Ti(50) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='6 Glass| Ni(5)|W(20) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='20 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='1 Glass| Ni(5)|Pt(3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='63 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='1 Glass| Ni(5)|Ti(20) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='6 Glass| Ni(5) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='7 Glass| Py(5)|W(3) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='2 Glass| Py(5)|Ti(3) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 Glass| Py(5)|Pt(3) 2.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='6 Glass| Ni(5)|W(2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='08 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='9 Si| Ni(5)|W(3) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='9 Si| Ni(5)|Ti(20) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='6 Si| Ti(50) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='5 Si| Ni(5)|Pt(3) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='4 Si| Ni(5)|W(20) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='3 Si| Ni(5) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='3 Si| Ni(5)|Ti(3) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='3 Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Optical properties of all studied samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' To obtain the absorbed fluence in the FM and PM layer, we assume imaginary parts of the dielectric constants at 800 nm of 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='07 for Ni, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='31 for Pt, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='41 for Ti and 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content='71 for W [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} +page_content=' Note that all films are additionally capped with 4 nm SiO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQf2Plc/content/2301.00747v1.pdf'} diff --git a/5dAyT4oBgHgl3EQfpfgA/vector_store/index.pkl b/5dAyT4oBgHgl3EQfpfgA/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..8f0f55b47b3250b014db407fd00877f85bc49e86 --- /dev/null +++ b/5dAyT4oBgHgl3EQfpfgA/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid 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mode 100644 index 0000000000000000000000000000000000000000..bf3e3408e546949570df92debd180c8f5aa833f6 --- /dev/null +++ b/6dAyT4oBgHgl3EQfQfZh/content/tmp_files/2301.00046v1.pdf.txt @@ -0,0 +1,783 @@ +A Bayesian treatment of the German tank problem +Cory M. Simon +School of Chemical, Biological, and Environmental Engineering. Oregon State +University. Corvallis, OR. USA. +cory.simon@oregonstate.edu +Abstract +The German tank problem has an interesting historical background and is an engaging +problem in statistical estimation for the classroom. The objective is to estimate the size +of a population of tanks inscribed with sequential serial numbers, from a random sample. +In this tutorial article, we outline the Bayesian approach to the German tank problem, +(i) whose solution assigns a probability to each tank population size, thereby quantifying +uncertainty, and (ii) which provides an opportunity to incorporate prior information and/or +beliefs about the tank population size into the solution. We illustrate with an example. +Finally, we survey other research problems that bear resemblance to the German tank +problem. +s1=15 +s2=14 +s3=3 +serial numbers +1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 +size of tank population, n +0 +10 +20 +30 +40 +probability +0.00 +0.05 +0.10 +0.15 +0.20 +prior +likelihood +posterior +1 +arXiv:2301.00046v1 [stat.OT] 30 Dec 2022 + +1 +Background +1.1 +History +To inform their military strategy during World War II (1939-1945), the Allies sought to es- +timate the rate of production of various military equipment (tanks, tires, rockets, etc.) by +Germany. Conventional methods to estimate armament production—including (i) extrapo- +lating data on prewar manufacturing capabilities, (ii) obtaining reports from secret sources, +and (iii) interrogating prisoners of war—were unreliable and/or contradictory. +In 1943, British and American economic intelligence agencies exploited a German man- +ufacturing practice in order to statistically estimate their armament production. Germany +marked their military equipment with serial numbers and codes for the date and/or place of +manufacture to handle spare parts and trace faulty/defective equipment/parts back to the +manufacturer for quality control. However, these markings on a captured sample of German +equipment provided the Allies information about Germany’s production of it. +To estimate Germany’s production of tanks, the Allies collected serial numbers on the +chassis, engines, gearboxes, and bogie wheels of samples of tanks by inspecting captured +tanks and examining captured records1. Despite lacking an exhaustive sample, the sequential +nature of2 and patterns in these samples of serial numbers enabled the Allies to estimate +Germany’s tank production—postwar, we know, much more accurately than conventional +American and British intelligence (Tab. 1). +See Ruggles and Brodie [1] for the detailed historical account of serial number analysis to +estimate German armament production during World War II. +Table 1: Monthly production of tanks by Germany. [1] +estimates +date +conventional +American +& British Intelligence +serial number analysis +German +records +June, 1940 +1000 +169 +122 +June, 1941 +1550 +244 +271 +August, 1942 +1550 +327 +342 +1Eg., captured records from tank repair depots listed serial numbers of the chassis and engine of repaired +tanks, and records from divisional headquarters listed chassis serial numbers of tanks held by a specific unit. +2Gearboxes on captured tanks, for example, were inscribed with serial numbers belonging to an unbroken +sequence. Chassis serial numbers, on the other hand, were broken into blocks to distinguish models/designs, +leaving gaps between the serial numbers assigned to them. +2 + +1.2 +The German tank problem +Simplification of the historical context to estimate German tank production via serial number +analysis [1] motivated the formulation of the textbook-friendly German tank problem [2]: +Problem statement +In the backdrop of World War II, the German military has n tanks. +Each tank is +inscribed with a unique serial number in {1, ..., n}. +As the Allies, we do not know n, but we captured (without replacement, of course) a +sample of k German tanks with inscribed serial numbers (s1, ..., sk). +s1 +s2 +· · · +sk +Assuming all tanks in the population were equally likely to be captured, our objective +is to estimate n in consideration of the data (s1, ..., sk). +In 1942, Alan Turing and Andrew Gleason discussed a variant of the German tank prob- +lem, “how to best to estimate the total number of taxicabs in a town, having seen a random +selection of their license numbers”, in a crowded restaurant in Washington DC [3,4]. Today, +with its interesting historical background [1], the German tank problem is still a suitable con- +versation topic for dinners and serves as an intellectually engaging, challenging, and enjoyable +problem to illustrate combinatorics and statistical estimation in the classroom [5–8]. +Uncertainty quantification. +Any estimate of the tank population size n from the data +(s1, ..., sk) is subject to uncertainty, since we (presumably) have not captured all of the tanks +(ie., k ̸= n, probably). Quantifying uncertainty in our estimate of the tank population size n +is important because high-stakes military decisions may be made on its basis. +Our contribution. +In this pedagogical article, we outline the Bayesian approach to the +German tank problem, (i) whose solution assigns a probability to each tank population size, +thereby quantifying uncertainty, and (ii) which provides an opportunity to incorporate prior +information and/or beliefs about the tank population size into the solution. +1.3 +Survey of previous work on the German tank problem +The frequentist approach. +Border [9] calls the German tank problem a ”weird case” in +frequentist estimation. The maximum likelihood estimator of the tank population size n is +3 + +the maximum serial number observed among the k captured tanks, m(k) := maxi∈{1,...,k} si. +This is a biased estimator, as certainly m(k) ≤ n. +Goodman [2, 10] derives the minimum-variance, unbiased point estimator of the tank +population size +ˆn = m(k) + +� +m(k) +k +− 1 +� +. +(1) +To intuit this estimator, note (i) n must be greater than or equal to m(k) and (ii) if we observe +large (small) gaps between the serial numbers (s1, ..., sk) after sorting them (incl. the gap +preceding the smallest serial number), then n is likely (unlikely) to be much greater than m(k). +The estimator of n in eqn. 1 quantifies how far beyond m(k) we should estimate the tank +population size, based on the gaps; m(k)/k − 1 is the average size of the gaps. Goodman +also derives a frequentist confidence interval for n. +Clark, Gonye, and Miller explore using simulations and linear regression to discover the +estimator in eqn. 1 [11]. +For pedagogy. +Champkin highlights the historical context of the German tank problem +as a ”great moment in statistics” [12]. Johnson lists and evaluates several intuitive point +estimators for the size of the tank population [5]. Scheaffer, Watkins, Gnanadesikan, and +Witmer [13] propose a hands-on learning activity to illustrate the German tank problem by +sampling chips, labeled with numbers from 1 to n, from a bowl. Berg [6] uses the German +tank problem as a competition in the classroom. +The Bayesian approach. +Closely related to our paper, Roberts [14], H¨ohle and Held [15], +and Linden, Dose, and Toussaint [16], and Cocco, Monasson, and Zamponi [17] provide a +Bayesian analysis of the German tank problem. They derive an analytical formula for the +mean of the posterior distribution of the tank population size under an improper, uniform +prior distribution. Andrews [18] outlines the Bayesian approach to the German tank problem +in a blog post containing code in the R language. +Generalizations/variants. +Goodman [2, 10] poses a variant of the German tank problem +where the initial serial number is not known; ie., where the n tanks are inscribed with serial +numbers {b + 1, ..., n + b} with b and n unknown. Lee and Miller generalize the German +tank problem to the settings where the serial numbers are continuous and/or lie in two +dimensions [19]. +1.4 +Overview of the Bayesian approach to the German tank problem +Under a Bayesian perspective [8,20,21], we treat the (unknown) total number of tanks as a +discrete random variable N (hence, capitalization) to model our uncertainty in it. A proba- +4 + +bility mass function of N assigns a probability to each possible tank population size n. This +probability is a measure of our degree of belief, perhaps with some basis in knowledge/data, +that the tank population size is n [22]. +Because the observed serial numbers (s1, ..., sk) provide information about the tank pop- +ulation size, the probability mass function of N differs before and after they are collected and +considered. Hence, N has a prior and posterior probability mass function. +The three inputs to a Bayesian treatment of the German tank problem are: +• the prior mass function of N, which expresses a combination of our subjective beliefs +and objective knowledge about the tank population size before we collect and consider +the sample of serial numbers. +• the data, the observed serial numbers (s1, ..., sk), viewed as realizations of random +variables owing to the stochasticity of tank-capturing. +• the likelihood function, giving the probability of the data (s1, ..., sk) under each tank +population size N = n, based on a probabilistic model of the tank-capturing process. +The output of a Bayesian treatment of the German tank problem is the posterior mass +function of the tank population size N, conditioned on the data (s1, ..., sk). The posterior +follows from Bayes’ theorem and can be viewed as an update to the prior in light of the +data. The posterior mass function of N assigns each possible tank population size n with a +probability according to a compromise between its (i) likelihood, which quantifies the support +the observed serial numbers (s1, ..., sk) lend to the tank population size being n according to +our probabilistic tank-capturing model, and (ii) prior probability, which quantifies how likely +we thought the tank population size might be n before the serial numbers (s1, ..., sk) were +collected and considered. [21] +The posterior mass function of N is the raw, uncertainty-quantifying, Bayesian solution +to the German tank problem. We may summarize the posterior by reporting its median and +the high-mass subset of the natural numbers that credibly contains the tank population size. +Also, we can use the posterior to answer questions such as, what is the probability that N +exceeds some threshold quantity n′ that would alter military strategy? +2 +A Bayesian approach to the German tank problem +We now tackle the German tank problem from a Bayesian standpoint. +For reference, the variables are listed in Tab. 2. We use upper- and lower-case letters to +represent random variables and realizations of them, respectively. Throughout, we employ +the indicator function IA(x) which maps its input x to 1 if x belongs to the set A and to 0 +otherwise (if x /∈ A). +5 + +Table 2: List of parameters/variables. +parameter/variable +∈ +description +n +N≥0 +size of population of tanks +k +N>0 +number of captured tanks +si +N>0 +serial number on captured tank i +s(k) +Nk +>0 +vector listing the serial numbers on the k captured tanks +m(k) +N>0 +maximum serial number among the k captured tanks +2.1 +The data, data-generating process, and likelihood function +The data. +The data we obtain in the German tank problem is the vector of serial numbers +inscribed on the k captured tanks +s(k) := (s1, ..., sk). +(2) +We view the data s(k) as a realization of the discrete random vector S(k) := (S1, ..., Sk). +Note, at this point, we are entertaining the possibility that the order in which tanks are +captured matters. +The data-generating process. +The stochastic data-generating process constitutes sequen- +tial capture of k tanks from a population of n tanks, without replacement, then inspecting +their serial numbers to construct s(k). We assume that each tank in the population is equally +likely to be captured at each step. Then, mathematically, the stochastic data-generating +process is sequential, uniform random selection of k integers, without replacement, from the +set {1, ..., n}. +The likelihood function. +The likelihood function specifies the probability of the data S(k) = +s(k) given each tank population size N = n. Each outcome s(k) in the sample space Ω(k) +n +is +equally likely, where +Ω(k) +n +:= {(s1, ..., sk)̸= : si ∈ {1, ..., n} for all i ∈ {1, ..., k}}, +(3) +with (· · · )̸= meaning the elements of the vector (· · · ) are unique. The number of outcomes in +the sample space, |Ω(k) +n |, is the number of distinct ordered arrangements of k distinct integers +from the set {1, ..., n}, given by the falling factorial: +(n)k := n(n − 1) · · · (n − k + 1) = n!/(n − k)!. +(4) +Under the data-generating process, then, the probability of observing data S(k) = s(k) given +the tank population size N = n is the uniform distribution: +πlikelihood(S(k) = s(k) | N = n) = +1 +(n)k +IΩ(k) +n +� +s(k)� +. +(5) +6 + +Interpretation. +We view πlikelihood(S(k) = s(k) | N = n) as a function of n, since in +practice we possess the data s(k) but not n. The likelihood quantifies the support the serial +numbers on the k captured tanks in s(k) lend for any particular tank population size n [21]. +The likelihood as a sequence of events. +Alternatively, we may arrive at eqn. 5 from +a perspective of sequential events S1 = s1, S2 = s2, ..., Sk = sk. First, the probability of a +given serial number on the ith captured tank, conditioned on the tank population size and +the outcomes of the previous serial numbers, is the uniform distribution +π(Si = si | N = n, S1 = s1, ..., Si−1 = si−1) = +1 +n − i + 1I{1,...,n}\{s1,...,si−1}(si) +(6) +since there are n − i + 1 tanks to choose from at uniform random. By the chain rule, the +joint probability +πlikelihood(S1 = s1, ..., Sk = sk | N = n) = +k� +i=1 +π(Si = si | N = n, S1 = s1, ..., Si−1 = si−1) +(7) +giving eqn. 5 after simplifying the product of indicator functions. +The likelihood function in terms of the maximum observed serial number. +We will +find in Sec. 2.3 that only two independent features of the data (s1, ..., sk) provide information +about the tank population size, N: its (i) size, k, and (ii) maximum observed serial number +m(k) = +max +i∈{1,...,k} si. +(8) +Thus, we also write a different likelihood: the probability of observing a maximum serial +number m(k) given the tank population size N = n, πlikelihood(M(k) = m(k) | N = n). +Because each outcome s(k) ∈ Ω(k) +n +is equally likely, πlikelihood(M(k) = m(k) | N = n) is the +fraction of sample space under population size n where the maximum serial number is m(k). +To count the outcomes (s1, ..., sk) ∈ Ω(k) +n +where the maximum serial number is m(k), consider +(i) one of the k captured tanks has serial number m(k) and (ii) the remaining k −1 tanks have +a serial number in {1, ..., m(k) − 1}. For each of the k possible positions of the maximum +serial number in the vector s(k), there are (m(k) − 1)k−1 distinct outcomes specifying the +other k − 1 entries. Thus: +πlikelihood(M(k) = m(k) | N = n) = k(m(k) − 1)k−1 +(n)k +I{k,...,n}(m(k)). +(9) +2.2 +The prior distribution +The prior probability mass function πprior(N = n) expresses a combination of our subjective +beliefs and objective knowledge about the total number of tanks N before the data (s1, ..., sk) +7 + +are collected and considered. Context-dependent, the prior mass function we impose on N +can vary in the amount of uncertainty it admits about the tank population size (measured by +eg. entropy [23]). +Prior distributions can be loosely classified as informative, weakly informative, or diffuse +[21]. If we do not possess prior information about the tank population size, we adopt the +principle of indifference and impose a diffuse prior to ”let the data speak for itself” [8], eg. +a uniform distribution over a set of feasible tank population sizes. On the other hand, an +informative prior might concentrate its mass around some estimate of the total number of +tanks obtained through other means. An informative prior will have a larger impact on the +posterior mass function of N than a diffuse one [21]. +Generally, as the number of captured tanks k increases (decreases), we expect the prior +mass function we impose to have a lesser (greater) influence on the posterior distribution [8]. +2.3 +The posterior distribution +The posterior probability mass function of N assigns a probability to each possible tank +population size n in consideration of its consistency with (1) the data (s1, ..., sk), according +to the likelihood in eqn. 5, and (2) our prior beliefs/knowledge encoded in πprior(N = n). +The posterior distribution is a conditional distribution related to the likelihood and prior +mass functions by Bayes’ theorem: +πposterior(N = n | S(k) = s(k)) = πlikelihood(S(k) = s(k) | N = n)πprior(N = n) +πdata(S(k) = s(k)) +, +(10) +where the denominator is the probability of the data s(k): +πdata(S(k) = s(k)) = +∞ +� +n′=0 +πlikelihood(S(k) = s(k) | N = n′)πprior(N = n′). +(11) +We view πposterior(N = n | S(k) = s(k)) as a probability mass function of N, since in practice +we have s(k). Then, πdata(S(k) = s(k)) is just a normalizing factor for the numerator in +eqn. 10. +Interpreting eqn. 10, the prior mass function of N is updated, in light of the data +(s1, ..., sk), to yield the posterior mass function of N. The posterior probability of N = n is +proportional to the product of the likelihood at and prior probability of N = n—a compromise +between the likelihood and prior. +We simplify the posterior mass function of N in eqn. 10 by (i) substituting eqn. 5, (ii) +restricting the sum in eqn. 11 to tank population sizes where the likelihood is nonzero, and +(iii) noting the only two features of the data (s1, ..., sk) that appear are (a) its size k and (b) +8 + +the maximum serial number m(k): +πposterior(N = n | M(k) = m(k)) = +(n)−1 +k πprior(N = n) +∞ +� +n′=m(k) +(n′)−1 +k πprior(N = n′) +I{m(k),m(k)+1,...}(n) +(12) +Note, we may arrive at eqn. 12 through eqn. 9 as well. +Interpretation. +The posterior probability mass function of N in eqn. 12 is our raw, uncertainty- +quantifying solution to the German tank problem. It assigns a probability to each tank popula- +tion size n in consideration of the serial numbers (s1, ..., sk) observed on the captured tanks, +our probabilistic model of the tank-capturing process, and our prior beliefs and knowledge +about the tank population size expressed in the prior mass function. +A remark on ”uncertainty”. +The spread of the posterior mass function of N in eqn. 12 +reflects epistemic [24] uncertainty about the tank population size, attributed to a lack of +complete data. Accounting for the data (s1, ..., sk) (probably) does not eliminate uncertainty +about the tank population size because we (presumably) have not captured all of the tanks +(ie. k < n) and observed their serial numbers. In practice, posterior uncertainty about the +tank population size also has a contribution from the possible inadequacy of the model of the +tank-capturing process (uniform sampling) in eqn. 5, which our analysis here neglects. +Summarizing the posterior mass function of N. +We may summarize the posterior mass +function of N with a point estimate of the tank population size and a credible subset of the +natural numbers that likely3 contains it. A suitable point estimate of the tank population +size is a median of the posterior mass function of N; by definition, the posterior probability +that the tank population size is greater (less) than or equal to a median is at least 0.5. A +suitable credible subset, which entertains multiple tank population sizes, is the α-high-mass +subset [25] +Hα := {n′ : πposterior(N = n′ | M(k) = m(k)) ≥ πα} +(13) +where πα is the largest mass to satisfy +πposterior(N ∈ Hα | M(k) = m(k)) ≥ 1 − α. +(14) +In words, the α-high-mass subset Hα is the smallest to (i) contain at least a fraction 1 − α +of the posterior mass of N and (ii) ensure every tank population size belonging to the subset +is more probable than all outside of it. +3Well, ”likely”, under our assumptions embedded in the likelihood and prior mass functions. +9 + +Querying the posterior distribution. +We may find the posterior probability that the tank +population size belongs to any set of interest by summing the posterior mass over it. Eg., +the probability the tank population size exceeds some number n′ is: +πposterior(N > n′ | M(k) = m(k)) = +∞ +� +n=n′+1 +πposterior(N = n | M(k) = m(k)). +(15) +2.3.1 +Posterior predictive checking +We may check the consistency of the data s(k) with the posterior mass function of N. +Conceptually, we can simulate new data ˜s(k) using the model of the tank-capturing process +under a sample of the tank population size from the posterior, then compare the simulated +data ˜s(k) to the real data s(k) [21,26]. More appropriately, we can compare the serial numbers +in the real data (s1, ..., sk) with the mass function giving the probability that the tank with +serial number ˜s would be captured under this process: +π(˜s ∈ ˜S(k)) = +∞ +� +n′=0 +k +n′ πposterior(N = n′ | S(k) = s(k))I{1,...,n′}(˜s), +(16) +since k/n′ is the probability any given viable serial number ˜s will be observed given the tank +population size N = n′. +3 +Example +We illustrate the Bayesian approach to the German tank problem through an example. +The prior probability mass function of N. +Suppose we have an upper bound nmax for the +possible number of tanks but no other information. Then, we may impose a diffuse prior, a +uniform prior probability mass function: +πprior(N = n) = +1 +nmax + 1I{0,...,nmax}(n). +(17) +This prior mass function expresses: in the absence of any data (s1, ..., sk) (ie., no serial +numbers, not k either), we believe the total number of tanks N is equally likely to be a value +in {0, ..., nmax}. Particularly, suppose nmax = 35. Fig. 1a visualizes πprior(N = n). +The data (s1, ..., sk). +Now suppose we capture k = 3 tanks, with serial numbers s(3) = +(15, 14, 3). See Fig. 1b. So, the maximum observed serial number is m(3) = 15. +10 + +size of tank population, n +0 +10 +20 +30 +40 +πprior(N=n) +0.00 +0.01 +0.02 +(a) prior mass function of N +s1=15 +s2=14 +s3=3 +serial numbers +1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 +(b) the data s(k=3) +tank population size, n +0 +10 +20 +30 +40 +πlikelihood(M(k=3)=15 | N=n) +0.00 +0.05 +0.10 +0.15 +0.20 +(c) the likelihood function +size of tank population, n +0 +10 +20 +30 +40 +πposterior(N=n | M(k = 3)=15) +0.00 +0.05 +0.10 +0.15 +nmax=35 +(d) posterior mass function of N +Figure 1: A Bayesian approach to the German tank problem. +(a, prior) The prior mass +function. (b, data) The data s(3), with maximum observed serial number m(3) = 15. (c, +likelihood) The likelihood function associated with the data s(3). (d, posterior) The posterior +mass function of N. H0.2 highlighted; median marked with vertical, dashed line. +11 + +The posterior probability mass function of N. +Under the uniform prior in eqn. 17, the +posterior probability mass function of N in eqn. 12 becomes: +πposterior(N = n | M(k) = m(k)) = +(n)−1 +k +nmax +� +n′=m(k) +(n′)−1 +k +I{m(k),m(k)+1,...,nmax}(n). +(18) +Fig. 1d visualizes the posterior probability mass function of N for the data s(3) in Fig. 1b and +the prior in eqn. 17 (nmax = 35). +Summarizing the posterior. +Summarizing the posterior mass function of N, its median +is 19 and its high-mass credible subset H0.2 = {15, ..., 25} (highlighted in Fig. 1d). For what +it’s worth, the data in Fig. 1b was generated from a tank population size of n = 20 (explaining +the choice of scale in Fig. 1b). +Querying the posterior. +Suppose our military strategy would change if the size of the +tank population exceeds 30. From the posterior distribution of N, we calculate πposterior(N > +30 | M(3) = 15) ≈ 0.066. +Posterior predictive checking. +As a posterior predictive check, Fig. 2a shows how the +observed serial numbers in the data s(3) compare with the probability of observing each serial +number under the posterior mass function of N, according to eqn. 16. +Sensitivity of the posterior to the prior. +Because of the subjectivity involved in construct- +ing the prior, checking the sensitivity of the posterior to the prior is good practice [21]. Fig. 2b +shows how the posterior mass function of N changes as we increase the upper-bound on the +tank population nmax we impose via the prior mass function of N in eqn. 17. The median of +the posterior under nmax ∈ {60, 70} is 20 (an increase of one compared to nmax = 35). The +maximum of the high-mass subset H0.2 increases to 29 for nmax = 70. +Capturing more tanks. +Suppose we capture an additional 9 tanks and re-run the Bayesian +analysis. Fig. 3 shows the updated posterior mass function of N. The high-mass credible +subset H0.2 shrinks considerably, to {19, 20}. This shows how more data—increasing the +number of tanks captured, k—generally reduces our uncertainty about the tank population +size. +4 +Discussion +Selection bias. +A strict assumption in the textbook-friendly German tank problem, which +enables us to estimate the size of the population of tanks from a random sample of their +12 + +serial number, s̃ +0 +10 +20 +30 +40 +probability +0.00 +0.05 +0.10 +0.15 +nmax=35 +data, s(k=3) +(a) posterior predictive check +size of tank population, n +0 +20 +40 +60 +πposterior(N=n | M(k = 3)=15) +0.00 +0.05 +0.10 +0.15 +nmax=50 +size of tank population, n +0 +20 +40 +60 +nmax=60 +size of tank population, n +0 +20 +40 +60 +nmax=70 +prior +posterior +(b) sensitivity of the posterior to the prior +Figure 2: Checking (a) the consistency of the data s(3) with the probability of the serial +numbers under the posterior mass function of N and (b) the sensitivity of the posterior mass +function of N to the upper bound nmax imposed by the prior mass function of N. +(sequential) serial numbers, is that sampling is uniform. To check consistency of the sample +with this assumption, Goodman [10] demonstrates a test of the hypothesis that the sample of +serial numbers is from a uniform distribution. Interesting extensions of the textbook German +tank problem could involve modeling selection bias in the tank-capturing process. Such bias +could arise eg. hypothetically, if older tanks with smaller serial numbers were more likely to be +deployed in the fronts opened earlier in the war, where capturing tanks is more difficult than +at less fortified fronts opened more recently. +The German tank problem in other contexts. +The Bayesian probability theory to solve +the German tank problem applies (perhaps, with modification) to many other contexts where +we wish to estimate the size of some finite, hidden set [27], eg.: the number of taxicabs in a +city [12], the number of accounts at a bank [15], the number of furniture pieces purchased +13 + +s1=15 +s2=14 +s3=3 +s4=6 +s5=2 +s6=10 +s7=5 +s8=16 +s9=8 +s10=1 +s11=4 +s12=19 +serial numbers +1 +2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 +(a) the updated data s(k=12) +size of tank population, n +0 +10 +20 +30 +40 +πposterior(N=n | M(k = 12)=19) +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +nmax=35 +(b) the updated posterior mass function of N +Figure 3: The updated posterior mass function of N (b) after we capture an additional 9 +tanks with serial numbers in (a). +by a university [10], the number of aircraft operations at an airport [28], the extent of leaked +classified government communications [29], the time needed to complete a project deadline +[30], the time-coverage of historical records of extreme events like floods [31], the length of +a short-tandem repeat allele [32], the size of a social network [33], the number of cases in +14 + +court [34], the lifetime of a flower of a plant [35], or the duration of existence of a species [36]. +Mark and recapture methods in ecology to estimate the size of an animal population [37,38] +are tangentially related to the German tank problem. +The practice of inscribing sequential serial numbers on military equipment. +Germany +adopted the practice of marking their military equipment with serial numbers and codes to +trace the equipment/parts/components back to the manufacturer. However, the sequential +nature of these serial numbers was exploited by the Allies to estimate their armament pro- +duction. To reduce vulnerability to serial number analysis for estimating production while +maintaining advantages of tracing equipment back to the manufacturer, serial numbers and +codes could instead be obfuscated by eg. chaffing [39]. +Data and code availability +The Julia [40] code to reproduce all of our visualizations drawn using Makie.jl [41] is available +on Github at github.com/SimonEnsemble/the˙German˙tank˙problem. +Acknowledgements +Thanks to Bernhard Konrad for providing detailed feedback on the first draft and to my +students Gbenga Fabusola, Adrian Henle, and Paul Morris for feedback on the introduction. +References +[1] Richard Ruggles and Henry Brodie. An empirical approach to economic intelligence in +World War II. Journal of the American Statistical Association, 42(237):72–91, 1947. +[2] Leo A Goodman. Serial number analysis. 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Journal of Open Source Software, 6(65):3349, 2021. +18 + diff --git a/6dAyT4oBgHgl3EQfQfZh/content/tmp_files/load_file.txt b/6dAyT4oBgHgl3EQfQfZh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4b309d2932f74ebcea82c7d2f1e10e9e5635f266 --- /dev/null +++ b/6dAyT4oBgHgl3EQfQfZh/content/tmp_files/load_file.txt @@ -0,0 +1,527 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf,len=526 +page_content='A Bayesian treatment of the German tank problem Cory M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Simon School of Chemical, Biological, and Environmental Engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Oregon State University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Corvallis, OR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' cory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='simon@oregonstate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='edu Abstract The German tank problem has an interesting historical background and is an engaging problem in statistical estimation for the classroom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' The objective is to estimate the size of a population of tanks inscribed with sequential serial numbers, from a random sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' In this tutorial article, we outline the Bayesian approach to the German tank problem, (i) whose solution assigns a probability to each tank population size, thereby quantifying uncertainty, and (ii) which provides an opportunity to incorporate prior information and/or beliefs about the tank population size into the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' We illustrate with an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Finally, we survey other research problems that bear resemblance to the German tank problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' s1=15 s2=14 s3=3 serial numbers 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 size of tank population, n 0 10 20 30 40 probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='20 prior likelihood posterior 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='00046v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='OT] 30 Dec 2022 1 Background 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='1 History To inform their military strategy during World War II (1939-1945), the Allies sought to es- timate the rate of production of various military equipment (tanks, tires, rockets, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=') by Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Conventional methods to estimate armament production—including (i) extrapo- lating data on prewar manufacturing capabilities, (ii) obtaining reports from secret sources, and (iii) interrogating prisoners of war—were unreliable and/or contradictory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' In 1943, British and American economic intelligence agencies exploited a German man- ufacturing practice in order to statistically estimate their armament production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Germany marked their military equipment with serial numbers and codes for the date and/or place of manufacture to handle spare parts and trace faulty/defective equipment/parts back to the manufacturer for quality control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' However, these markings on a captured sample of German equipment provided the Allies information about Germany’s production of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' To estimate Germany’s production of tanks, the Allies collected serial numbers on the chassis, engines, gearboxes, and bogie wheels of samples of tanks by inspecting captured tanks and examining captured records1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Despite lacking an exhaustive sample, the sequential nature of2 and patterns in these samples of serial numbers enabled the Allies to estimate Germany’s tank production—postwar, we know, much more accurately than conventional American and British intelligence (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' See Ruggles and Brodie [1] for the detailed historical account of serial number analysis to estimate German armament production during World War II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Table 1: Monthly production of tanks by Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' [1] estimates date conventional American & British Intelligence serial number analysis German records June, 1940 1000 169 122 June, 1941 1550 244 271 August, 1942 1550 327 342 1Eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', captured records from tank repair depots listed serial numbers of the chassis and engine of repaired tanks, and records from divisional headquarters listed chassis serial numbers of tanks held by a specific unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 2Gearboxes on captured tanks, for example, were inscribed with serial numbers belonging to an unbroken sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Chassis serial numbers, on the other hand, were broken into blocks to distinguish models/designs, leaving gaps between the serial numbers assigned to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='2 The German tank problem Simplification of the historical context to estimate German tank production via serial number analysis [1] motivated the formulation of the textbook-friendly German tank problem [2]: Problem statement In the backdrop of World War II, the German military has n tanks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Each tank is inscribed with a unique serial number in {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' As the Allies, we do not know n, but we captured (without replacement, of course) a sample of k German tanks with inscribed serial numbers (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', sk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' s1 s2 · · sk Assuming all tanks in the population were equally likely to be captured, our objective is to estimate n in consideration of the data (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', sk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' In 1942, Alan Turing and Andrew Gleason discussed a variant of the German tank prob- lem, “how to best to estimate the total number of taxicabs in a town, having seen a random selection of their license numbers”, in a crowded restaurant in Washington DC [3,4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Today, with its interesting historical background [1], the German tank problem is still a suitable con- versation topic for dinners and serves as an intellectually engaging, challenging, and enjoyable problem to illustrate combinatorics and statistical estimation in the classroom [5–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Uncertainty quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Any estimate of the tank population size n from the data (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', sk) is subject to uncertainty, since we (presumably) have not captured all of the tanks (ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', k ̸= n, probably).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Quantifying uncertainty in our estimate of the tank population size n is important because high-stakes military decisions may be made on its basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Our contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' In this pedagogical article, we outline the Bayesian approach to the German tank problem, (i) whose solution assigns a probability to each tank population size, thereby quantifying uncertainty, and (ii) which provides an opportunity to incorporate prior information and/or beliefs about the tank population size into the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='3 Survey of previous work on the German tank problem The frequentist approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Border [9] calls the German tank problem a ”weird case” in frequentist estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' The maximum likelihood estimator of the tank population size n is 3 the maximum serial number observed among the k captured tanks, m(k) := maxi∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=',k} si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' This is a biased estimator, as certainly m(k) ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Goodman [2, 10] derives the minimum-variance, unbiased point estimator of the tank population size ˆn = m(k) + � m(k) k − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' (1) To intuit this estimator, note (i) n must be greater than or equal to m(k) and (ii) if we observe large (small) gaps between the serial numbers (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', sk) after sorting them (incl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' the gap preceding the smallest serial number), then n is likely (unlikely) to be much greater than m(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' The estimator of n in eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 1 quantifies how far beyond m(k) we should estimate the tank population size, based on the gaps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' m(k)/k − 1 is the average size of the gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Goodman also derives a frequentist confidence interval for n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Clark, Gonye, and Miller explore using simulations and linear regression to discover the estimator in eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 1 [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' For pedagogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Champkin highlights the historical context of the German tank problem as a ”great moment in statistics” [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Johnson lists and evaluates several intuitive point estimators for the size of the tank population [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Scheaffer, Watkins, Gnanadesikan, and Witmer [13] propose a hands-on learning activity to illustrate the German tank problem by sampling chips, labeled with numbers from 1 to n, from a bowl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Berg [6] uses the German tank problem as a competition in the classroom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' The Bayesian approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Closely related to our paper, Roberts [14], H¨ohle and Held [15], and Linden, Dose, and Toussaint [16], and Cocco, Monasson, and Zamponi [17] provide a Bayesian analysis of the German tank problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' They derive an analytical formula for the mean of the posterior distribution of the tank population size under an improper, uniform prior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Andrews [18] outlines the Bayesian approach to the German tank problem in a blog post containing code in the R language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Generalizations/variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Goodman [2, 10] poses a variant of the German tank problem where the initial serial number is not known;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', where the n tanks are inscribed with serial numbers {b + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', n + b} with b and n unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Lee and Miller generalize the German tank problem to the settings where the serial numbers are continuous and/or lie in two dimensions [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='4 Overview of the Bayesian approach to the German tank problem Under a Bayesian perspective [8,20,21], we treat the (unknown) total number of tanks as a discrete random variable N (hence, capitalization) to model our uncertainty in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' A proba- 4 bility mass function of N assigns a probability to each possible tank population size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' This probability is a measure of our degree of belief, perhaps with some basis in knowledge/data, that the tank population size is n [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Because the observed serial numbers (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', sk) provide information about the tank pop- ulation size, the probability mass function of N differs before and after they are collected and considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Hence, N has a prior and posterior probability mass function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' The three inputs to a Bayesian treatment of the German tank problem are: the prior mass function of N, which expresses a combination of our subjective beliefs and objective knowledge about the tank population size before we collect and consider the sample of serial numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' the data, the observed serial numbers (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', sk), viewed as realizations of random variables owing to the stochasticity of tank-capturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' the likelihood function, giving the probability of the data (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', sk) under each tank population size N = n, based on a probabilistic model of the tank-capturing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' The output of a Bayesian treatment of the German tank problem is the posterior mass function of the tank population size N, conditioned on the data (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', sk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' The posterior follows from Bayes’ theorem and can be viewed as an update to the prior in light of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' The posterior mass function of N assigns each possible tank population size n with a probability according to a compromise between its (i) likelihood, which quantifies the support the observed serial numbers (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', sk) lend to the tank population size being n according to our probabilistic tank-capturing model, and (ii) prior probability, which quantifies how likely we thought the tank population size might be n before the serial numbers (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', sk) were collected and considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' [21] The posterior mass function of N is the raw, uncertainty-quantifying, Bayesian solution to the German tank problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' We may summarize the posterior by reporting its median and the high-mass subset of the natural numbers that credibly contains the tank population size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Also, we can use the posterior to answer questions such as, what is the probability that N exceeds some threshold quantity n′ that would alter military strategy?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 2 A Bayesian approach to the German tank problem We now tackle the German tank problem from a Bayesian standpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' For reference, the variables are listed in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' We use upper- and lower-case letters to represent random variables and realizations of them, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Throughout, we employ the indicator function IA(x) which maps its input x to 1 if x belongs to the set A and to 0 otherwise (if x /∈ A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 5 Table 2: List of parameters/variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' parameter/variable ∈ description n N≥0 size of population of tanks k N>0 number of captured tanks si N>0 serial number on captured tank i s(k) Nk >0 vector listing the serial numbers on the k captured tanks m(k) N>0 maximum serial number among the k captured tanks 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='1 The data, data-generating process, and likelihood function The data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' The data we obtain in the German tank problem is the vector of serial numbers inscribed on the k captured tanks s(k) := (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', sk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' (2) We view the data s(k) as a realization of the discrete random vector S(k) := (S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', Sk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Note, at this point, we are entertaining the possibility that the order in which tanks are captured matters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' The data-generating process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' The stochastic data-generating process constitutes sequen- tial capture of k tanks from a population of n tanks, without replacement, then inspecting their serial numbers to construct s(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' We assume that each tank in the population is equally likely to be captured at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Then, mathematically, the stochastic data-generating process is sequential, uniform random selection of k integers, without replacement, from the set {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' The likelihood function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' The likelihood function specifies the probability of the data S(k) = s(k) given each tank population size N = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Each outcome s(k) in the sample space Ω(k) n is equally likely, where Ω(k) n := {(s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', sk)̸= : si ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', n} for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', k}}, (3) with (· · · )̸= meaning the elements of the vector (· · · ) are unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' The number of outcomes in the sample space, |Ω(k) n |, is the number of distinct ordered arrangements of k distinct integers from the set {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', n}, given by the falling factorial: (n)k := n(n − 1) · · · (n − k + 1) = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='/(n − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='. (4) Under the data-generating process, then, the probability of observing data S(k) = s(k) given the tank population size N = n is the uniform distribution: πlikelihood(S(k) = s(k) | N = n) = 1 (n)k IΩ(k) n � s(k)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' (5) 6 Interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' We view πlikelihood(S(k) = s(k) | N = n) as a function of n, since in practice we possess the data s(k) but not n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' The likelihood quantifies the support the serial numbers on the k captured tanks in s(k) lend for any particular tank population size n [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' The likelihood as a sequence of events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Alternatively, we may arrive at eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 5 from a perspective of sequential events S1 = s1, S2 = s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', Sk = sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' First, the probability of a given serial number on the ith captured tank, conditioned on the tank population size and the outcomes of the previous serial numbers, is the uniform distribution π(Si = si | N = n, S1 = s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', Si−1 = si−1) = 1 n − i + 1I{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=',n}\\{s1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=',si−1}(si) (6) since there are n − i + 1 tanks to choose from at uniform random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' By the chain rule, the joint probability πlikelihood(S1 = s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', Sk = sk | N = n) = k� i=1 π(Si = si | N = n, S1 = s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', Si−1 = si−1) (7) giving eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 5 after simplifying the product of indicator functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' The likelihood function in terms of the maximum observed serial number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' We will find in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='3 that only two independent features of the data (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', sk) provide information about the tank population size, N: its (i) size, k, and (ii) maximum observed serial number m(k) = max i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=',k} si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' (8) Thus, we also write a different likelihood: the probability of observing a maximum serial number m(k) given the tank population size N = n, πlikelihood(M(k) = m(k) | N = n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Because each outcome s(k) ∈ Ω(k) n is equally likely, πlikelihood(M(k) = m(k) | N = n) is the fraction of sample space under population size n where the maximum serial number is m(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' To count the outcomes (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', sk) ∈ Ω(k) n where the maximum serial number is m(k), consider (i) one of the k captured tanks has serial number m(k) and (ii) the remaining k −1 tanks have a serial number in {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', m(k) − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' For each of the k possible positions of the maximum serial number in the vector s(k), there are (m(k) − 1)k−1 distinct outcomes specifying the other k − 1 entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Thus: πlikelihood(M(k) = m(k) | N = n) = k(m(k) − 1)k−1 (n)k I{k,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=',n}(m(k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' (9) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='2 The prior distribution The prior probability mass function πprior(N = n) expresses a combination of our subjective beliefs and objective knowledge about the total number of tanks N before the data (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', sk) 7 are collected and considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Context-dependent, the prior mass function we impose on N can vary in the amount of uncertainty it admits about the tank population size (measured by eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' entropy [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Prior distributions can be loosely classified as informative, weakly informative, or diffuse [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' If we do not possess prior information about the tank population size, we adopt the principle of indifference and impose a diffuse prior to ”let the data speak for itself” [8], eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' a uniform distribution over a set of feasible tank population sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' On the other hand, an informative prior might concentrate its mass around some estimate of the total number of tanks obtained through other means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' An informative prior will have a larger impact on the posterior mass function of N than a diffuse one [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Generally, as the number of captured tanks k increases (decreases), we expect the prior mass function we impose to have a lesser (greater) influence on the posterior distribution [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='3 The posterior distribution The posterior probability mass function of N assigns a probability to each possible tank population size n in consideration of its consistency with (1) the data (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', sk), according to the likelihood in eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 5, and (2) our prior beliefs/knowledge encoded in πprior(N = n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' The posterior distribution is a conditional distribution related to the likelihood and prior mass functions by Bayes’ theorem: πposterior(N = n | S(k) = s(k)) = πlikelihood(S(k) = s(k) | N = n)πprior(N = n) πdata(S(k) = s(k)) , (10) where the denominator is the probability of the data s(k): πdata(S(k) = s(k)) = ∞ � n′=0 πlikelihood(S(k) = s(k) | N = n′)πprior(N = n′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' (11) We view πposterior(N = n | S(k) = s(k)) as a probability mass function of N, since in practice we have s(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Then, πdata(S(k) = s(k)) is just a normalizing factor for the numerator in eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Interpreting eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 10, the prior mass function of N is updated, in light of the data (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', sk), to yield the posterior mass function of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' The posterior probability of N = n is proportional to the product of the likelihood at and prior probability of N = n—a compromise between the likelihood and prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' We simplify the posterior mass function of N in eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 10 by (i) substituting eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 5, (ii) restricting the sum in eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 11 to tank population sizes where the likelihood is nonzero, and (iii) noting the only two features of the data (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', sk) that appear are (a) its size k and (b) 8 the maximum serial number m(k): πposterior(N = n | M(k) = m(k)) = (n)−1 k πprior(N = n) ∞ � n′=m(k) (n′)−1 k πprior(N = n′) I{m(k),m(k)+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='}(n) (12) Note, we may arrive at eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 12 through eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 9 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' The posterior probability mass function of N in eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 12 is our raw, uncertainty- quantifying solution to the German tank problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' It assigns a probability to each tank popula- tion size n in consideration of the serial numbers (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', sk) observed on the captured tanks, our probabilistic model of the tank-capturing process, and our prior beliefs and knowledge about the tank population size expressed in the prior mass function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' A remark on ”uncertainty”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' The spread of the posterior mass function of N in eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 12 reflects epistemic [24] uncertainty about the tank population size, attributed to a lack of complete data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Accounting for the data (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', sk) (probably) does not eliminate uncertainty about the tank population size because we (presumably) have not captured all of the tanks (ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' k < n) and observed their serial numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' In practice, posterior uncertainty about the tank population size also has a contribution from the possible inadequacy of the model of the tank-capturing process (uniform sampling) in eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 5, which our analysis here neglects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Summarizing the posterior mass function of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' We may summarize the posterior mass function of N with a point estimate of the tank population size and a credible subset of the natural numbers that likely3 contains it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' A suitable point estimate of the tank population size is a median of the posterior mass function of N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' by definition, the posterior probability that the tank population size is greater (less) than or equal to a median is at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' A suitable credible subset, which entertains multiple tank population sizes, is the α-high-mass subset [25] Hα := {n′ : πposterior(N = n′ | M(k) = m(k)) ≥ πα} (13) where πα is the largest mass to satisfy πposterior(N ∈ Hα | M(k) = m(k)) ≥ 1 − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' (14) In words, the α-high-mass subset Hα is the smallest to (i) contain at least a fraction 1 − α of the posterior mass of N and (ii) ensure every tank population size belonging to the subset is more probable than all outside of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 3Well, ”likely”, under our assumptions embedded in the likelihood and prior mass functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 9 Querying the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' We may find the posterior probability that the tank population size belongs to any set of interest by summing the posterior mass over it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', the probability the tank population size exceeds some number n′ is: πposterior(N > n′ | M(k) = m(k)) = ∞ � n=n′+1 πposterior(N = n | M(k) = m(k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' (15) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='1 Posterior predictive checking We may check the consistency of the data s(k) with the posterior mass function of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Conceptually, we can simulate new data ˜s(k) using the model of the tank-capturing process under a sample of the tank population size from the posterior, then compare the simulated data ˜s(k) to the real data s(k) [21,26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' More appropriately, we can compare the serial numbers in the real data (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', sk) with the mass function giving the probability that the tank with serial number ˜s would be captured under this process: π(˜s ∈ ˜S(k)) = ∞ � n′=0 k n′ πposterior(N = n′ | S(k) = s(k))I{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=',n′}(˜s), (16) since k/n′ is the probability any given viable serial number ˜s will be observed given the tank population size N = n′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 3 Example We illustrate the Bayesian approach to the German tank problem through an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' The prior probability mass function of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Suppose we have an upper bound nmax for the possible number of tanks but no other information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Then, we may impose a diffuse prior, a uniform prior probability mass function: πprior(N = n) = 1 nmax + 1I{0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=',nmax}(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' (17) This prior mass function expresses: in the absence of any data (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', sk) (ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', no serial numbers, not k either), we believe the total number of tanks N is equally likely to be a value in {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', nmax}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Particularly, suppose nmax = 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 1a visualizes πprior(N = n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' The data (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', sk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Now suppose we capture k = 3 tanks, with serial numbers s(3) = (15, 14, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' So, the maximum observed serial number is m(3) = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 10 size of tank population, n 0 10 20 30 40 πprior(N=n) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='02 (a) prior mass function of N s1=15 s2=14 s3=3 serial numbers 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 (b) the data s(k=3) tank population size, n 0 10 20 30 40 πlikelihood(M(k=3)=15 | N=n) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='20 (c) the likelihood function size of tank population, n 0 10 20 30 40 πposterior(N=n | M(k = 3)=15) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='15 nmax=35 (d) posterior mass function of N Figure 1: A Bayesian approach to the German tank problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' (a, prior) The prior mass function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' (b, data) The data s(3), with maximum observed serial number m(3) = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' (c, likelihood) The likelihood function associated with the data s(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' (d, posterior) The posterior mass function of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='2 highlighted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' median marked with vertical, dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 11 The posterior probability mass function of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Under the uniform prior in eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 17, the posterior probability mass function of N in eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 12 becomes: πposterior(N = n | M(k) = m(k)) = (n)−1 k nmax � n′=m(k) (n′)−1 k I{m(k),m(k)+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=',nmax}(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' (18) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 1d visualizes the posterior probability mass function of N for the data s(3) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 1b and the prior in eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 17 (nmax = 35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Summarizing the posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Summarizing the posterior mass function of N, its median is 19 and its high-mass credible subset H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='2 = {15, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=', 25} (highlighted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' For what it’s worth, the data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 1b was generated from a tank population size of n = 20 (explaining the choice of scale in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Querying the posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Suppose our military strategy would change if the size of the tank population exceeds 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' From the posterior distribution of N, we calculate πposterior(N > 30 | M(3) = 15) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='066.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Posterior predictive checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' As a posterior predictive check, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 2a shows how the observed serial numbers in the data s(3) compare with the probability of observing each serial number under the posterior mass function of N, according to eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Sensitivity of the posterior to the prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Because of the subjectivity involved in construct- ing the prior, checking the sensitivity of the posterior to the prior is good practice [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 2b shows how the posterior mass function of N changes as we increase the upper-bound on the tank population nmax we impose via the prior mass function of N in eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' The median of the posterior under nmax ∈ {60, 70} is 20 (an increase of one compared to nmax = 35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' The maximum of the high-mass subset H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='2 increases to 29 for nmax = 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Capturing more tanks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Suppose we capture an additional 9 tanks and re-run the Bayesian analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 3 shows the updated posterior mass function of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' The high-mass credible subset H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='2 shrinks considerably, to {19, 20}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' This shows how more data—increasing the number of tanks captured, k—generally reduces our uncertainty about the tank population size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 4 Discussion Selection bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' A strict assumption in the textbook-friendly German tank problem, which enables us to estimate the size of the population of tanks from a random sample of their 12 serial number, s̃ 0 10 20 30 40 probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='15 nmax=35 data, s(k=3) (a) posterior predictive check size of tank population, n 0 20 40 60 πposterior(N=n | M(k = 3)=15) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='15 nmax=50 size of tank population, n 0 20 40 60 nmax=60 size of tank population, n 0 20 40 60 nmax=70 prior posterior (b) sensitivity of the posterior to the prior Figure 2: Checking (a) the consistency of the data s(3) with the probability of the serial numbers under the posterior mass function of N and (b) the sensitivity of the posterior mass function of N to the upper bound nmax imposed by the prior mass function of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' (sequential) serial numbers, is that sampling is uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' To check consistency of the sample with this assumption, Goodman [10] demonstrates a test of the hypothesis that the sample of serial numbers is from a uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Interesting extensions of the textbook German tank problem could involve modeling selection bias in the tank-capturing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Such bias could arise eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' hypothetically, if older tanks with smaller serial numbers were more likely to be deployed in the fronts opened earlier in the war, where capturing tanks is more difficult than at less fortified fronts opened more recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' The German tank problem in other contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' The Bayesian probability theory to solve the German tank problem applies (perhaps, with modification) to many other contexts where we wish to estimate the size of some finite, hidden set [27], eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' : the number of taxicabs in a city [12], the number of accounts at a bank [15], the number of furniture pieces purchased 13 s1=15 s2=14 s3=3 s4=6 s5=2 s6=10 s7=5 s8=16 s9=8 s10=1 s11=4 s12=19 serial numbers 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 (a) the updated data s(k=12) size of tank population, n 0 10 20 30 40 πposterior(N=n | M(k = 12)=19) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='6 nmax=35 (b) the updated posterior mass function of N Figure 3: The updated posterior mass function of N (b) after we capture an additional 9 tanks with serial numbers in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' by a university [10], the number of aircraft operations at an airport [28], the extent of leaked classified government communications [29], the time needed to complete a project deadline [30], the time-coverage of historical records of extreme events like floods [31], the length of a short-tandem repeat allele [32], the size of a social network [33], the number of cases in 14 court [34], the lifetime of a flower of a plant [35], or the duration of existence of a species [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Mark and recapture methods in ecology to estimate the size of an animal population [37,38] are tangentially related to the German tank problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' The practice of inscribing sequential serial numbers on military equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Germany adopted the practice of marking their military equipment with serial numbers and codes to trace the equipment/parts/components back to the manufacturer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' However, the sequential nature of these serial numbers was exploited by the Allies to estimate their armament pro- duction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' To reduce vulnerability to serial number analysis for estimating production while maintaining advantages of tracing equipment back to the manufacturer, serial numbers and codes could instead be obfuscated by eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' chaffing [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Data and code availability The Julia [40] code to reproduce all of our visualizations drawn using Makie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='jl [41] is available on Github at github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='com/SimonEnsemble/the˙German˙tank˙problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Acknowledgements Thanks to Bernhard Konrad for providing detailed feedback on the first draft and to my students Gbenga Fabusola, Adrian Henle, and Paul Morris for feedback on the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' References [1] Richard Ruggles and Henry Brodie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' An empirical approach to economic intelligence in World War II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Journal of the American Statistical Association, 42(237):72–91, 1947.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' [2] Leo A Goodman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Serial number analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Journal of the American Statistical Association, 47(260):622–634, 1952.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' [3] Andrew Hodges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Alan Turing: the enigma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' In Alan Turing: The Enigma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Princeton University Press, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' [4] Marshall Hall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Alan Turing, Marshall Hall, and the Alignment of WW2 Japanese Naval Intercepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Notices of the AMS, 61(3), 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' [5] Roger W Johnson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Estimating the size of a population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Teaching Statistics, 16(2):50–52, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' [6] Arthur Berg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Bayesian modeling competitions for the classroom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Revista Colombiana de Estad´ıstica, 44(2):243–252, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 15 [7] Frederick Mosteller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Fifty challenging problems in probability with solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Courier Corporation, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' [8] Allen B Downey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Think Bayes 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' https://allendowney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='github.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' [41] Simon Danisch and Julius Krumbiegel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Makie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content='jl: Flexible high-performance data visual- ization for julia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' Journal of Open Source Software, 6(65):3349, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} +page_content=' 18' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfQfZh/content/2301.00046v1.pdf'} diff --git a/6dAzT4oBgHgl3EQfvP2u/content/tmp_files/2301.01704v1.pdf.txt b/6dAzT4oBgHgl3EQfvP2u/content/tmp_files/2301.01704v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..966389129f4807b081963898e1ffcfc60f81f01c --- /dev/null +++ b/6dAzT4oBgHgl3EQfvP2u/content/tmp_files/2301.01704v1.pdf.txt @@ -0,0 +1,965 @@ +Error Tolerant Multi-Robot System for Roadside +Trash Collection +1st Lee Milburn +College of Engineering +Northeastern University +Boston, Massachusetts +milburn.l@northeastern.edu +2nd John Chiaramonte +College of Engineering +Northeastern University +Boston, Massachusetts +chiaramonte.j@northeastern.edu +3rd Jack Fenton +College of Engineering +Northeastern University +Boston, Massachusetts +fenton.j@northeastern.edu +Abstract—In this paper, we present the first iteration of an +error-tolerant, autonomous, multi-robot system that monitors +highway road verges and identifies and collects roadside litter. It +is designed to use an aerial vehicle that can rapidly cover a vast +area and collect data on the road verge. This data is then passed +to a ground vehicle that constructs a map of the road verge and +uses a trained Convolutional Neural Network (CNN) to identify +pieces of litter. After the pieces of litter are identified on the +map of the road verge, the ground robot navigates to each piece +of trash, re-evaluates the area, and performs a ”greedy pickup” +procedure. This final stage accounts for any error in the map’s +construction or the identified trash’s location. We found that +ending the robotic system’s control flow with a greedy pickup +procedure can retroactively account for processing errors of the +system as it runs. This increases the system’s fault tolerance and +allows for the use of cheaper equipment since pinpoint accuracy +is not always necessary. In this paper, we present the feasibility of +this system by testing in simulation and later using real robotic +hardware. We show that the system is effective enough to iterate +on its design principles to create a more sophisticated system. +Index Terms—Autonomous trash collection, Environmental +monitoring, Error tolerance, Multi-robot system +I. INTRODUCTION +Roadside trash is a massive issue currently managed by +manual labor - a woefully inadequate solution [7]. Despite +being a nationwide issue, the task of waste management is +mostly under the jurisdiction of municipalities and it garners +little to no attention or investment. +To estimate the amount of litter along roadways, a research +team selected a random sample of 240 roadway segments, +stratified by type and by rural/urban areas [2]. The results +indicate that there are 51.2 billion pieces of litter on roadways +nationwide. Of this, the majority (91%, or 46.6 billion pieces) +is less than four inches. +In Monterey City, California, complaints about trash have +increased since the start of the COVID-19 pandemic. Officials +say this is not due to increased littering, but rather due to +the inability to clean it up. Monterey County Public Works +Maintenance Manager Shawn Atkins stated that his cleanup +crew was so busy cleaning up from illegal dumpsites that they +did not have time to walk the shoulders of their roads to pick +up loose trash [14]. Caltrans, California’s public transportation +department, has been faced with the same problem. Kevin +Drabinski, public information officer for Caltrans District 5, +said it’s important to Caltrans that they manage litter because +of safety and environmental concerns. Caltrans spends $50 +million annually on litter cleanup [14]. +To address this issue, our multi-robot system uses a three- +stage approach to autonomously map, identify, and pick up +trash. The three modes are Mapping, Navigation, and Greedy +Pickup. We would first have a lightweight drone fly over a +specified area on the road and stream its visuals to a ground +robot. That robot would generate a map using the drone’s +input and identify trash pieces on that map. The ground robot +would then navigate near each piece of identified trash and +then switch to the greedy pickup mode where it scans the +area for the suspected piece of trash. After a piece of trash +is re-identified locally, the robot moves and collects it. Once +either collected or not found, the ground robot then moves to +the next piece on its map. +Fig. 1. System Design Overview +Our approach allows for accurate pickup without the need +for massive processing power or overly expensive sensors. +The system’s configuration used the open-source convolutional +neural network You Only Look Once (YoLOv4) for image +identification [24], the open-source visual SLAM solution +ORBSLAM-2 for map-building [19], and open-source ROS +navigation software for path planning and navigation. We +simulated this system using Gazebo [13] and after receiving +consistent results, tested it in a real-world environment. Our +arXiv:2301.01704v1 [cs.RO] 4 Jan 2023 + +Step 1: Mapping Robot identifies trash and maps +Step 2: Mapping Robot transfers map data to UGV +the surrounding area +over Wi-Fi +Step 3: UGV takes most efficient path to pick up +Step 4: Human operator removes storage bin and +trash and clear area. +dumps trash into larger containerreal-world results, with a relatively low-powered system, in- +dicate that our approach is a proof-of-concept for a scalable +and viable solution to the growing worldwide litter problem. +II. RELATED WORK +There has been research into multi-robot systems used +for environmental monitoring. The research for these sys- +tems finds that multi-robot systems pose a more effective +solution to surveying an environment than static monitoring +[6]. There is also research into multi-robot systems that do +autonomous trash collection [16]. This research concludes +that for maximum efficiency, robots should be aware of their +environment when trying to collect trash as opposed to making +decisions based solely on their field of view (FOV). Therefore, +these two systems, monitoring an environment, and using a +collection algorithm to pick up trash in a dynamically changing +environment could be combined to create the most effective +version of an autonomous collection system. A version of this +system has been created to autonomously collect and monitor +plastics in rivers [9]. The system includes a central processor +that takes in the necessary tasks of the environment and assigns +those tasks to underwater autonomous vehicles that then pick +up the plastics. This system concluded that a Multi-robot task +allocation architecture [11] with a controlling center increased +the efficiency of the system, but the hardware for working +effectively in that environment would have to be improved for +more effective use. Our research team structured our multi- +robot system design to have a robot monitor the environment +and wirelessly transmit its environmental depiction to a UGV +that would collect the trash. +For our system to be cheap and lightweight, existing soft- +ware was needed that works in real-time on standard CPUs +in a wide variety of environments. The Robotic Operating +System (ROS) [17] is an open-source robotics framework that +allowed each of our hardware and software components to +communicate freely in real-time, and each software component +used was compatible with this framework. ORB-SLAM2 was +the ideal solution for mapping [18][15]. It uses differing +angles of static environmental features to create a map and a +keyframe-based SLAM approach that reduces the overall data +size of the SLAM map considerably [1]. Since the system is +designed with visual sensors, a software to visually identify +trash was necessary. YoLOv4 is a CNN model trained from +annotated images to place bounding boxes around specified +objects in RBG images. Adaptive Monte Carlo Localization +(AMCL) is the method of navigation used as well as the +name of a compatible software stack used for navigation +provided by ROS [8]. AMCL takes in odometry feedback +from the robot’s wheels and scan data derived from the RGB- +D Camera to navigate. After some static conversions from +ORB-SLAM2’s native map format to a 2D occupancy grid, +AMCL can autonomously navigate around an environment. +These existing software stacks served as the framework for +the multi-robot system to be built. +III. SYSTEM DESIGN +A. System Overview +Fig. 2. Systems’ Communication Flow +After initialization by a human operator, the mapping robot +will scan an area with a visual sensor. This sensor data will be +compiled using Simultaneous Location and Mapping (SLAM) +technology to create a continuous digital map of the target area +which will then be wirelessly transmitted to the unmanned +ground vehicle (UGV). The UGV will identify pieces of +trash in the environment using computer vision algorithms +and construct a two-dimensional map populated with target +coordinates of identified trash. The UGV will then create an +efficient path between the target coordinates in the map. Once +the UGV sets off on the calculated path, it will confirm the +trash location using an onboard visual sensor and proceed to +pick it up. Once the UGV has completed its rounds or the bin +is detected as full, it will return home, and a human operator +will empty the bin. +B. Mode Controller +Fig. 3. Mode Controller Flow +The “Mode Controller” was created to switch between the +three separate software components of the system: Mapping, +Navigation, and Greedy Pickup. The Mode Controller is a +ROS node that communicates with the Mapping, Navigation, +and Greedy Pickup nodes, turning them on or off as needed. +The Mode Controller starts in idle before putting the system +into the Mapping mode. Once mapping finishes, the mode +controller turns off Mapping mode. The map is then passed on + +Stage 1 +Mapping With +ORBslam +Realsense +map +Stage 2 +General +RGB + depth camera feed +Mode Control +Navigation +Stage 3 +Throughput +Greedy Pickup +YOLO +Control1.Mapping & +2. Navigation +TrashID +3. Greedy +Idle +Pickupto the Navigation mode alongside the coordinates of identified +trash. The Mode Controller then turns on Navigation mode. +Once a trash coordinate has been reached by the Navigation +mode, the Mode Controller next turns Navigation off and +Greedy Pickup on, picking up the trash. Navigation mode is +once again activated. Navigation and Greedy Pickup modes +will alternate until all trash is removed from the environment. +Once all the trash is picked up and no marked coordinates +remain on the map, the Mode Controller turns back to idle +and awaits further instruction. +C. Mapping +The system navigates the surrounding area and maps its +environment using ORB-SLAM2. This repository is designed +to be used within ROS as a ROS node. In our default RGB-D +configuration, the node subscribes to 2 topics (RGB and depth +image topics) and in turn, publishes all necessary data built +by the ORB-SLAM2 system. This includes a point cloud of +all map key points, the current camera pose, the full camera +path trajectory, and a morphologically transformed version of +the projected occupancy grid [21]. In experimentation, the +maps were initially filled with noise that led to an inability +to navigate the space, figure 4. Morphological operations +are commonly used tools in image processing to clean up +an image. By “eroding” and “opening” the space, errant +data points that were being misidentified as occupied were +removed. By “closing” the space gaps caused by the sparse +data, holes in our map were closed, and smooth, continuous +maps were generated, figure 5. The product was an occupancy +grid very close to real-world surroundings with a real-time, +lightweight mapping solution. +D. Trash Identification +Simultaneously, as an area is being mapped, the system also +detects trash. To recognize where on the map a piece of trash +is, the mapping robot first finds the location of a piece of +trash relative to itself. The system to locate trash was devised +using multiple components: YOLOv4, the odometry data of +the robot, and the depth camera feed provided by the Realsense +RGB-D camera, figure 7. +The first step in the trash identification pipeline is image +identification using YOLOv4. YOLOv4 is a convolutional +neural network that we trained with a custom dataset of +over 1000 images, each taken of varying pieces of trash +from the perspective of the robot. Each image was hand- +labeled and fed into the machine learning model using an +80-15-5 split between training, validation, and testing sets. +The model was trained and runs in our software stack using +a customized open-source ROS wrapper for YOLOv4 [24]. +The image identification model runs simultaneously while the +environment is being mapped using the RGB camera feed and +returns “bounding boxes” around identified trash pieces in the +image, providing coordinates relative to the camera’s image +frame, figure 6. +These bounding boxes provide 2D pixel locations for the +trash in the image but do not contain any information about +Fig. 4. Raw Occupancy Grid +Fig. 5. Morphologically Transformed Occupancy Grid +where the trash lies in the environment. Therefore, the next +step is to identify the angle of the closest piece of trash relative +to the camera. This is accomplished by using the center pixel +x-coordinate of an identified piece of trash. Using the field of +view of the camera, an imaginary triangle can be created to +discover the angle of the trash relative to the camera in the +real world by using pixels as the coordinate system. +The FOV angle is 69.4 degrees, its opposite side is 640 +pixels, and it is known to be an isosceles triangle, the re- +maining side lengths and angles can be extrapolated as this +is considered a trigonometrically “solved” triangle. Using this +triangle the angle of the identified trash piece is calculated +using the inverse tangent function, as shown in the following +equation and in figure 9. +θ = tan−1 trashx − 320 +462.139 +(1) +The next step in the localization process is to determine the +distance between the camera and the piece of trash. This is +accomplished using the depth camera feed provided by the +RGB-D camera. This camera outputs a grayscale image in +which each pixel is a 16-bit value representing the distance to +that pixel in millimeters directly from the center of the camera. +The depth picture can be indexed as a matrix using the 2D +coordinates given by YOLO’s bounding box to determine the +exact distance between the camera and any piece of trash. +Once the distance between the camera and the trash has been +calculated, all information necessary to localize the piece of + +Fig. 6. Trash Bounding Boxes +Fig. 7. Depth Feed +trash relative to the robot has been acquired. Using a second +triangle with coordinates in meters, both the angle of the trash +relative to the robot as well as the distance between the trash +and the robot can be extrapolated. +The first unknown variable encountered is the distance +between the trash and the center of the robot base, d. Using the +distance between the camera and the center of the robot base +s, as well as the distance between the trash and the camera +taken from the depth camera feed (depth), d can be solved +using the Law of Cosines as shown below. +c2& = a2 + b2 + 2ab cos(c) +(2) +c& = +� +a2 + b2 + 2ab cos(c) +(3) +d& = +� +(depth2 + r2 + 2(depth)(s)(cos(180◦ − θ)) +(4) +Once d is known, the final variable which needs solving is +β. This can be solved using the Law of Sines. +sin X +x +& = sin Y +y +(5) +sin(β) +depth & = sin(180◦ − θ) +d +(6) +β& = sin−1 +�depth sin(180◦ − θ) +d +� +(7) +Once the angle to the piece of trash relative to the robot and +the distance between these two points became known, these +values were added to the robot’s current position to realize the +piece of trash on the map. However, some difficulties arose +when the computer did not process the images fast enough. +Algorithm 1 Mapping Trash to a Map +Input: Robot’s path r in the map m, YOLO Bounding +Box b +Output: Pose of piece of trash in the map p, orientation +o of robot relative to p +confident pieces cp ← empty +▷ init array to hold all +confident pieces +for every item in b do +if items i’s trash confidence is greater than ct then +cp ← i +end if +end for +yolotimestamp yt ← b[0].timestamp +for pose pr in r do +time difference td ← abs(yt − pr) +if you don’t have a closest pose timestamp cpr to the +yolo timestamp yt then +cpr ← pr +smallest time difference std ← td +else +if td < std then +std ← td +end if +end if +end for +for each timestamp, image in depth camera history do +Find the closest depthimage di taken to b +end for +Set robots orientation o from when the picture was +taken +for trashpiece tp in cp do +get distance d of tp from di +get tp’s angle theta from robot base +trash x distance tdx ← o + (d ∗ cos θ) +trash y distance tdy ← o + (d ∗ sin θ) +p ← tdx, tdy +end for +YOLOv4, when run on the Intel NUC, processed images at a +throughput of 0.5-0.8 FPS with about 4-5 seconds of latency +from when the image was originally taken. This created a +large gap between the time when the image was taken and the +current position of the robot. To account for the processing +latency, the path of the robot as it was mapping is logged +with timestamps for every position in its path from ORB- +SLAM2. Once the mapping robot received a successful trash +detection, the ROS timestamp given from YOLOv4 from when +that image was taken was passed to the path, and a Pose is +output. It is from this Pose that distance d and angle β are +added to localize the piece of trash relative to the map itself. +In the figure 11, the thin blue line is the path of the robot +as it maps the area. The red arrow is the current position of +the robot in the map. The cyan arrow is the Pose where the +robot was when the YOLOv4 image was taken. From this cyan +Pose, a red trash detection is then finally placed on the map. + +rubbish:0.99Fig. 8. FOV Diagram +Fig. 9. FOV Trignometric Calculations +Fig. 10. Robot-Trash Trignometric Calculations +Every trash detection is plotted, and a separate anti-clustering +node averages these together, getting an approximate location +of the piece of trash. +E. Anti-clustering +Initially, we found consistency issues with the trash identifi- +cation. Either images of the same piece of trash were processed +more than once, or the trash’s estimated position became +inaccurate as the SLAM map updated. This problem led to +a large amount of noise, causing up to and exceeding thirty +detections for two pieces of trash in one single trial. In some +limited cases, our YoLOv4 model would also erroneously +classify a random background object as trash. To sort through +the noisy detections, each new trash detection was run through +Fig. 11. Robot Trajectory in the map +a filter. Every time a piece of trash was detected, a ROS +subscriber would listen to the detection and determine if it was +a new piece or detection of a piece of trash already found. A +clustered piece of trash is denoted by the green mark in figure +11. +To accomplish this, all the detected pieces of trash were +stored at their initial positions. If any new trash detection +was within a set radius of a previously detected piece of +trash, the new trash detection became combined with the +established piece by taking a rolling average of the detections. +The calculations are seen in the equation 8, where p1x/y is the +existing trash detection’s respective x and y coordinate, p2x/2y +is the new trash detection’s x and y coordinates, and a is the +amount of times p1 has been averaged to that point. +p1x = p1xa + p2x +a + 1 +p1y = p1ya + p2y +a + 1 +(8) +This anti-clustering algorithm decreased the total amount of +detections to accurately reflect the number of trash pieces seen +in the environment. To avoid erroneous trash detections, aver- +age trash locations without enough detections were determined +as “noisy” and filtered out. Trash points were only published to +the navigation stack if it had three or more detections averaged +to one point. +F. General Navigation and Path Planning +General navigation consists of two parts: localization and +path planning. The Robot first receives the 2D occupancy +grid from our mapping software, alongside the trash detection +coordinates. Once these data are received, the robot then +navigates to within two meters of the nearest detected trash +point. Navigation only needs to navigate near a trash location +since greedy pickup is routinely effective within a two meter +distance, and the anti-clustering algorithm accounts for noise +in our trash detections. +To get the goal pose g you need it’s orientation, and x/y +coordinates. The equation 9 describes how to get the angle +for the pose where p1 is the starting robot pose and p2 the +trash pose. In equation 10 the distance the goal pose is from +the robot is calculated to have it within Greedy Pickup range, + +-ldentifiedTrashPiece +trash_depth (m) +180°-0 +0.6096m +KobukiRobot--ldentifiedTrashPiece +640Pixels +RGBCameraFrame +CameraFOV- +69.4° +RealsenseD435 +RGB-DCamera-ldentifiedTrashPiece +trash x +320-(640-trash_x)- +462.139 PixelsAlgorithm 2 Anti-Clustering Algorithm +Input: Trash Pose tp +Output: Poses of clustered trash pieces ctp +averaged Trash poses atp tupled with times averaged ta +(atp, ta) ← empty +if atp = 0 then +atp ← tp +ta ← 1 +else +for every pose tuple pt in atpt do +get x and y bottom and top around the trash’s +location +xb, xt, yb, yt +if xb ≤ tpx ≤ xt & yx ≤ tpy ≤ yt then +pose tuple x ptx = (ptx * ta + tpx) / (ta + 1.0) +pose tuple y pty = (pty * ta + tpy) / (ta + 1.0) +ta + 1 +end if +end for +if tp is not in atp then +atp ← tp +end if +ctp ← all trash poses tp in atp where tpi’s tai > 2 +times +end if +where d3 is the goal distance, d1 is the distance between +the trash and the robot, and d3 is Greedy Pickup’s range. In +equation 11 the coordinates of the goal pose is found where +gx/y is represents the coordinates respectively. +θ = arctan(p1y − p2y +p1x − p2x +) +(9) +d3 = d1 − d2 +(10) +gx = d3 + p1x cos(θ)gy = d3 + p2x sin(θ) +(11) +Fig. 12. Robot Path Planning in Rviz +The Navigation and path planning stack was based on the +ROS-provided open-source AMCL software stack. This soft- +ware loads and localizes the robot in a mapped environment +and its DWA planner creates a path between the identified +points of trash. We created another software module that feeds +our target coordinates from our trash detection phase into +AMCL’s path planner to follow the most efficient path between +the robot’s current location and the nearest possible trash point, +figure 12. +G. Greedy Pickup +Once the Navigation portion of the software stack places +the robot within 2 meters of the piece of trash, Greedy Pickup +is activated. Greedy Pickup is an asynchronous algorithm that +ignores all navigation and map factors and solely focuses on +seeking out the nearby trash directly. +When the greedy pickup is activated, it rotates in a given +direction to look for trash using YoLOv4. Once it receives a +trash detection, it calculates the position of the trash using the +same algorithm explained in the Trash Identification section. +After the robot localizes the trash, it turns back towards the +trash at its precise angle, turns on the collection mechanism’s +motor, and moves exactly 0.2m past the trash’s location to +ensure proper collection. Once this occurs, the motor turns off +and navigation resumes. +Algorithm 3 Greedy Pickup +Input: Detected Trash pose list tpl +set timeout time to +set confidence threshold ct +for Trash pose tp in tpl do +Determine whether tp is to the left or right of the robot +in the Map +Spin in direction of tp, scanning for confirmation +if Robot gets tp ≥ ct then +Robot Stop spinning +Use III-D algorithm to find relevant Robot orienta- +tion and trash pose +Get destination angle +Robot turns to destination angle +Robot Turn on collection mechanism and drive over +detected piece of trash +else +Robot exceeded to looking for trash +end if +end for +Output: Poses of clustered trash pieces ctp +For the collection mechanism to turn on or off, the NUC +sends a serial packet to the connected Arduino with only three +bytes in sequence, either [0x59, 0x59, 0x59] to start the motor +or [0x4E, 0x4E, 0x4E] to stop the motor. Once the Arduino +receives a start packet, it then outputs a PWM signal to two +of its GPIO pins which control the L928N motor controller. +The PWM signal gradually increases from a low duty cycle to +a higher duty cycle to control the current spikes on the 12V +line from the Robot base to the Motor. In the initial design on +bench power, starting the motor from 0 to full power produced +an initial current spike of approximately 1.9A before settling +around 0.9-1.1A when in normal motion or picking up an + +object. The initial spike was over the 1.5A limit provided by +the 12V port accessible on the robot base. To eliminate this +spike, a slow ramp-up of the duty cycle of the PWM signal +was introduced from 20% duty cycle to a maximum of 80% +linearly over the course of 5 seconds. This removes the initial +current spike and ensures that the motor can both properly +power the collection mechanism and does not exceed the 1.5A +current limit. +H. ROS Middleware +ROS was used to connect all the functions of this system. +Every design block in figure 2 functions as a node, or multiple +nodes, which subscribe and publish information to the other +nodes. ROS would also be used to network between the +different robots in the multi-robot system over WiFi or other +wireless protocols. +IV. EXPERIMENT +A. Goal +The purpose of this experiment is to assert the feasibility +of this system design before iterating on the hardware used to +make it scalable and adaptable. The robot was evaluated on +its ability to accurately map the enclosure, identify, and mark +the pieces of trash, choose an efficient path, and pick the trash +up. +The robotic system is meant to clear trash as large and heavy +as an average 600mL Spring Valley Water bottle weighing ap- +proximately 0.64 kg. The system’s robotic base, the TurtleBot +2, has a load limitation of approximately 5 kilograms [20], +which presents an upper weight limit on the total load. Since +the robot’s additional components (external frame, storage +container, etc.) are estimated to weigh approximately 3 kg, +the trash load must weigh at most 2 kg. +The UGV operates in narrow environmental parameters. The +weather must be clear with no rain since the electronic systems +onboard the UGV are non-weatherproofed. In addition, due to +the wheels that come default with the robotic base (Kobuki +Mobile Base), our prototype can only operate on relatively flat, +smooth, evenly colored surfaces, with no extreme movement +in the background. +B. Testing In Simulation +Simulated testing was done in the Gazebo Robotics simu- +lator, 13. This simulator was included in the base Turtlebot +SDK and includes a near true-to-life recreation of the entire +turtlebot system. The use of ROS allows for the navigation +stack and greedy pickup to be run against the simulator and +behave exactly identically to reality. This simulator was instru- +mental in the initial testing of movement and navigation as it +allowed our team to test many different speed parameters and +movement algorithms without risking any physical damage to +the robot. All data associated with movement and mapping +were recorded and played back in a simulated environment to +recreate and reevaluate our physical testing. This allowed for +useful visualizations and assessments of what the robot was +processing at any given time. +Fig. 13. Simulation Testing Still +C. Testing the Hardware +1) Experiment System: The system’s design is meant to +function with a mapping robot and a ground vehicle. To +simulate the mapping vehicle in this experiment the UGV +plays both roles. The UGV first passed through the area to +get a map and identify trash. Then taking that information to +navigate and pick up the trash. +2) The Robot: +The hardware design uses a modified +Turtlebot-2 as a base design, figure 15, that has four main +components: the depth camera (480p RGB-D Intel Realsense +D435) [10], the computer (7-year-old Intel NUC with 8GB +of RAM) which runs a GNU-Linux OS along with ROS to +manage sensor data collection and real-time processing, the +mobile robotic base (Kobuki Mobile Base) [12], and a custom- +designed collection mechanism. The camera relays RGB and +depth images which are processed to identify and target trash. +The Kobuki’s motor and wheels relay odometric feedback +that helps confirm the UGV’s current location. The collection +mechanism attaches to the front of the Kobuki Mobile base, +plugs into power and data ports on the robot, and uses a rotary +brush to pick up the trash. +3) The Collection Mechanism: The collection mechanism +is a custom-designed addition to the Turtlebot Robot. Its +mechanical construction consists of 20-20 aluminum bars +connected by 90-degree brackets. To allow free range of +motion, caster wheels were affixed to the bottom of the frame. +When the collection mechanism motor is activated, it sweeps +trash up a ramp into a plastic storage container. A camera +mount was printed so the Realsense could be attached to the +front of mechanism [23]. A funnel was added to the front of +the collection mechanism to rein in the trash that may have +been missed slightly, figure 18. +The design of the electronics system for the collection +mechanism is an Arduino Mega that is connected to the NUC +using a USB cable, figure 17. The Arduino is then connected +to an L982N Motor driver breakout board over PWM-enabled +GPIO pins (5V). This motor driver breakout board is supplied +with 12V by the Kobuki base from a 12V, 1.5A Max port. The +output of the L982N is the DC motor which drives the chain +and the brush. + +Fig. 14. Hardware Overview Diagram +Fig. 15. Turtlebot2 Base System +The brush itself was hand-designed and fabricated since +there is no commercial brush model that fit the design speci- +fications for the mobile robot, figure 16. +D. The Environment +The environment where the robot was tested was a room +with a random configuration of chairs and obstacles put around +an open space. Therefore, the map would be created each +time in a dynamic environment and the system would have +to account for a new configuration. In that space, trash was +put in different locations for all tests. We tested up to four +pieces of trash in the environment at a time. +E. Tests +1) Testing Mapping/ Image Identification accuracy/ Ac- +counting for Latency: To test the trash detection and how +well the processing latency was accounted for, expected maps +with the approximate trash locations were checked against the +created ones. +2) Testing the Greedy Pickup Algorithm: To test Greedy +Pickup, the mode was enabled with varying pieces of trash +within two meters of the robot. The piece of trash would start +out of the UGV’s perception range. The UGV would have to +scan for the trash, identify it and then collect it. This trial was +Fig. 16. Custom Designed Brush +Fig. 17. Motor Circuitry +Fig. 18. Final Construction +run with trash at 2 meters, 1 meter, and half a meter distance +from the UGV. +3) Full System Trials: Full system trials were then con- +ducted, starting with the mapping of an environment and +labeling trash points in that environment. Then to navigating +to the labeled pieces of trash in the environment and collecting +the pieces of trash using Greedy Pickup. +V. RESULTS +A. Accounting for Processing Latency +The map creation and the image identification, figure 23, +were shown to be an accurate, figure 19, fast and lightweight +way of monitoring an area and identifying pieces of trash. +B. Greedy Pickup Results +Greedy pickup is shown to be accurate to an extent with +a success rate of 77.78%, figure 20. There were multiple +failures in these trials that were caused by the design of +the collection mechanism. The collection mechanism was not +equipped with odometric wheels that could relay its’ location +so, at times it would overturn, and the caster wheels introduced +an error the system was not aware of and could not account + +Intel Realsense +Camera +RGB + depth images +collection +Intel NUC +mechanism +motor feedback + control +Kobuki +Mobile Base12VDCMotor +CURRENT SENSING A +CURRENT SENSING +OUTPUT +OUTPUT 2 +OUTPUT +GNT +SUPPLY VOLTAGE VS +INPUT +ENABLE +INPUT +ENABLEA +LOGIC SUPPLY VOLTAGE VSS +INPUT +INPUT 2 +GND +OME +L928N Motor Driver +TXO +8 +12V +REOET +RX3 +TX3 +MEGA +91 +ROTAFig. 19. Accurate Map Creation with Identified Trash Points Result +Fig. 20. Greedy Pickup Results +Fig. 21. Full System Run-through Results +Fig. 22. Real Time Trash Identification and Map Creation +for. This caused the UGV to over or under turn occasionally +and not successfully collect the trash. The Greedy Pickup +algorithm has shown the ability to increase the error tolerance +of the system, however, there are more improvements to the +algorithm that could be made to make it more error-tolerant. +For example, an added “lock on” mechanism that when a piece +of trash is in sight it would keep the piece of trash in the center +of its FOV as it moves forward to collect the piece of trash. +Ensuring that the failure-causing edge cases are accounted for. +Fig. 23. Still of UGV Picking up Water Bottle in System Test +C. Full System Results +These results, with a 68% success rate, figure 21, show +that there are possible errors that can be introduced to this +system and that they compound on themselves as the task gets +more complex. During these trials, the collection mechanism +introduced errors during navigation and greedy pickup. The +collection mechanism introduced errors by being slightly out +of position. Since the Collection mechanism did not have +sensor feedback to tell the UGV it was not in the correct +position the little errors compounded into failures to collect +trash. +VI. CONCLUSION +Our contribution is a multi-robot system design applied +to monitoring and managing roadside litter. We developed a +system that can map, identify, and pick up pieces of trash. The +system is designed to be relatively cheap and scalable which +could be done because we introduced different algorithms that +account for errors in a system that has less precise sensors. +Our Greedy Pickup algorithm accounts for errors in trash iden- +tification, and trash can be accurately identified in a system +with low processing power. Our custom-designed collection +mechanism, designed to pick up the main offending types of +trash found on the side of the road, was also introduced. +This system builds off previously known algorithms: AMCL +and DWA for navigation, Orbslam-2 for map creation, and +YOLO.v4 for trash identification. It also build off multi-robot +systems to monitor a dynamic environment and combines that +with information with a dynamic collection algorithm, Greedy +Pickup, to create an efficient collection process. +The next steps for this system would to be refine the +mechanical design of the UGV, such as adding odometry +sensors onto the collection mechanism’s wheels and improving +the wheels to be able to drive over more difficult terrains. +There are also improvements to be made to the Greedy Pickup +algorithm such as adding a “Lock on” capability. Better logic +could be applied to the navigation, for example, a future +iteration of this system could use a graph search algorithm + +Mapping and Trash Identification Results +8% +92% + Successful Maps +■ Failed MapsGreedy Pickup Success +16 +14 +12 +10 +8 +6 +4 +2 +0 +2 meter +1 meter +0.5 meter +Robot's Distance from Trash +Total Trials +Successful trialsFull System Tests +Aount of Trash in the Environment +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +Successful Trials + Total Trials[ Interact Move Camera + Select +Focus CameraMeasure2D Pose Estimate2DNavGoalPublishPoint +Q +RGBImage +RGB View +ORB-SLAM2Ima +RViz +SLAMView +OOYOLOV4 +Map View +YOLOView +199.22 +Wall Elapsed:416.55 + Experimentalto find the most efficient path between every piece of trash. +An aerial robot can also be introduced as the mapping robot in +the next iteration of this system for testing, to more accurately +represent the issues that would arise from changes in the +perspective of the system. More replicas of the UGV could be +added into the system as well, so there would be multiple trash +collectors in one environment at the same time. The system +could also have mapping and trash collection happening in +parallel instead of the modes being sequential. +VII. ACKNOWLEDGMENTS +This is a video of a full run-through our system did with +two pieces of trash in the environment [5]. This is the central +repository in our organization that sets up the system [4]. +This is the Greedy Pickup repository which holds all the +functions laid out in this paper [3]. We would like to thank +the contributions of Divya Venkatraman, Jared Raines and +Catherine Ellingham for their work on gathering data, editing +the paper and system networking. We’d also like to thank Dr. +Taskin Padir and his Robotics and Intelligent Ground Vehicle +Research Laboratory (RIVeR) [22] for allowing us to use their +hardware to test our design. +REFERENCES +[1] +Yong-bao Ai et al. “Visual SLAM in dynamic environ- +ments based on object detection”. In: Defence Technol- +ogy (2020). ISSN: 2214-9147. DOI: https://doi.org/10. +1016/j.dt.2020.09.012. URL: https://www.sciencedirect. +com/science/article/pii/S2214914720304402. +[2] +Keep Louisiana Beautiful. Executive Summary: Litter in +America. URL: https://keeplouisianabeautiful.org/wp- +content/uploads/2015/09/Litter-in-America-Executive +Summary - FINAL.pdf. (Accessed: 10-June-2021). +[3] +John Chiaramonte, Jack Fenton, and Lee Milburn. cave- +man mode. URL: https://github.com/Capstone- W3/ +caveman mode. (Accessed: 20-April-2021). +[4] +John Chiaramonte et al. Capstone-W3. URL: https:// +github.com/Capstone-W3. (Accessed: 11-June-2021). +[5] +John Chiaramonte et al. T.R.A.S.H. Demonstration. +URL: https : / / www . youtube . com / watch ? v = +cIwk0Kh9k7E. (Accessed: 5-July-2021). +[6] +Maria Valera Espina et al. “Multi-robot Teams for +Environmental Monitoring”. In: Innovations in Defence +Support Systems – 3: Intelligent Paradigms in Secu- +rity. Ed. by Paolo Remagnino, Dorothy N. Monekosso, +and Lakhmi C. Jain. Berlin, Heidelberg: Springer +Berlin Heidelberg, 2011, pp. 183–209. ISBN: 978-3- +642-18278-5. DOI: 10.1007/978- 3- 642- 18278- 5 8. +URL: https://doi.org/10.1007/978-3-642-18278-5 8. +[7] +Conserve Energy Future. The Catastrophic Effects of +Littering on Humans. URL: https : / / www. conserve - +energy-future.com/littering-effects-humans-animals- +environment.php. (Accessed: 14-June-2021). +[8] +Brian Gerkey. AMCL Package Summary. URL: https: +//wiki.ros.org/amcl?distro=melodic. (Accessed: 10- +March-2022). +[9] +Le Hong, Weicheng Cui, and Hao Chen. “A Novel +Multi-Robot Task Allocation Model in Marine Plastics +Cleaning Based on Replicator Dynamics”. In: Journal +of Marine Science and Engineering 9.8 (2021). ISSN: +2077-1312. DOI: 10.3390/jmse9080879. URL: https: +//www.mdpi.com/2077-1312/9/8/879. +[10] +Intel. Intel® RealSense™ Depth Camera D435. URL: +https://ark.intel.com/content/www/us/en/ark/products/ +128255 / intel - realsense - depth - camera - d435 . html. +(Accessed: 5-July-2021). +[11] +Alaa Khamis, Ahmed Hussein, and Ahmed Elmogy. +“Multi-robot Task Allocation: A Review of the State- +of-the-Art”. In: vol. 604. May 2015, pp. 31–51. ISBN: +978-3-319-18299-5. DOI: 10.1007/978-3-319-18299- +5 2. +[12] +Iclebo Kobuki. Kobuki User Guide. URL: http://kobuki. +yujinrobot . com / wiki / online - user- guide/. (Accessed: +11-June-2021). +[13] +Nathan Koenig and Andrew Howard. “Design and Use +Paradigms for Gazebo, An Open-Source Multi-Robot +Simulator”. In: (). +[14] +Stephanie Melchor. Roadside Trash A Growing Prob- +lem. URL: https://www.montereyherald.com/2021/01/ +23/roadside-trash-a-growing-problem/. (Accessed: 10- +June-2021). +[15] +Ra´ul Mur-Artal, J. M. M. Montiel, and Juan D. Tard´os. +“ORB-SLAM: A Versatile and Accurate Monocular +SLAM System”. In: IEEE Transactions on Robotics +31.5 (2015), pp. 1147–1163. DOI: 10.1109/TRO.2015. +2463671. +[16] +Shinkyu Park, Yaofeng Desmond Zhong, and Naomi +Ehrich Leonard. “Multi-Robot Task Allocation Games +in Dynamically Changing Environments”. In: 2021 +IEEE International Conference on Robotics and Au- +tomation (ICRA). 2021, pp. 8678–8684. DOI: 10.1109/ +ICRA48506.2021.9561809. +[17] +Morgan Quigley et al. “ROS: an open-source Robot +Operating System”. In: Proc. of the IEEE Intl. Conf. +on Robotics and Automation (ICRA) Workshop on Open +Source Robotics. Kobe, Japan, May 2009. +[18] +raulmur. ORB SLAM2. URL: https : / / github . com / +raulmur/ORB SLAM2. (Accessed: 11-June-2021). +[19] +rayvburn. ORBSLAM2 ROS. URL: https://github.com/ +rayvburn / ORB - SLAM2 ROS. (Accessed: 11-June- +2021). +[20] +Clearpath Robotics. TurtleBot 2 - Open Source Personal +Research Robot. URL: https://clearpathrobotics.com/ +turtlebot - 2 - open - source - robot. (Accessed: 10-June- +2021). +[21] +Adrian Rosebrock. OpenCV Morphological Operations. +URL: https://pyimagesearch.com/2021/04/28/opencv- +morphological-operations/. (Accessed: 11-June-2021). +[22] +Padir Taskin. RIVeR. URL: https : / / robot . neu . edu/. +(Accessed: 7-July-2021). + +[23] +Makerbot Thingiverse. Intel RealSense D435 Camera +Mount for OpenBuilds 20x20mm Extrusion. URL: https: +//www.thingiverse.com/thing:2965905. +[24] +Tossy0423. darknet ros. URL: https : / / github . com / +Tossy0423 / darknet ros / tree / master/. (Accessed: 11- +June-2021). + diff --git a/6dAzT4oBgHgl3EQfvP2u/content/tmp_files/load_file.txt b/6dAzT4oBgHgl3EQfvP2u/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7dbffb2faf133d47a41794a6c323a6e1e9892560 --- /dev/null +++ b/6dAzT4oBgHgl3EQfvP2u/content/tmp_files/load_file.txt @@ -0,0 +1,530 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf,len=529 +page_content='Error Tolerant Multi-Robot System for Roadside Trash Collection 1st Lee Milburn College of Engineering Northeastern University Boston, Massachusetts milburn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='l@northeastern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='edu 2nd John Chiaramonte College of Engineering Northeastern University Boston, Massachusetts chiaramonte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='j@northeastern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='edu 3rd Jack Fenton College of Engineering Northeastern University Boston, Massachusetts fenton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='j@northeastern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='edu Abstract—In this paper, we present the first iteration of an error-tolerant, autonomous, multi-robot system that monitors highway road verges and identifies and collects roadside litter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' It is designed to use an aerial vehicle that can rapidly cover a vast area and collect data on the road verge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' This data is then passed to a ground vehicle that constructs a map of the road verge and uses a trained Convolutional Neural Network (CNN) to identify pieces of litter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' After the pieces of litter are identified on the map of the road verge, the ground robot navigates to each piece of trash, re-evaluates the area, and performs a ”greedy pickup” procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' This final stage accounts for any error in the map’s construction or the identified trash’s location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' We found that ending the robotic system’s control flow with a greedy pickup procedure can retroactively account for processing errors of the system as it runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' This increases the system’s fault tolerance and allows for the use of cheaper equipment since pinpoint accuracy is not always necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' In this paper, we present the feasibility of this system by testing in simulation and later using real robotic hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' We show that the system is effective enough to iterate on its design principles to create a more sophisticated system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Index Terms—Autonomous trash collection, Environmental monitoring, Error tolerance, Multi-robot system I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' INTRODUCTION Roadside trash is a massive issue currently managed by manual labor - a woefully inadequate solution [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Despite being a nationwide issue, the task of waste management is mostly under the jurisdiction of municipalities and it garners little to no attention or investment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' To estimate the amount of litter along roadways, a research team selected a random sample of 240 roadway segments, stratified by type and by rural/urban areas [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The results indicate that there are 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='2 billion pieces of litter on roadways nationwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Of this, the majority (91%, or 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='6 billion pieces) is less than four inches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' In Monterey City, California, complaints about trash have increased since the start of the COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Officials say this is not due to increased littering, but rather due to the inability to clean it up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Monterey County Public Works Maintenance Manager Shawn Atkins stated that his cleanup crew was so busy cleaning up from illegal dumpsites that they did not have time to walk the shoulders of their roads to pick up loose trash [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Caltrans, California’s public transportation department, has been faced with the same problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Kevin Drabinski, public information officer for Caltrans District 5, said it’s important to Caltrans that they manage litter because of safety and environmental concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Caltrans spends $50 million annually on litter cleanup [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' To address this issue, our multi-robot system uses a three- stage approach to autonomously map, identify, and pick up trash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The three modes are Mapping, Navigation, and Greedy Pickup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' We would first have a lightweight drone fly over a specified area on the road and stream its visuals to a ground robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' That robot would generate a map using the drone’s input and identify trash pieces on that map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The ground robot would then navigate near each piece of identified trash and then switch to the greedy pickup mode where it scans the area for the suspected piece of trash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' After a piece of trash is re-identified locally, the robot moves and collects it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Once either collected or not found, the ground robot then moves to the next piece on its map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' System Design Overview Our approach allows for accurate pickup without the need for massive processing power or overly expensive sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The system’s configuration used the open-source convolutional neural network You Only Look Once (YoLOv4) for image identification [24], the open-source visual SLAM solution ORBSLAM-2 for map-building [19], and open-source ROS navigation software for path planning and navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' We simulated this system using Gazebo [13] and after receiving consistent results, tested it in a real-world environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Our arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='01704v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='RO] 4 Jan 2023 Step 1: Mapping Robot identifies trash and maps Step 2: Mapping Robot transfers map data to UGV the surrounding area over Wi-Fi Step 3: UGV takes most efficient path to pick up Step 4: Human operator removes storage bin and trash and clear area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' dumps trash into larger containerreal-world results, with a relatively low-powered system, in- dicate that our approach is a proof-of-concept for a scalable and viable solution to the growing worldwide litter problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' RELATED WORK There has been research into multi-robot systems used for environmental monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The research for these sys- tems finds that multi-robot systems pose a more effective solution to surveying an environment than static monitoring [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' There is also research into multi-robot systems that do autonomous trash collection [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' This research concludes that for maximum efficiency, robots should be aware of their environment when trying to collect trash as opposed to making decisions based solely on their field of view (FOV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Therefore, these two systems, monitoring an environment, and using a collection algorithm to pick up trash in a dynamically changing environment could be combined to create the most effective version of an autonomous collection system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' A version of this system has been created to autonomously collect and monitor plastics in rivers [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The system includes a central processor that takes in the necessary tasks of the environment and assigns those tasks to underwater autonomous vehicles that then pick up the plastics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' This system concluded that a Multi-robot task allocation architecture [11] with a controlling center increased the efficiency of the system, but the hardware for working effectively in that environment would have to be improved for more effective use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Our research team structured our multi- robot system design to have a robot monitor the environment and wirelessly transmit its environmental depiction to a UGV that would collect the trash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' For our system to be cheap and lightweight, existing soft- ware was needed that works in real-time on standard CPUs in a wide variety of environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The Robotic Operating System (ROS) [17] is an open-source robotics framework that allowed each of our hardware and software components to communicate freely in real-time, and each software component used was compatible with this framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' ORB-SLAM2 was the ideal solution for mapping [18][15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' It uses differing angles of static environmental features to create a map and a keyframe-based SLAM approach that reduces the overall data size of the SLAM map considerably [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Since the system is designed with visual sensors, a software to visually identify trash was necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' YoLOv4 is a CNN model trained from annotated images to place bounding boxes around specified objects in RBG images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Adaptive Monte Carlo Localization (AMCL) is the method of navigation used as well as the name of a compatible software stack used for navigation provided by ROS [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' AMCL takes in odometry feedback from the robot’s wheels and scan data derived from the RGB- D Camera to navigate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' After some static conversions from ORB-SLAM2’s native map format to a 2D occupancy grid, AMCL can autonomously navigate around an environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' These existing software stacks served as the framework for the multi-robot system to be built.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' SYSTEM DESIGN A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' System Overview Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Systems’ Communication Flow After initialization by a human operator, the mapping robot will scan an area with a visual sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' This sensor data will be compiled using Simultaneous Location and Mapping (SLAM) technology to create a continuous digital map of the target area which will then be wirelessly transmitted to the unmanned ground vehicle (UGV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The UGV will identify pieces of trash in the environment using computer vision algorithms and construct a two-dimensional map populated with target coordinates of identified trash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The UGV will then create an efficient path between the target coordinates in the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Once the UGV sets off on the calculated path, it will confirm the trash location using an onboard visual sensor and proceed to pick it up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Once the UGV has completed its rounds or the bin is detected as full, it will return home, and a human operator will empty the bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Mode Controller Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Mode Controller Flow The “Mode Controller” was created to switch between the three separate software components of the system: Mapping, Navigation, and Greedy Pickup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The Mode Controller is a ROS node that communicates with the Mapping, Navigation, and Greedy Pickup nodes, turning them on or off as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The Mode Controller starts in idle before putting the system into the Mapping mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Once mapping finishes, the mode controller turns off Mapping mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The map is then passed on Stage 1 Mapping With ORBslam Realsense map Stage 2 General RGB + depth camera feed Mode Control Navigation Stage 3 Throughput Greedy Pickup YOLO Control1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='Mapping & 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Navigation TrashID 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Greedy Idle Pickupto the Navigation mode alongside the coordinates of identified trash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The Mode Controller then turns on Navigation mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Once a trash coordinate has been reached by the Navigation mode, the Mode Controller next turns Navigation off and Greedy Pickup on, picking up the trash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Navigation mode is once again activated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Navigation and Greedy Pickup modes will alternate until all trash is removed from the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Once all the trash is picked up and no marked coordinates remain on the map, the Mode Controller turns back to idle and awaits further instruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Mapping The system navigates the surrounding area and maps its environment using ORB-SLAM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' This repository is designed to be used within ROS as a ROS node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' In our default RGB-D configuration, the node subscribes to 2 topics (RGB and depth image topics) and in turn, publishes all necessary data built by the ORB-SLAM2 system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' This includes a point cloud of all map key points, the current camera pose, the full camera path trajectory, and a morphologically transformed version of the projected occupancy grid [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' In experimentation, the maps were initially filled with noise that led to an inability to navigate the space, figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Morphological operations are commonly used tools in image processing to clean up an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' By “eroding” and “opening” the space, errant data points that were being misidentified as occupied were removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' By “closing” the space gaps caused by the sparse data, holes in our map were closed, and smooth, continuous maps were generated, figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The product was an occupancy grid very close to real-world surroundings with a real-time, lightweight mapping solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Trash Identification Simultaneously, as an area is being mapped, the system also detects trash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' To recognize where on the map a piece of trash is, the mapping robot first finds the location of a piece of trash relative to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The system to locate trash was devised using multiple components: YOLOv4, the odometry data of the robot, and the depth camera feed provided by the Realsense RGB-D camera, figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The first step in the trash identification pipeline is image identification using YOLOv4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' YOLOv4 is a convolutional neural network that we trained with a custom dataset of over 1000 images, each taken of varying pieces of trash from the perspective of the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Each image was hand- labeled and fed into the machine learning model using an 80-15-5 split between training, validation, and testing sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The model was trained and runs in our software stack using a customized open-source ROS wrapper for YOLOv4 [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The image identification model runs simultaneously while the environment is being mapped using the RGB camera feed and returns “bounding boxes” around identified trash pieces in the image, providing coordinates relative to the camera’s image frame, figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' These bounding boxes provide 2D pixel locations for the trash in the image but do not contain any information about Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Raw Occupancy Grid Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Morphologically Transformed Occupancy Grid where the trash lies in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Therefore, the next step is to identify the angle of the closest piece of trash relative to the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' This is accomplished by using the center pixel x-coordinate of an identified piece of trash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Using the field of view of the camera, an imaginary triangle can be created to discover the angle of the trash relative to the camera in the real world by using pixels as the coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The FOV angle is 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='4 degrees, its opposite side is 640 pixels, and it is known to be an isosceles triangle, the re- maining side lengths and angles can be extrapolated as this is considered a trigonometrically “solved” triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Using this triangle the angle of the identified trash piece is calculated using the inverse tangent function, as shown in the following equation and in figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' θ = tan−1 trashx − 320 462.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='139 (1) The next step in the localization process is to determine the distance between the camera and the piece of trash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' This is accomplished using the depth camera feed provided by the RGB-D camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' This camera outputs a grayscale image in which each pixel is a 16-bit value representing the distance to that pixel in millimeters directly from the center of the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The depth picture can be indexed as a matrix using the 2D coordinates given by YOLO’s bounding box to determine the exact distance between the camera and any piece of trash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Once the distance between the camera and the trash has been calculated, all information necessary to localize the piece of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Trash Bounding Boxes Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Depth Feed trash relative to the robot has been acquired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Using a second triangle with coordinates in meters, both the angle of the trash relative to the robot as well as the distance between the trash and the robot can be extrapolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The first unknown variable encountered is the distance between the trash and the center of the robot base, d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Using the distance between the camera and the center of the robot base s, as well as the distance between the trash and the camera taken from the depth camera feed (depth), d can be solved using the Law of Cosines as shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' c2& = a2 + b2 + 2ab cos(c) (2) c& = � a2 + b2 + 2ab cos(c) (3) d& = � (depth2 + r2 + 2(depth)(s)(cos(180◦ − θ)) (4) Once d is known, the final variable which needs solving is β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' This can be solved using the Law of Sines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' sin X x & = sin Y y (5) sin(β) depth & = sin(180◦ − θ) d (6) β& = sin−1 �depth sin(180◦ − θ) d � (7) Once the angle to the piece of trash relative to the robot and the distance between these two points became known, these values were added to the robot’s current position to realize the piece of trash on the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' However, some difficulties arose when the computer did not process the images fast enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Algorithm 1 Mapping Trash to a Map Input: Robot’s path r in the map m, YOLO Bounding Box b Output: Pose of piece of trash in the map p, orientation o of robot relative to p confident pieces cp ← empty ▷ init array to hold all confident pieces for every item in b do if items i’s trash confidence is greater than ct then cp ← i end if end for yolotimestamp yt ← b[0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='timestamp for pose pr in r do time difference td ← abs(yt − pr) if you don’t have a closest pose timestamp cpr to the yolo timestamp yt then cpr ← pr smallest time difference std ← td else if td < std then std ← td end if end if end for for each timestamp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' image in depth camera history do Find the closest depthimage di taken to b end for Set robots orientation o from when the picture was taken for trashpiece tp in cp do get distance d of tp from di get tp’s angle theta from robot base trash x distance tdx ← o + (d ∗ cos θ) trash y distance tdy ← o + (d ∗ sin θ) p ← tdx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' tdy end for YOLOv4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' when run on the Intel NUC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' processed images at a throughput of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='5-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='8 FPS with about 4-5 seconds of latency from when the image was originally taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' This created a large gap between the time when the image was taken and the current position of the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' To account for the processing latency, the path of the robot as it was mapping is logged with timestamps for every position in its path from ORB- SLAM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Once the mapping robot received a successful trash detection, the ROS timestamp given from YOLOv4 from when that image was taken was passed to the path, and a Pose is output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' It is from this Pose that distance d and angle β are added to localize the piece of trash relative to the map itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' In the figure 11, the thin blue line is the path of the robot as it maps the area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The red arrow is the current position of the robot in the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The cyan arrow is the Pose where the robot was when the YOLOv4 image was taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' From this cyan Pose, a red trash detection is then finally placed on the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' rubbish:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='99Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' FOV Diagram Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' FOV Trignometric Calculations Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Robot-Trash Trignometric Calculations Every trash detection is plotted, and a separate anti-clustering node averages these together, getting an approximate location of the piece of trash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Anti-clustering Initially, we found consistency issues with the trash identifi- cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Either images of the same piece of trash were processed more than once, or the trash’s estimated position became inaccurate as the SLAM map updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' This problem led to a large amount of noise, causing up to and exceeding thirty detections for two pieces of trash in one single trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' In some limited cases, our YoLOv4 model would also erroneously classify a random background object as trash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' To sort through the noisy detections, each new trash detection was run through Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Robot Trajectory in the map a filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Every time a piece of trash was detected, a ROS subscriber would listen to the detection and determine if it was a new piece or detection of a piece of trash already found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' A clustered piece of trash is denoted by the green mark in figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' To accomplish this, all the detected pieces of trash were stored at their initial positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' If any new trash detection was within a set radius of a previously detected piece of trash, the new trash detection became combined with the established piece by taking a rolling average of the detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The calculations are seen in the equation 8, where p1x/y is the existing trash detection’s respective x and y coordinate, p2x/2y is the new trash detection’s x and y coordinates, and a is the amount of times p1 has been averaged to that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' p1x = p1xa + p2x a + 1 p1y = p1ya + p2y a + 1 (8) This anti-clustering algorithm decreased the total amount of detections to accurately reflect the number of trash pieces seen in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' To avoid erroneous trash detections, aver- age trash locations without enough detections were determined as “noisy” and filtered out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Trash points were only published to the navigation stack if it had three or more detections averaged to one point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' General Navigation and Path Planning General navigation consists of two parts: localization and path planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The Robot first receives the 2D occupancy grid from our mapping software, alongside the trash detection coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Once these data are received, the robot then navigates to within two meters of the nearest detected trash point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Navigation only needs to navigate near a trash location since greedy pickup is routinely effective within a two meter distance, and the anti-clustering algorithm accounts for noise in our trash detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' To get the goal pose g you need it’s orientation, and x/y coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The equation 9 describes how to get the angle for the pose where p1 is the starting robot pose and p2 the trash pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' In equation 10 the distance the goal pose is from the robot is calculated to have it within Greedy Pickup range, ldentifiedTrashPiece trash_depth (m) 180°-0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='6096m KobukiRobot--ldentifiedTrashPiece 640Pixels RGBCameraFrame CameraFOV- 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='4° RealsenseD435 RGB-DCamera-ldentifiedTrashPiece trash x 320-(640-trash_x)- 462.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='139 PixelsAlgorithm 2 Anti-Clustering Algorithm Input: Trash Pose tp Output: Poses of clustered trash pieces ctp averaged Trash poses atp tupled with times averaged ta (atp, ta) ← empty if atp = 0 then atp ← tp ta ← 1 else for every pose tuple pt in atpt do get x and y bottom and top around the trash’s location xb, xt, yb, yt if xb ≤ tpx ≤ xt & yx ≤ tpy ≤ yt then pose tuple x ptx = (ptx * ta + tpx) / (ta + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='0) pose tuple y pty = (pty * ta + tpy) / (ta + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='0) ta + 1 end if end for if tp is not in atp then atp ← tp end if ctp ← all trash poses tp in atp where tpi’s tai > 2 times end if where d3 is the goal distance, d1 is the distance between the trash and the robot, and d3 is Greedy Pickup’s range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' In equation 11 the coordinates of the goal pose is found where gx/y is represents the coordinates respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' θ = arctan(p1y − p2y p1x − p2x ) (9) d3 = d1 − d2 (10) gx = d3 + p1x cos(θ)gy = d3 + p2x sin(θ) (11) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Robot Path Planning in Rviz The Navigation and path planning stack was based on the ROS-provided open-source AMCL software stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' This soft- ware loads and localizes the robot in a mapped environment and its DWA planner creates a path between the identified points of trash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' We created another software module that feeds our target coordinates from our trash detection phase into AMCL’s path planner to follow the most efficient path between the robot’s current location and the nearest possible trash point, figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Greedy Pickup Once the Navigation portion of the software stack places the robot within 2 meters of the piece of trash, Greedy Pickup is activated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Greedy Pickup is an asynchronous algorithm that ignores all navigation and map factors and solely focuses on seeking out the nearby trash directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' When the greedy pickup is activated, it rotates in a given direction to look for trash using YoLOv4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Once it receives a trash detection, it calculates the position of the trash using the same algorithm explained in the Trash Identification section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' After the robot localizes the trash, it turns back towards the trash at its precise angle, turns on the collection mechanism’s motor, and moves exactly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='2m past the trash’s location to ensure proper collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Once this occurs, the motor turns off and navigation resumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Algorithm 3 Greedy Pickup Input: Detected Trash pose list tpl set timeout time to set confidence threshold ct for Trash pose tp in tpl do Determine whether tp is to the left or right of the robot in the Map Spin in direction of tp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' scanning for confirmation if Robot gets tp ≥ ct then Robot Stop spinning Use III-D algorithm to find relevant Robot orienta- tion and trash pose Get destination angle Robot turns to destination angle Robot Turn on collection mechanism and drive over detected piece of trash else Robot exceeded to looking for trash end if end for Output: Poses of clustered trash pieces ctp For the collection mechanism to turn on or off,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' the NUC sends a serial packet to the connected Arduino with only three bytes in sequence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' either [0x59,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' 0x59,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' 0x59] to start the motor or [0x4E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' 0x4E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' 0x4E] to stop the motor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Once the Arduino receives a start packet, it then outputs a PWM signal to two of its GPIO pins which control the L928N motor controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The PWM signal gradually increases from a low duty cycle to a higher duty cycle to control the current spikes on the 12V line from the Robot base to the Motor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' In the initial design on bench power, starting the motor from 0 to full power produced an initial current spike of approximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='9A before settling around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='9-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='1A when in normal motion or picking up an object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The initial spike was over the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='5A limit provided by the 12V port accessible on the robot base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' To eliminate this spike, a slow ramp-up of the duty cycle of the PWM signal was introduced from 20% duty cycle to a maximum of 80% linearly over the course of 5 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' This removes the initial current spike and ensures that the motor can both properly power the collection mechanism and does not exceed the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='5A current limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' ROS Middleware ROS was used to connect all the functions of this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Every design block in figure 2 functions as a node, or multiple nodes, which subscribe and publish information to the other nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' ROS would also be used to network between the different robots in the multi-robot system over WiFi or other wireless protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' EXPERIMENT A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Goal The purpose of this experiment is to assert the feasibility of this system design before iterating on the hardware used to make it scalable and adaptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The robot was evaluated on its ability to accurately map the enclosure, identify, and mark the pieces of trash, choose an efficient path, and pick the trash up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The robotic system is meant to clear trash as large and heavy as an average 600mL Spring Valley Water bottle weighing ap- proximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='64 kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The system’s robotic base, the TurtleBot 2, has a load limitation of approximately 5 kilograms [20], which presents an upper weight limit on the total load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Since the robot’s additional components (external frame, storage container, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=') are estimated to weigh approximately 3 kg, the trash load must weigh at most 2 kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The UGV operates in narrow environmental parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The weather must be clear with no rain since the electronic systems onboard the UGV are non-weatherproofed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' In addition, due to the wheels that come default with the robotic base (Kobuki Mobile Base), our prototype can only operate on relatively flat, smooth, evenly colored surfaces, with no extreme movement in the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Testing In Simulation Simulated testing was done in the Gazebo Robotics simu- lator, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' This simulator was included in the base Turtlebot SDK and includes a near true-to-life recreation of the entire turtlebot system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The use of ROS allows for the navigation stack and greedy pickup to be run against the simulator and behave exactly identically to reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' This simulator was instru- mental in the initial testing of movement and navigation as it allowed our team to test many different speed parameters and movement algorithms without risking any physical damage to the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' All data associated with movement and mapping were recorded and played back in a simulated environment to recreate and reevaluate our physical testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' This allowed for useful visualizations and assessments of what the robot was processing at any given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Simulation Testing Still C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Testing the Hardware 1) Experiment System: The system’s design is meant to function with a mapping robot and a ground vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' To simulate the mapping vehicle in this experiment the UGV plays both roles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The UGV first passed through the area to get a map and identify trash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Then taking that information to navigate and pick up the trash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' 2) The Robot: The hardware design uses a modified Turtlebot-2 as a base design, figure 15, that has four main components: the depth camera (480p RGB-D Intel Realsense D435) [10], the computer (7-year-old Intel NUC with 8GB of RAM) which runs a GNU-Linux OS along with ROS to manage sensor data collection and real-time processing, the mobile robotic base (Kobuki Mobile Base) [12], and a custom- designed collection mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The camera relays RGB and depth images which are processed to identify and target trash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The Kobuki’s motor and wheels relay odometric feedback that helps confirm the UGV’s current location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The collection mechanism attaches to the front of the Kobuki Mobile base, plugs into power and data ports on the robot, and uses a rotary brush to pick up the trash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' 3) The Collection Mechanism: The collection mechanism is a custom-designed addition to the Turtlebot Robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Its mechanical construction consists of 20-20 aluminum bars connected by 90-degree brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' To allow free range of motion, caster wheels were affixed to the bottom of the frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' When the collection mechanism motor is activated, it sweeps trash up a ramp into a plastic storage container.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' A camera mount was printed so the Realsense could be attached to the front of mechanism [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' A funnel was added to the front of the collection mechanism to rein in the trash that may have been missed slightly, figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The design of the electronics system for the collection mechanism is an Arduino Mega that is connected to the NUC using a USB cable, figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The Arduino is then connected to an L982N Motor driver breakout board over PWM-enabled GPIO pins (5V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' This motor driver breakout board is supplied with 12V by the Kobuki base from a 12V, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='5A Max port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The output of the L982N is the DC motor which drives the chain and the brush.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Hardware Overview Diagram Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Turtlebot2 Base System The brush itself was hand-designed and fabricated since there is no commercial brush model that fit the design speci- fications for the mobile robot, figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The Environment The environment where the robot was tested was a room with a random configuration of chairs and obstacles put around an open space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Therefore, the map would be created each time in a dynamic environment and the system would have to account for a new configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' In that space, trash was put in different locations for all tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' We tested up to four pieces of trash in the environment at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Tests 1) Testing Mapping/ Image Identification accuracy/ Ac- counting for Latency: To test the trash detection and how well the processing latency was accounted for, expected maps with the approximate trash locations were checked against the created ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' 2) Testing the Greedy Pickup Algorithm: To test Greedy Pickup, the mode was enabled with varying pieces of trash within two meters of the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The piece of trash would start out of the UGV’s perception range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The UGV would have to scan for the trash, identify it and then collect it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' This trial was Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Custom Designed Brush Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Motor Circuitry Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Final Construction run with trash at 2 meters, 1 meter, and half a meter distance from the UGV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' 3) Full System Trials: Full system trials were then con- ducted, starting with the mapping of an environment and labeling trash points in that environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Then to navigating to the labeled pieces of trash in the environment and collecting the pieces of trash using Greedy Pickup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Accounting for Processing Latency The map creation and the image identification, figure 23, were shown to be an accurate, figure 19, fast and lightweight way of monitoring an area and identifying pieces of trash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Greedy Pickup Results Greedy pickup is shown to be accurate to an extent with a success rate of 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='78%, figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' There were multiple failures in these trials that were caused by the design of the collection mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The collection mechanism was not equipped with odometric wheels that could relay its’ location so,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' at times it would overturn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' and the caster wheels introduced an error the system was not aware of and could not account Intel Realsense Camera RGB + depth images collection Intel NUC mechanism motor feedback + control Kobuki Mobile Base12VDCMotor CURRENT SENSING A CURRENT SENSING OUTPUT OUTPUT 2 OUTPUT GNT SUPPLY VOLTAGE VS INPUT ENABLE INPUT ENABLEA LOGIC SUPPLY VOLTAGE VSS INPUT INPUT 2 GND OME L928N Motor Driver TXO 8 12V REOET RX3 TX3 MEGA 91 ROTAFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Accurate Map Creation with Identified Trash Points Result Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Greedy Pickup Results Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Full System Run-through Results Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Real Time Trash Identification and Map Creation for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' This caused the UGV to over or under turn occasionally and not successfully collect the trash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The Greedy Pickup algorithm has shown the ability to increase the error tolerance of the system, however, there are more improvements to the algorithm that could be made to make it more error-tolerant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' For example, an added “lock on” mechanism that when a piece of trash is in sight it would keep the piece of trash in the center of its FOV as it moves forward to collect the piece of trash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Ensuring that the failure-causing edge cases are accounted for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Still of UGV Picking up Water Bottle in System Test C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Full System Results These results, with a 68% success rate, figure 21, show that there are possible errors that can be introduced to this system and that they compound on themselves as the task gets more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' During these trials, the collection mechanism introduced errors during navigation and greedy pickup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The collection mechanism introduced errors by being slightly out of position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Since the Collection mechanism did not have sensor feedback to tell the UGV it was not in the correct position the little errors compounded into failures to collect trash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' CONCLUSION Our contribution is a multi-robot system design applied to monitoring and managing roadside litter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' We developed a system that can map, identify, and pick up pieces of trash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The system is designed to be relatively cheap and scalable which could be done because we introduced different algorithms that account for errors in a system that has less precise sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Our Greedy Pickup algorithm accounts for errors in trash iden- tification, and trash can be accurately identified in a system with low processing power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Our custom-designed collection mechanism, designed to pick up the main offending types of trash found on the side of the road, was also introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' This system builds off previously known algorithms: AMCL and DWA for navigation, Orbslam-2 for map creation, and YOLO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='v4 for trash identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' It also build off multi-robot systems to monitor a dynamic environment and combines that with information with a dynamic collection algorithm, Greedy Pickup, to create an efficient collection process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The next steps for this system would to be refine the mechanical design of the UGV, such as adding odometry sensors onto the collection mechanism’s wheels and improving the wheels to be able to drive over more difficult terrains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' There are also improvements to be made to the Greedy Pickup algorithm such as adding a “Lock on” capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Better logic could be applied to the navigation, for example, a future iteration of this system could use a graph search algorithm Mapping and Trash Identification Results 8% 92% Successful Maps ■ Failed MapsGreedy Pickup Success 16 14 12 10 8 6 4 2 0 2 meter 1 meter 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content="5 meter Robot's Distance from Trash Total Trials Successful trialsFull System Tests Aount of Trash in the Environment 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='5 Successful Trials Total Trials[ Interact Move Camera Select Focus CameraMeasure2D Pose Estimate2DNavGoalPublishPoint Q RGBImage RGB View ORB-SLAM2Ima RViz SLAMView OOYOLOV4 Map View YOLOView 199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='22 Wall Elapsed:416.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='55 Experimentalto find the most efficient path between every piece of trash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' An aerial robot can also be introduced as the mapping robot in the next iteration of this system for testing, to more accurately represent the issues that would arise from changes in the perspective of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' More replicas of the UGV could be added into the system as well, so there would be multiple trash collectors in one environment at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The system could also have mapping and trash collection happening in parallel instead of the modes being sequential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' ACKNOWLEDGMENTS This is a video of a full run-through our system did with two pieces of trash in the environment [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' This is the central repository in our organization that sets up the system [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' This is the Greedy Pickup repository which holds all the functions laid out in this paper [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' We would like to thank the contributions of Divya Venkatraman, Jared Raines and Catherine Ellingham for their work on gathering data, editing the paper and system networking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' We’d also like to thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Taskin Padir and his Robotics and Intelligent Ground Vehicle Research Laboratory (RIVeR) [22] for allowing us to use their hardware to test our design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' REFERENCES [1] Yong-bao Ai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' “Visual SLAM in dynamic environ- ments based on object detection”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' In: Defence Technol- ogy (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' ISSN: 2214-9147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' DOI: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' 1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='09.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' URL: https://keeplouisianabeautiful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='org/wp- content/uploads/2015/09/Litter-in-America-Executive Summary - FINAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' (Accessed: 10-June-2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' [3] John Chiaramonte, Jack Fenton, and Lee Milburn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' cave- man mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' URL: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='com/Capstone- W3/ caveman mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' (Accessed: 20-April-2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' [4] John Chiaramonte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Capstone-W3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' URL: https:// github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='com/Capstone-W3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' (Accessed: 11-June-2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' [5] John Chiaramonte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Demonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' URL: https : / / www .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' youtube .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' com / watch ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' v = cIwk0Kh9k7E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' (Accessed: 5-July-2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' [6] Maria Valera Espina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' “Multi-robot Teams for Environmental Monitoring”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' In: Innovations in Defence Support Systems – 3: Intelligent Paradigms in Secu- rity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' by Paolo Remagnino, Dorothy N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Monekosso, and Lakhmi C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Jain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Berlin, Heidelberg: Springer Berlin Heidelberg, 2011, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' 183–209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' ISBN: 978-3- 642-18278-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='1007/978- 3- 642- 18278- 5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' URL: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='1007/978-3-642-18278-5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' [7] Conserve Energy Future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' The Catastrophic Effects of Littering on Humans.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' “Multi-robot Task Allocation: A Review of the State- of-the-Art”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' In: vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' 604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' May 2015, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' 31–51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' ISBN: 978-3-319-18299-5.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' com / wiki / online - user- guide/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' (Accessed: 11-June-2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' [13] Nathan Koenig and Andrew Howard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' “Design and Use Paradigms for Gazebo, An Open-Source Multi-Robot Simulator”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' In: ().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' [14] Stephanie Melchor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Roadside Trash A Growing Prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' URL: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='montereyherald.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='com/2021/01/ 23/roadside-trash-a-growing-problem/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' (Accessed: 10- June-2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' [15] Ra´ul Mur-Artal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Montiel, and Juan D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Tard´os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' “ORB-SLAM: A Versatile and Accurate Monocular SLAM System”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' In: IEEE Transactions on Robotics 31.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='com/ rayvburn / ORB - SLAM2 ROS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' (Accessed: 11-June- 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' [20] Clearpath Robotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' TurtleBot 2 - Open Source Personal Research Robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' URL: https://clearpathrobotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='com/ turtlebot - 2 - open - source - robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' (Accessed: 10-June- 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' [21] Adrian Rosebrock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' OpenCV Morphological Operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' URL: https://pyimagesearch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='com/2021/04/28/opencv- morphological-operations/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' (Accessed: 11-June-2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' [22] Padir Taskin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' RIVeR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' URL: https : / / robot .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' neu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' edu/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' (Accessed: 7-July-2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' [23] Makerbot Thingiverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' Intel RealSense D435 Camera Mount for OpenBuilds 20x20mm Extrusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' URL: https: //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='thingiverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content='com/thing:2965905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' [24] Tossy0423.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' darknet ros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' URL: https : / / github .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' com / Tossy0423 / darknet ros / tree / master/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} +page_content=' (Accessed: 11- June-2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQfvP2u/content/2301.01704v1.pdf'} diff --git a/8dFRT4oBgHgl3EQfpTfl/content/tmp_files/2301.13613v1.pdf.txt b/8dFRT4oBgHgl3EQfpTfl/content/tmp_files/2301.13613v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7568a3bec66ffeb75d63d6f110ea45d0c1e046e7 --- /dev/null +++ b/8dFRT4oBgHgl3EQfpTfl/content/tmp_files/2301.13613v1.pdf.txt @@ -0,0 +1,1377 @@ +Geometry-based approximation of waves propagating +through complex domains∗ +Davide Pradovera† +Monica Nonino† +Ilaria Perugia† +February 1, 2023 +Abstract +We consider wave propagation problems over 2-dimensional domains with piecewise-linear bound- +aries, possibly including scatterers. Under the assumption that the initial conditions and forcing terms +are radially symmetric and compactly supported (which is common in applications), we propose an ap- +proximation of the propagating wave as the sum of some special nonlinear space-time functions: each +term in this sum identifies a particular ray, modeling the result of a single reflection or diffraction ef- +fect. We describe an algorithm for identifying such rays automatically, based on the domain geometry. +To showcase our proposed method, we present several numerical examples, such as waves scattering off +wedges and waves propagating through a room in presence of obstacles. +Keywords: wave propagation, model reduction, scattering, geometrical optics, diffraction +AMS subject classifications: 35L05, 35Q60, 65M25, 78A45, 78M34 +1 +Introduction +The discretization of numerical models for the simulation of complex phenomena results in high-dimensional +systems to be solved, usually at an extremely high cost in terms of computational time and storage memory. +Among these models, wave propagation problems represent an extremely interesting topic: relevant applica- +tions can be found, e.g., in the field of array imaging, where acoustic, electromagnetic, and elastic waves in +scattering media are modeled by the reflectivity coefficient, which is often unknown. Some examples in this +direction can be found in [5, 6, 7, 30], where inverse scattering problems are used to infer the reflectivity of +one or more scatterers embedded either in a known and smooth medium, or in a randomly inhomogeneous +medium. Another example of application of wave propagation problems is numerical acoustics, where the +goal is to simulate the propagation of sound in a room, in presence of obstacles and walls with different +absorbing and/or reflecting properties, see [28]. +Wave propagation problems in the time-harmonic setting (the Helmholtz problem, cast in the frequency +domain) have been widely studied. See, e.g., [4, 13, 19, 24, 25, 27, 28]. However, our focus here are problems +in the time domain, whose numerical simulation is expensive, mainly because one needs to use both a fine +spatial mesh and a carefully chosen time step in order to satisfy the CFL condition [11, 16]. In the interest +of making these simulations feasible, model order reduction (MOR) [3, 9, 14, 17] represents a promising +framework, whose goal is to reduce the computational cost of solving the problem of interest. +In this context, it is well known [12, 15] that wave propagation problems are characterized by a slowly +decaying Kolmogorov n-width. +Because of this, classical linear-subspace MOR methods are not able to +reproduce the behavior of the wave propagation without relying on a very high-dimensional linear manifold. +This makes linear surrogate models unappealing, since they do not yield significant speed-ups. In recent +years, many approaches have been proposed to overcome the intrinsic “difficulty” of problems with slowly +∗M. Nonino and I. Perugia have been funded by the Austrian Science Fund (FWF) through project F 65 “Taming Complexity +in Partial Differential Systems” and project P 33477. +†Faculty +of +Mathematics, +University +of +Vienna, +Oskar-Morgenstern-Platz +1, +1090 +Vienna, +Austria +(da- +vide.pradovera@univie.ac.at, monica.nonino@univie.ac.at, ilaria.perugia@univie.ac.at). +1 +arXiv:2301.13613v1 [math.NA] 31 Jan 2023 + +D. Pradovera, M. Nonino, and I. Perugia +Geometry-based approximation of waves in complex domains +decaying Kolmogorov n-width, with the target of making MOR more efficient. To this end, such methods +rely on nonlinear and/or hybrid space-time approaches. For more details, we refer to [8, 10, 18, 21, 26, 29, 31]. +In this work, we focus on wave propagation over 2-dimensional spatial domains, possibly including ob- +stacles. We limit our investigation to domains with piecewise-linear boundaries and a constant wave speed. +The initial conditions and forcing terms are assumed to be compactly supported and radially symmetric +around a “source point”. This situation arises in many of the above-mentioned applications. Under these +assumptions, we propose to approximate the solution of the problem of interest with the sum of some special +nonlinear space-time functions, which we call “rays”. Each ray models a reflection or diffraction effect, and +is composed of different parts: +• the free-space radially symmetric solution of the wave equation, modeling the space-time propagation +of the ray; +• a spatial indicator function, determining the light cone of each ray; +• a nonlinear spatial term encoding the angular modulation of the ray, which is crucial when modeling +diffraction effects. +The number of terms appearing in the sum is determined by the number of reflection and diffraction effects +that are required to faithfully approximate the target wave, which ultimately depends on the geometry of +the physical domain. +Among the advantages of the proposed approach, we mention the fact that each ray is separable into +time-radial and angular components (in the “polar coordinates” sense). As we will see, we can leverage this +to reduce drastically the computational cost and the storage memory that are required by our approximation, +with respect to competitor methods. +The rest of the paper is structured as follows. In Section 1.1 we present the problem of interest. In +Section 2 we introduce the main ingredients of our method, and we describe the “training phase” of the +algorithm, i.e., the construction of the approximated wave. In Sections 3 and 4 we detail how we model +reflection and diffraction effects, respectively. The latter section is rather extensive, since diffraction is much +harder to model than reflection, and requires special care. In Section 5 we present some numerical results to +showcase our method. Both simple benchmarks (wedges) and more complicated tests (2D room model with +scatterers) are considered. Some final considerations follow in Section 6. +1.1 +Target problem +We are interested in the numerical approximation of the solution of the wave equation in complex domains. +In this work, we consider 2-dimensional domains only. However, most of our discussion generalizes to 3D. +We defer a discussion on this till Section 6. +We denote by Ω ⊂ R2 the physical domain in which the wave equation is considered. We assume that +Ω is either a closed polygon or a set-subtraction of polygons (to allow for multiply connected domains). +We denote by ne and nv the number of edges and vertices of ∂Ω, respectively. We study the propagation +of waves in Ω over a given time interval of interest [0, T]. The model problem is the wave equation with +constant (unit) wave speed: +� +� +� +� +� +� +� +� +� +∂ttu(x, t) = ∆u(x, t) + f(x, t) +for (x, t) ∈ Ω × (0, T), +u(x, 0) = u0(x) +for x ∈ Ω, +∂tu(x, 0) = u1(x) +for x ∈ Ω, +∂νu(x, t) = 0 +for (x, t) ∈ ∂Ω × (0, T], +(1) +with ∆ the Laplacian operator, defined, in 2 dimensions, as ∆ = �2 +j=1 ∂xjxj. The homogeneous Neumann +condition (i.e., the last equation above) models the whole boundary ∂Ω as sound-hard [11]. More generally, +all or parts of ∂Ω may be modeled as sound-soft via a Dirichlet-type condition: u(x, t) = 0. +We assume that the initial conditions u0 and u1, as well as the forcing term f, have radial symmetry +around a given point. Without loss of generality, we will take such point to be the origin of R2: +u0(x) = η0(∥x∥), u1(x) = η1(∥x∥), f(x, t) = η2(∥x∥ , t) +∀(x, t) ∈ Ω × (0, T), +(2) +2 + +D. Pradovera, M. Nonino, and I. Perugia +Geometry-based approximation of waves in complex domains +with ∥x∥2 = �2 +j=1 x2 +j. We further assume that the functions ηj have compact support, namely, that there +exist R > 0 such that ηj(ρ) = 0 for all ρ > R and j = 0, 1, 2. Moreover, to avoid incompatibilities with the +boundary conditions, for simplicity we will only consider the situation where the supports of the functions +ηj are fully contained in Ω. +2 +Approximation framework +Before we can model boundary effects (reflection and diffraction), we need to understand how the solution +u would behave if no boundary were present. To this aim, we consider the wave equation in free space +� +� +� +� +� +∂ttU(x, t) = ∆U(x, t) + f(x, t) +for (x, t) ∈ R2 × (0, ∞), +U(x, 0) = u0(x) +for x ∈ R2, +∂tU(x, 0) = u1(x) +for x ∈ R2, +(3) +which we have obtained from (1) by replacing Ω with the whole plane. +Due to radial symmetry (of the initial conditions and of the forcing term), we can recast the problem in +polar coordinates. This allows us to define the free-space solution in the radial coordinate Ψ, as the solution +of +� +� +� +� +� +� +� +� +� +∂ttΨ(ρ, t) = �∆Ψ(ρ, t) + η2(ρ, t) +for (ρ, t) ∈ (0, ∞) × (0, ∞), +Ψ(ρ, 0) = η0(ρ) +for ρ ∈ [0, ∞), +∂tΨ(ρ, 0) = η1(ρ) +for ρ ∈ [0, ∞), +∂ρΨ(0, t) = 0 +for t ∈ (0, ∞), +(4) +where �∆ is the Laplace operator in polar coordinates (under radial symmetry), i.e. +�∆ = ∂ρρ + 1 +ρ∂ρ, and +U(x, t) = Ψ(∥x∥ , t) for all x ∈ R2. Note that, by the compact support of the initial conditions and of +the forcing term, and by the finite (unit) speed of propagation of the wave equation, we have Ψ(ρ, t) = 0 +whenever ρ > t + R. +Remark 2.1. Generally, the free-space solution Ψ is not available analytically, except for very simple choices +of initial conditions and forcing term. Accordingly, in most applications, the function Ψ will need to be +replaced with a suitable approximation. To this effect, one could discretize (4), e.g., with a finite element +approximation (in space) and some timestepping scheme (in time). See Section 5 for more details on how +this can be carried out. +Our goal is to approximate, for all (x, t) ∈ Ω × [0, T], the solution u(x, t) of the wave equation problem +(1) with the following sum of special functions: +u(x, t) ≈ �u(x, t) = +N +� +n=1 +Ψ(∥x − ξn∥ + rn, t)1Ωn(x)ζn(x − ξn) +� +�� +� +�un(x,t) +. +(5) +Each term �un is what we will call a “ray”. Therein, Ψ is the above-mentioned free-space radially symmetric +solution of (4), and 1A denotes the indicator function with support A, i.e., +1A(y) = +� +1 +if y ∈ A, +0 +if y /∈ A. +(6) +Moreover, in (5), we have introduced the following quantities: +• N is the number of rays used in the approximation. +• ξn is the location of the new source. +• rn ≥ 0 is a spatial delay, which will be used for the synchronization of diffraction effects. +• Ωn ⊂ Ω is the light cone (the spatial support) of a term of the sum. +3 + +D. Pradovera, M. Nonino, and I. Perugia +Geometry-based approximation of waves in complex domains +• ζn : R2 \ {0} → R is a weight function describing the angular modulation. We require that ζn be a +positive-homogeneous functions, i.e., ζn(y) = ζn(τy) for all τ > 0 and y ∈ R2. +Note that, due to the finite speed of propagation of the free-space solution Ψ, we have that a generic +term �un(x, t) is zero whenever t < ∥x − ξn∥ + rn − R, i.e., for t small enough, depending on x. +The number of rays N in the sum (5) will be determined based on how many boundary effects (reflections +and diffractions) need to be included in �u in order to have a good approximation of the target wave u. We +describe a strategy for automatically identifying a good N in the next section. See, e.g., Remark 2.2. +2.1 +Building the low-rank skeleton +Recalling that u solves the wave equation (1) in the domain Ω, we use the first term in (5), namely, �u1, to +approximate the outgoing component of u, ignoring any effect due to the boundary ∂Ω, except for shadows. +Then, given such �u1, we use the other terms �u2, . . . , �uN to correct this first approximation. Each extra term +models a single effect due to a certain portion of the boundary, specifically, an edge (reflection off that edge) +or a vertex (diffraction about that vertex). +Going back to the first ray �u1, let us define it, by providing its “ingredients” ξ1, r1, Ω1, and ζ1, cf. (5). +We set ξ1 = 0, the center of the initial condition, as well as r1 = 0, since no delay is necessary for this first +term. Then, leveraging symmetry, we set ζ1 ≡ 1, which corresponds to the (physical) assumption that the +propagation of �u1 is purely radial. Finally, we set Ω1 (the light cone around 0) as the set of points that can +be reached from 0 via a straight line without going outside ∂Ω, i.e., +Ω1 = {x ∈ Ω : τx ∈ Ω ∀0 ≤ τ ≤ 1} . +(7) +In summary, the first term of �u is +�u1(x, t) = Ψ(∥x∥ , t)1Ω1(x). +(8) +Then we can move to the subsequent terms �un, n ≥ 2. +Their expressions depend on our choice of +reflection and diffraction modeling, and will be provided in the upcoming sections. Instead, in the rest of +the present section we focus on understanding how large N should be, in order for �u to provide a faithful +approximation of u. Equivalently, we want to count the number of times the wave gets reflected or diffracted +at the boundary ∂Ω. This is done incrementally, starting from the initial value N = 1 (no boundary effects) +and then updating this guess as more and more boundary effects get “discovered”. +To help us in this endeavor, we employ what we call a timetable, which, in this work, is simply a list of +vectors, each with size ne + nv. The timetable is built incrementally starting from an empty list, appending +one new vector every time a new term is added in the sum (5), starting from �u1. The entries of the n-th +timetable vector are the waiting times before �un comes in contact with an edge or a vertex of ∂Ω. If it is +impossible for �un to “cast light” (along a straight path) onto a certain edge or vertex, then the corresponding +entry in the timetable is set to ∞. After this, it suffices to look for the smallest not-yet-explored entry of the +timetable to identify what the next term of the approximation �u should be. Once the entry in the timetable +has been explored, its value is set to ∞. +We start by describing how the first vector a1 ∈ Rne+nv of the timetable (corresponding to �u1) is +computed, and how a1 allows us to identify the (geometric) features of �u2. The vector a1 can be partitioned +into edges-related part (the first ne entries) and vertices-related part (the last nv entries). +• Edge times. Given a generic edge γj ⊂ ∂Ω (j = 1, . . . , ne) belonging to the domain boundary, we +define the corresponding entry of a1 as +(a1)j = +� +r1 + inf +� +∥x − ξ1∥ : x ∈ γj ∩ Ω1 +� +if the set is non-empty, +∞ +otherwise. +(9) +Note that we have taken the shortest path from ξ1 to γj, and that we have denoted the closure of the +light cone Ω1 as Ω1. +• Vertex times. Given a generic vertex yj ⊂ ∂Ω (j = 1, . . . , nv) of the domain boundary, we set +(a1)ne+j = +� +r1 + ∥yj − ξ1∥ +if yj ∈ Ω1, +∞ +otherwise. +(10) +4 + +D. Pradovera, M. Nonino, and I. Perugia +Geometry-based approximation of waves in complex domains +Note that we have included the delay r1 (which is actually zero here) as a way to streamline Eqs. (9) and (10) +for the upcoming section. See Fig. 1 for a diagram showcasing these formulas. +(a1)1 +(a1)2 +(a1)5 +(a1)17 +Ω1 +Ω \ Ω1 +γ1 +γ2 +γ3 +γ4 +γ5 +y3 +y4 +y6 +ξ1 +(a1)3 = ∞ +(a1)4 = ∞ +(a1)14 = ∞ +(a1)15 = ∞ +... +Figure 1: Computation of some timetable entries. The boundary ∂Ω has 11 sides, so that, e.g., (a1)14 is +related to y3 and (a1)17 is related to y6. The shadowed area Ω \ Ω1 is in darker grey. +The smallest entry of a1 is the time at which the first “boundary event” (reflection or diffraction) can +happen1. The index of the smallest entry tells us whether the event is a reflection (index 1 ≤ j ≤ ne) or a +diffraction (index ne + 1 ≤ j ≤ ne + nv), and also what edge/vertex causes the event. From here, we use the +models described in Sections 3 and 4 to build �u2, by computing ξ2, r2, Ω2, and ζ2. +Then, the second timetable vector a2 can be computed by replacing all subscripts “1” by “2” in Eqs. (9) +and (10). This is followed by the construction of �u3, and so on. The process continues until all not-yet- +explored entries of the timetable are larger than T +R. Indeed, starting from this time instant, the would-be +next terms of �u do not affect the approximation anymore, since, due to the finite speed of wave propagation, +they only act (on Ω) after the end of the time horizon, i.e., for t > T. The total number of rays N is simply +the number of vectors in the timetable. +We summarize the overall procedure for the construction of the rays �un in Algorithm 1. For ease of +presentation, once an entry of the timetable has been explored, it is set to ∞ as a way for the algorithm to +ignore it from that point forward. +Algorithm 1 Step-by-step construction of the surrogate model +Set N ← 1, find Ω1 as in (7), and define �u1 as in (8) +Define a1 ∈ Rne+nv using Eqs. (9) and (10) +Set i ← 1 and j ← arg minj=1,...,ne+nv (a1)j +while (ai)j ≤ T + R do +Set (ai)j ← ∞ and N ← N + 1 +if j ≤ ne then +Find ξN, rN, ΩN, and ζN as in Section 3 +← Reflection from edge j +else +Find vertex index j′ ← j − ne +Find ξN, rN, ΩN, and ζN as in Section 4 +← Diffraction from vertex j′ +end if +Define �uN from ξN, rN, ΩN, and ζN, as in (5) +Define aN ∈ Rne+nv using Eqs. (9) and (10), with “N” replacing “1” in subscripts +Set (i, j) ← arg mini=1,...,N,j=1,...,ne+nv (ai)j +end while +1We say “can happen” since not all vertices can cause diffraction, when hit from a certain point source. +This issue is +discussed in Section 4, cf. Assumption 4.3. +5 + +D. Pradovera, M. Nonino, and I. Perugia +Geometry-based approximation of waves in complex domains +γ +ξi +ξn +y(x) +x +θr +θi +φi(y(x)) +φn(x) +βγ +ξn +ξi +y(x) +x +Figure 2: Graphical representation of a reflection off edge γ. On the left, the law of reflection prescribes +θr = θi. We show the straight line �γ supporting γ with a dotted stroke. For a given observation point x, +y(x) denotes the point of incidence of the reflected ray. On the right, computation of the light cone Ωn +(light-grey area) and its complementary shadow zone Ω \ Ωn (dark-grey area) for the reflected ray, in the +presence of a rectangular obstacle. The dashed portion of edge γ denotes the shadow γ \ γ(i). The shadow +zone consists of two connected components. +Remark 2.2. In trapping domains, see, e.g., Section 5.2, the number of terms N might be rather large +due to waves repeatedly “bouncing back and forth” between two or more edges/vertices. A large N, although +necessary for a good approximation of all wavefronts, is undesirable since it increases the computational cost +of both the construction of the surrogate �u and its evaluation. +As a compromise, one could remove all terms �un that are smaller than a certain tolerance tol, uniformly +over x and t. This can be done as a post-processing step (thus speeding up the evaluation of �u but not +its construction) or while building the surrogate itself. This can be achieved with a simple modification of +Algorithm 1, by introducing a test on the magnitude of each soon-to-be-added wave contribution �un, discarding +terms that are too small. +3 +Modeling reflection +We now present the strategy for modeling reflection due to an edge γ of the domain boundary ∂Ω. We +rely on the well-known geometrical optics model, which describes wave propagation in terms of rays, not +accounting for any diffraction [23]. We assume that we are adding a new ray �un to the surrogate model (5), +due to a reflection phenomenon caused by ray �ui. Specifically, we assume that a ray coming from source +point ξi hits the edge γ ⊂ ∂Ω, i.e., that γ ∩ Ωi ̸= ∅. We need to prescribe several ingredients. +Spatial correction rn. +We just transfer rn over from the incoming wave: rn = ri. Indeed, as we will see +in Section 4, we require the term rn only when modeling diffraction. +Source point ξn. +We use the method of images, which gives the position of ξn as the reflection of ξi with +respect to the edge γ: +ξn = 2 arg min +z∈�γ +∥z − ξi∥ − ξi, +(11) +where �γ ⊂ R2 is the straight line on which edge γ lies. See Fig. 2 (left). +Weight function ζn. +Let x − ξn be a generic point where we wish to evaluate the weight function ζn. +We define the incidence point y(x) as the intersection (if any) between edge γ and the segment from ξn to +x. See Fig. 2 (left). According to the method of images, the amplitude of the reflected wave is equal (up to +sign) to the amplitude of the incoming wave: +ζn(x − ξn) = (2σγ − 1)ζi(y(x) − ξi). +(12) +6 + +D. Pradovera, M. Nonino, and I. Perugia +Geometry-based approximation of waves in complex domains ++ += +Figure 3: Example of reflection off an edge in the presence of an obstacle, from Fig. 2. Neumann conditions +are imposed on all edges. Source wave (left), reflected wave (middle), and superimposition of the two (right). +Note how the obstacle creates a shadow zone for source and reflected waves. For simplicity, in this plot we +are not showing any reflection or diffraction effects due to the rectangular obstacle, since they would be +modeled at different stages of the algorithm. +In the equation above, the quantity σγ is related to the kind of boundary conditions that are imposed on γ: +if γ is an edge with Neumann boundary conditions, we set σγ = 1 (ζn and ζi have the same sign), whereas +we set σγ = 0 if we have Dirichlet boundary conditions on γ (ζn and ζi have opposite signs). +Now, recall that we are assuming all weight functions to be positive-homogeneous: ζi(x − ξi) = ζi(τ(x − +ξi)), for all τ > 0. Accordingly, as we are in 2D, ζi(x − ξi) is only a function of the direction (with sign) +υi(x) = (x − ξi)/ ∥x − ξi∥, or, equivalently, of the angle φi(x) between υi(x) and the positive x1-axis. See +Fig. 2 (left) for a graphical depiction. Specifically, with an abuse of notation, let ζi(x − ξi) = ζi(φi(x)) and +ζn(x − ξn) = ζn(φn(x)), where the “new” angle-dependent functions ζi and ζn are 2π-periodic. By (12), we +deduce the property +ζn(φn(x)) = (2σγ − 1)ζi(φi(y(x))) = (2σγ − 1)ζi(2βγ − φn(y(x))) = (2σγ − 1)ζi(2βγ − φn(x)), +(13) +where βγ is the angle between edge γ and the positive x1-axis. This uniquely identifies ζn given ζi and βγ. +Light cone Ωn. +We first identify what portion of γ is actually “lit” by �ui: γ(i) = γ ∩ Ωi. Note that we +may have γ ̸= γ(i), for instance when obstacles are present between ξi and γ. See Fig. 2 (right) for an +illustration. Then, roughly speaking, we define the new light cone Ωn as the union of all rays from ξn that +pass through γ(i). To be more precise, given x ∈ Ω, let y(x) be the intersection (if any) between γ and the +line segment from ξn to x. Also, if y(x) exists, we define τ0(x) = ∥y(x) − ξn∥ / ∥x − ξn∥ ∈ (0, 1), which +satisfies y(x) = ξn + τ0(x)(x − ξn). The new light cone is defined as +Ωn = +� +x ∈ Ω : y(x) ∈ γ(i) and ξn + τ(x − ξn) ∈ Ω ∀τ0(x) < τ ≤ 1 +� +. +(14) +Figure 3 represents a possible output of the numerical algorithm. In this case, we simulate only the +reflections, thus discarding, for the time being, any effect due to diffraction. It is clear that, by modeling +reflection effects only, we may obtain a discontinuous approximation of the solution of our target problem, +where the discontinuity happens exactly at the shadow boundaries (the boundaries of light cones). As we +will see in the next section, introducing diffraction in our approximation will allow us to obtain a continuous +approximation �u. +4 +Modeling diffraction +Here, we describe a strategy for modeling waves diffracted by a vertex of the domain boundary ∂Ω. This is +required in building a new ray �un whenever the smallest unexplored entry of the timetable is related to a +vertex, i.e., j > ne in Algorithm 1, cf. Section 2. We need to identify several ingredients. +Source point ξn. +We employ the (standard, see, e.g., [23]) assumption that diffraction emerges as a wave +outgoing from a point source located at the diffraction point yj′ = yj−ne (we are employing the notation of +Algorithm 1). This motivates the choice of the center ξn = yj′. +7 + +D. Pradovera, M. Nonino, and I. Perugia +Geometry-based approximation of waves in complex domains +∂Ω +γ′ +γ +ξ +α +φ +π − θ +θ +3π − 2α − θ +∂Ω +γ′ +γ +ξ +α +θ − π +3π − 2α − θ +θ +Figure 4: Diagrams for the two cases of scattering for concave corners (0 < α < π): without (left plot) and +with shadow zone (right plot). The dashed lines are reflection boundaries. The dash-dotted line is a shadow +boundary. Shadow regions are absent if and only if π − α ≤ θ ≤ π. The angular coordinate 0 < φ < 2π − α +is measured starting from one of the two adjacent edges of ∂Ω. +∂Ω +γ′ +γ +ξ +α +φ +π + θ − α +θ +Figure 5: Diagrams for the scattering at convex corners (π < α < 2π). The source point ξ is virtual, being +used to model reflection off of edge γ. The dash-dotted line is the shadow boundary due to edge γ′. The +shadow region is present if and only if α − π < θ < π. The angular coordinate 0 < φ < 2π − α is measured +starting from one of the two adjacent edges of ∂Ω. +Light cone Ωn. +Since the diffracted wave propagates in all geometrically allowed directions, we define the +support Ωn as the set of all points that are visible (along straight-line paths) from ξn, i.e., +Ωn = {x ∈ Ω : ξn + τ(x − ξn) ∈ Ω ∀0 < τ ≤ 1} . +(15) +Modeling diffraction is substantially more complicated than modeling reflection. For this reason, before +we can describe how the remaining unknown quantities rn and ζn are defined, cf. (5), we need to introduce +several assumptions. +Assumption 4.1 (Separability). Diffracted waves are separable into radial-temporal and angular compo- +nents around the diffraction point ξn. +Otherwise stated, �un(x, t) can be expressed (at least locally) as +ψn(∥x − ξn∥ , t)ζn(x − ξn), where ζn is positive-homogeneous, i.e., ζn(z) is independent of ∥z∥ (as long +as z ̸= 0). Using an abuse of notation, we will express ζn as a function of φ only, with φ defined as the +angular coordinate around ξn. See Figs. 4 and 5. +This, together with the following assumption on the angular component ζn, will allow us to recover the +approximation structure presented in Section 2. +Assumption 4.2 (Piecewise-linear angular component). The angular component ζn is a piecewise-linear +function of the angular coordinate φ, with discontinuities at all reflection and shadow boundaries. Using the +geometrical optics approximation, we can explicitly compute the locations of such discontinuities: +• at concave corners (see, e.g., Fig. 4), φ1 = |π − θ| = max{π−θ, θ−π} and φ2 = 2π−α−|π − α − θ| = +min {θ + π, 3π − 2α − θ}; +• at convex corners (see, e.g., Fig. 5), φ3 = π + θ − α. +Now we are ready to describe our full diffraction model, which satisfies Assumptions 4.1 and 4.2, as well +as the following three standard requirements. +8 + +D. Pradovera, M. Nonino, and I. Perugia +Geometry-based approximation of waves in complex domains +Assumption 4.3 (Characterization of diffracting vertices). A vertex ξn emits a diffraction wave in “re- +sponse” to �ui only if both following conditions are met: +• ξn is visible from ξi, i.e., ξn ∈ Ωi; +• one of the following is true: +– the domain Ω is locally concave near ξn, with ξi being located on the “concave side” of ξn, i.e., +0 < α < π and 0 ≤ θ ≤ 2π − α in Fig. 4; +or +– the domain Ω is locally convex near ξn and a “shadow zone” is present, i.e., π < α < 2π and +π − α < θ < π in Fig. 5. +Assumption 4.4 (Continuity of the full approximation). The full wave approximation �u is continuous, in +particular across reflection and shadow boundaries. +Assumption 4.5 (Conservation of mass). Diffracted waves have zero “net mass”, i.e., +� +R2 �un(x, t)dx = 0, +leading to mass conservation of the full wave approximation �u. (Note that we are stating mass conservation +in free space to ignore further reflections and diffractions of �un, which are also assumed to conserve mass.) +Spatial correction rn. +As in Algorithm 1, let i be the index of the term �ui that causes the diffraction +�un. With the objective of satisfying (5) and Assumption 4.4, we define the radial component ψn as +ψn(∥x − ξn∥ , t) = Ψ(∥x − ξn∥ + ∥ξn − ξi∥ + ri +� +�� +� +=:rn +, t). +(16) +By direct inspection of this definition, we can see that, by our choice of rn, we are “aligning” the wavefronts +of the diffracted waves with the wavefronts of the reflected wave at the reflection boundaries (the shadow +boundary of the reflected waves, if any) and with the wavefronts of the incoming wave �ui at its shadow +boundary (if any). For instance, it is easy to see that, using (16), a point close to the diffraction point +(x ≈ ξn) is within the support of the diffracted wave �un only for t ≥ rn − R, i.e., only when the wave �ui has +crossed the distance from ξi to ξn. +Weight function ζn. +According to Assumption 4.2, we define the discontinuous piecewise-linear function +ζn : [0, 2π − α] → R as +ζn(φ) = +� +� +� +� +� +z1(φ1−φ)+z2φ +φ1 +for 0 < φ < φ1 := |π − θ| , +z3(φ2−φ)+z4(φ−φ1) +φ2−φ1 +for φ1 < φ < φ2 := 2π − α − |π − α − θ| , +z5(2π−α−φ)+z6(φ−φ2) +2π−α−φ2 +for φ2 < φ < 2π − α, +(17) +for concave corners, and +ζn(φ) = +� z1(φ3−φ)+z7φ +φ3 +for 0 < φ < φ3 := π + θ − α, +z8(2π−α−φ)+z6(φ−φ3) +2π−α−φ3 +for φ3 < φ < 2π − α, +(18) +for convex corners. The scalars z1, . . . , z8 are nodal values of ζn: ζn(0) = z1, ζn(φ+ +1 ) = z3 for concave corners, +ζn(φ− +3 ) = z7 for convex corners, etc. These values are chosen so as to satisfy: +• The boundary conditions at the edges ending at ξn, i.e., γ and γ′. +• Assumption 4.4 at the discontinuity angles φ1, φ2, and φ3. To this aim, we prescribe values for the +jumps (z3 − z2), (z5 − z4), and (z8 − z7). +• Assumption 4.5. Given the radial-angular decomposition of �un from Assumption 4.1, this is equivalent +to the condition +� 2π−α +0 +ζn(φ)dφ = 0. +9 + +D. Pradovera, M. Nonino, and I. Perugia +Geometry-based approximation of waves in complex domains ++ += ++ += +Figure 6: Examples of diffraction at the concave (top) and convex (bottom) corners from Fig. 4 (right) and +Fig. 5. Neumann conditions are imposed on all edges. In each row of plots, we have: discontinuous wave +without diffraction (left), diffraction wave (middle), and continuous wave with diffraction (right). Note that, +in the convex case, we are not showing the wave �ui that causes the reflection off edge γ nor the reflection +and scattering of �ui off edge γ′. +In the case of a convex corner, this set of condition uniquely identifies the four degrees of freedom. See +Section 4.1 for the formulas and for their derivation. +However, in the case concave case, an additional +condition is required. In this work, we set this last condition as described in Section 4.2. We show in Fig. 6 +the results obtained with our diffraction modeling in two simple illustrative cases. +Before proceeding further, we deem it important to make the following remark. +Remark 4.6. Our proposed strategy is able to deliver only a fairly crude approximation of diffraction effects. +(We refer to Section 5.1 for a validation of our model.) However, it has the great advantage of being extremely +simple to build and to evaluate. Thanks to the modularity of our approach, it would be surely possible to +replace our diffraction model with more sophisticated ones (e.g., removing Assumptions 4.1 and 4.2), in the +interest of achieving a better approximation of the exact solution. To this aim, we mention that a wide body +of works has been dedicated to the modeling of diffraction in the time-harmonic (Helmholtz) setting: among +others, we name the geometrical [20] and uniform [23] theories of diffraction. However, the authors have +not been able to find any satisfactory all-purpose time-domain diffraction modeling in the literature. +4.1 +Convex diffraction coefficients +Consider the situation depicted in Fig. 5 and the notation introduced therein. Also, we rely on the quantities +ξn, rn, Ωn, and i introduced in Section 4. For diffraction to happen, cf. Assumption 4.3, �ui must be a wave +reflected off either edge γ or γ′. Indeed, a convex vertex ξn cannot be “hit” from outside the domain Ω by +the source wave �u1, nor by any wave reflected off a different edge, nor by any diffracted wave centered at +some vertex of ∂Ω. +For this reason, the shadow boundary {φ = φ3} must belong to ∂Ωi (the boundary of the light cone Ωi), +at least locally around ξn. Without loss of generality, we assume that �ui is a wave reflected off edge γ, so +that Ωi consists (locally) of point whose angular coordinate is 0 < φ < φ3. This means that φ3 < φ < 2π −α +is a shadow zone. The alternative case (of reflection off edge γ′) can be obtained by symmetry. +Let σγ = 0 if γ is a Dirichlet edge and σγ = 1 if it is a Neumann edge. Define σγ′ similarly for edge γ′. +To satisfy the conditions described in Section 4, the quantities z1, z7, z8, z6 appearing in the angular weight +ζn, cf. (18), must satisfy the conditions: +• Boundary condition at γ: z1 = σγz7. +• Boundary condition at γ′: z6 = σγ′z8. +• For �u to be continuous at φ = φ3, there must be a jump to account for the fact that �ui is nonzero for +φ → φ− +3 but zero for φ → φ+ +3 : given the angular component of �ui at the shadow boundary, namely, +10 + +D. Pradovera, M. Nonino, and I. Perugia +Geometry-based approximation of waves in complex domains +hi := ζi(ξn − ξi), we impose z8 − z7 = hi. +• Conservation of mass: for all t > 0, +0 = +� +R2 �un(x, t)dx = +� ∞ +0 +� 2π−α +0 +ψn(ρ, t)ζn(φ)ρdφdρ += +�� 2π−α +0 +ζn(φ)dφ +� �� ∞ +0 +ψn(ρ, t)ρdρ +� += +�z1 + z7 +2 +φ3 + z8 + z6 +2 +(2π − α − φ3) +� �� ∞ +0 +ψn(ρ, t)ρdρ +� +, +which leads to the condition z1+z7 +2 +φ3 + z8+z6 +2 +(2π − α − φ3) = 0. +With simple algebra, we now obtain +z7 = +(σγ′ + 1)hi(φ3 + α − 2π) +(σγ′ + 1)(2π − α) + (σγ − σγ′)φ3 +, +z8 = +(σγ + 1)hiφ3 +(σγ′ + 1)(2π − α) + (σγ − σγ′)φ3 +, +(19) +as well as z1 = σγz7 and z6 = σγ′z8. See Fig. 6 (bottom) for an example of the resulting diffraction wave. +4.2 +Concave diffraction coefficients +Consider the setup depicted in Fig. 4 and the notation introduced therein. Also, we rely on the quantities +ξn, rn, Ωn, and i introduced in Section 4. Let σγ = 0 if γ is a Dirichlet edge and σγ = 1 if it is a Neumann +edge. Define σγ′ similarly for edge γ′. Without loss of generality, we assume that 0 < θ ≤ π − α +2 , since the +other case can be easily obtained by symmetry. In this setting, γ is in the light cone of �ui (at least locally +around ξn). Accordingly, let �ui′ be the wave component obtained by reflection of �ui off edge γ. +If 0 < θ ≤ π − α, γ′ is not in the light cone of �ui, so that: +• for 0 < φ < φ1 = π − θ, both �ui and �ui′ are present; +• for φ1 < φ < φ2 = π + θ, only �ui is present, since the light cone Ωi′ ends at {φ = φ1}; +• for φ2 < φ < 2π − α, neither �ui nor �ui′ is present, i.e., we have a shadow zone, since the light cone Ωi +ends at {φ = φ2}. +Otherwise, assume that π − α < θ ≤ π − α +2 . In this case, γ′ is also in the light cone of �ui (at least locally +around ξn). We denote the wave component obtained by reflection of �ui off edge γ′ by �ui′′. Then: +• for 0 < φ < φ1 = π − θ, both �ui and �ui′ are present; +• for φ1 < φ < φ2 = 3π − 2α − θ, only �ui is present, since the light cones Ωi′ and Ωi′′ end at {φ = φ1} +and at {φ = φ2}, respectively; +• for φ2 < φ < 2π − α, both �ui and �ui′′ are present. +To satisfy the conditions described in Section 4, the quantities z1, . . . , z6 appearing in the angular weight +ζn, cf. (17), must satisfy the conditions: +• Boundary condition at γ: z1 = σγz2. +• Boundary condition at γ′: z6 = σγ′z5. +• For �u to be continuous at φ = φ1, there must be a jump whose height is the angular component of �ui′ +at φ = φ1, i.e., hi′ := ζi′(ξn −ξi′); we impose z3 −z2 = hi′. Note that, by the law of reflection, cf. (11), +hi′ = τhi, with τ = 2σγ − 1 and hi := ζi(ξn − ξi). +• For �u to be continuous at φ = φ2, there must be a jump whose height depends on whether θ ≤ π − α +or not: +11 + +D. Pradovera, M. Nonino, and I. Perugia +Geometry-based approximation of waves in complex domains +– If 0 < θ ≤ π − α, the jump equals the angular component of �ui at φ = φ2, i.e., hi := ζi(ξn − ξi); +we impose z5 − z4 = hi. +– If π − α < θ ≤ π − α +2 , the jump equals minus the angular component of �ui′′ at φ = φ2, i.e., +hi′′ := ζi′′(ξn − ξi′′); we impose z5 − z4 = −hi′′. Note that, by the law of reflection, cf. (11), +hi′′ = (2σγ′ − 1)hi, with hi := ζi(ξn − ξi). +In summary, z5 − z4 = τ ′hi, with τ ′ = 1 if 0 < θ ≤ π − α and τ ′ = 1 − 2σγ′ if π − α < θ ≤ π − α +2 . +• Conservation of mass: for all t > 0, +0 = +� +R2 �un(x, t)dx = +� ∞ +0 +� 2π−α +0 +ψn(ρ, t)ζn(φ)ρdφdρ += +�� 2π−α +0 +ζn(φ)dφ +� �� ∞ +0 +ψn(ρ, t)ρdρ +� +� +�� +� +=:C(t) += +�z1 + z2 +2 +φ1 + z3 + z4 +2 +(φ2 − φ1) + z5 + z6 +2 +(2π − α − φ2) +� +C(t), +which leads to the condition z1+z2 +2 +φ1 + z3+z4 +2 +(φ2 − φ1) + z5+z6 +2 +(2π − α − φ2) = 0. +One constraint is missing for the values z1, . . . , z6 to be uniquely determined. Specifically, some simple +algebra shows that, for any δ ∈ R, the following set of values satisfies all the above conditions: +z2 = +τhi + δ +(σγφ1 + φ2)/(φ1 − φ2), +z5 = +τ ′hi + δ +(φ1 + σγ′φ2 − (σγ′ + 1)(2π − α))/(φ1 − φ2), +(20) +together with z1 = σγz2, z3 = z2 + τhi, z4 = z5 − τ ′hi, and z6 = σγ′z5. Note that the jump heights τhi and +τ ′hi appear in the numerators of z2 and z5, respectively. +Our diffraction model is simply the one given by δ = 0. Intuitively, this corresponds to a “balanced” +partitioning of the mass of the diffracted wave into the components related to the two (reflection and/or +shadow) boundaries φ1 and φ2. See Fig. 6 (top) for an example of the resulting diffraction wave. +To further highlight the (mostly beneficial) effects of the choice δ = 0, we note that: +• in the symmetric case θ = π − α +2 , δ = 0 leads to a symmetric ζn: (1 − 2σγ′)z2 = (1 − 2σγ)z5; +• in the case θ = π − α, the second transition happens at γ′, i.e., φ2 = 2π − α, and the choice δ = 0 +yields z4 = 0, so that the diffraction wave vanishes at γ′ (which is physically sound); +• in the “grazing incidence” case θ = 0, the two transition coalesce into one, i.e., φ1 = φ2, and the choice +δ = 0 leads to ζn(φ) = 0 for all φ, which is unphysical; see the following remark for a possible solution. +Remark 4.7. In the “grazing incidence” case θ = 0 (which corresponds to φ1 = φ2 = π), the diffraction +wave �un does not cure the discontinuity of �u at the boundary {φ = π}. This is because, in some sense, the +two discontinuities of �un at φ1 and φ2 cancel each other out. For a similar reason, a small θ ≈ 0 will result +in a continuous total wave, but a sharp gradient will be present for φ1 < φ < φ2. +By tweaking the value of δ, we can obtain an alternative diffraction model, which guarantees Assump- +tion 4.4 even in the case of grazing incidence. To this aim, we can set +δ = +� +(σγ + 1) σγφ1 + φ2 − (σγ′ + 1)(2π − α) +(σγ − σγ′)φ1 + (σγ′ + 1)(2π − α)φ1τ+ ++(σγ′ + 1) +φ1 + σγφ2 +(σγ − σγ′)φ2 + (σγ′ + 1)(2π − α)(φ2 − 2π + α)τ ′ +� +hi +φ1 − φ2 +. +(21) +Roughly speaking, this corresponds to a different “balancing” of the mass of the diffracted wave into the +components related to the two boundaries φ1 and φ2. +This being said, in our numerical tests, such alternative model, albeit recovering a continuous total wave, +resulted in a reduced accuracy of approximation. Specifically, using the value of δ above, we have observed +an exaggerated magnitude of the diffraction wave, especially in the shadow zone. +12 + +D. Pradovera, M. Nonino, and I. Perugia +Geometry-based approximation of waves in complex domains +Example +exterior +incidence +∥�u(·, T)∥L2(Ω) +∥�u(·, T) − uFEM(·, T)∥L2(Ω) +index +angle α +angle ω +#1 +4.712 +0.984 +2.50 · 10−1 +3.65 · 10−4 +#2 +5.093 +0.603 +2.02 · 10−3 +#3 +2.761 +1.363 +1.05 · 10−2 +#4 +2.761 +3.277 +2.00 · 10−2 +Table 1: Setup for the four wedge examples. The angle ω is as in Fig. 7. +ω +ω +ω +ω +−1 +0 +1 +Figure 7: Initial conditions for the wedge examples, indexed #1 through #4 from left to right. The (dashed) +distance between the center of the Gaussian and the boundary vertex is 4 units in all cases. +5 +Numerical results +In our experiments, we require a “reference” solution of (1) to validate our results. To this effect, we use the +solution uFEM obtained by discretizing (1) with: +• the P1-finite element method (P1-FEM) with mass-lumping, over a regular triangulation (mesh) of the +physical domain Ω; +• explicit leapfrog timestepping with a uniform time step that satisfies the CFL condition on the chosen +mesh. +See [11, 16] for more details on this discretization strategy. +If the domain Ω is unbounded, we first need to truncate it in such a way that reflections from the non- +physical truncation boundary do not affect the solution in the region of interest for t < T. Recalling that +the problem data are supported in a ball of radius R and center 0, this can be done, e.g., by truncating Ω +at the sphere with radius R + T and center 0. In our tests, we rely on FEniCS [1] to carry out the P1-FEM +discretization on 2-dimensional domains Ω. +Instead, note that with our proposed approach, modeling unbounded domains is straightforward. Indeed, +we can simply ignore any reflection or diffraction from its “infinitely far” vertices/edges. +All our tests are performed in Python 3.8 on a machine with an 8-core 3.60 GHz Intel® processor +and 64 GB of RAM. For reproducibility, our code is made available at https://github.com/pradovera/ +ray-wave-2d. +5.1 +Some simple wedges +As a way to assess our proposed method in simple settings, we consider four different “wedge” domains. +Similarly to the diagrams in Figs. 4 and 5, we define Ω to be one of the portions of the plane R2 delimited by +straight lines intersecting at a point. Locally around such point, Ω “looks” like either Fig. 4 or Fig. 5, with +α being the outer angle. The specific choices of wedge angles α are reported in Table 1 for the four cases. +We set up a problem of the form (1), with u0 an isotropic Gaussian with standard deviation 0.2. The +center of u0 is at a point located at a 4-unit distance from the wedge vertex, in the direction determined by +the “incidence angle” ω. See Fig. 7 for a representation of the initial conditions in the four cases. We set +u1 = f = 0, we enforce Neumann boundary conditions on the whole ∂Ω, and we seek the solution at the +final time T = 5, i.e., 1 time unit after the wave crest has reached the wedge vertex. +13 + +D. Pradovera, M. Nonino, and I. Perugia +Geometry-based approximation of waves in complex domains +0 +1 +2 +3 +4 +5 +0 +2 +4 +6 +t +ρ +−1 +−0.5 +0 +0.5 +1 +Figure 8: Free-space solution Ψ. The dashed line denotes the upper bound of the “causality cone” of Ψ, i.e., +ρ = t + R, with R = 1. +�u +uFEM +�u − uFEM +Example #1 +Example #2 +Example #3 +Example #4 +−0.1 +0 +0.1 +−2 · 10−4 +0 +2 · 10−4 +−0.1 +0 +0.1 +−5 · 10−3 +0 +5 · 10−3 +−0.1 +0 +0.1 +−1 · 10−2 +0 +1 · 10−2 +−0.1 +0 +0.1 +−5 · 10−2 +0 +5 · 10−2 +Figure 9: Results for the four wedge examples. Each row pertains to a different example. In each row, from +left to right: surrogate solution, FE solution, and error. The color scales for the first two columns are the +same. All results are shown at the final time t = T. +14 + +AD. Pradovera, M. Nonino, and I. Perugia +Geometry-based approximation of waves in complex domains +To this aim, we employ our proposed approach, see Section 2. First, we compute an approximation of +the free-space solution Ψ, which solves (4), by employing P1-FEM with explicit leapfrog timestepping. Note +that, since (4) is cast in polar coordinates, we only need to discretize a 1D interval with P1-FEM. Since the +initial condition u0 is supported within the unit disk, we have R = 1, and it suffices to approximate Ψ(ρ, t) +for (ρ, t) ∈ [0, T + R] × [0, T]. Since this space-time domain is only 2-dimensional, we can afford even a very +fine discretization. In our experiments, we employ a 1001 × 2001 uniform Cartesian space-time grid, i.e., +the mesh size is δx = T +R +1000 and the time step is δt = +T +2000. This satisfies the CFL condition. We show the +resulting Ψ (which, in fact, we should denote by ΨFEM) in Fig. 8. +After this preliminary step, we use the timetable-based strategy from Section 2 to identify reflection +and scattering effects, which are then added up to give the final approximation �u. We show the resulting +�u(·, T) in Fig. 9. In this figure, we also display a reference solution uFEM(·, T), which we obtain by direct +discretization of (1) by P1-FEM and leapfrog timestepping, as described at the beginning of Section 5. +In all four examples, we see that �u and the reference uFEM seem qualitatively close. Notably, we can +observe a good representation of the most prominent wavefronts, which are due to propagation of either the +main “free-space” wave or to its reflections. Indeed, those wave contributions are reconstructed exactly: the +only errors are the ones due to FE approximation and timestepping, which affect both uFEM and �u (the +latter through the approximation of Ψ). Instead, some differences are present when comparing diffraction +effects, which arise as circular waves about the wedge vertex. We can quantitatively observe this in the last +column of both Table 1 and Fig. 9. +In example #1, we observe a very small error, which, in fact, is simply the (FEM and timestepping) +discretization error. This is related to the fact that the wedge has exterior angle α = +3 +2π, which makes +diffraction unnecessary in approximating the wave u: reflections are enough2. +In the other examples, diffraction effects are necessary to correctly identify u. While a good qualitative +behavior can be observed in Fig. 9, we can see in Table 1 that a modest error is present. Specifically, we +report the L2(Ω)-norm of �u and of the error �u − uFEM at the final time t = T, defined as +∥v∥L2(Ω) = +�� +Ω +v(x)2dx +�1/2 +. +(22) +We see the largest error in example #4, where the relative L2(Ω)-approximation error amounts to 8%. +This was to be expected, since this last example is rather close to the setting of grazing incidence (α+ω ≈ 2π), +which, as discussed in Section 4.2, is approximated rather poorly by our diffraction model. Qualitatively, +the bad approximation quality is apparent in the form of a rather sharp gradient in the corresponding plot +of �u in Fig. 9 (bottom left). +5.1.1 +Building a cavity out of wedges +As a slightly more complicated example, we now combine the four wedges from the previous section to obtain +the open cavity represented in Fig. 10. In this case, more reflection and diffraction effects will arise, due +to the trapping nature of the domain. Our initial conditions and forcing term are the same as before, but +now all edges are sound-soft. Accordingly, we model them using Dirichlet boundary conditions. The time +horizon is T = 9. +Using our strategy from Section 2, we build the approximation �u, which contains 47 wave terms (1 source +wave, 32 reflected waves, and 14 diffraction waves). We compare the approximation �u with the reference +solution uFEM, obtained as described at the beginning of Section 5. +We show the results of the comparison in Fig. 10, at 4 time instants t ∈ {0, 3, 6, 9}. Once more, we see +a good qualitative agreement between �u and uFEM, with the most important features of u being identified +well. +Up to t = 3, only reflections have happened, so that the error �u − uFEM consists only of FEM +and timestepping errors. On the other hand, for larger times, we see “error waves” of small-to-moderate +amplitude propagating from the 3 vertices of Ω that generate diffraction effects. These correspond to errors +in diffraction modeling. +2To intuitively understand why, let γ1 and γ2 be the two sides forming ∂Ω. The domain Ω is partitioned exactly into (i) +the light cone of the reflection off γ1 and then off γ2 and (ii) the light cone of the reflection off γ2 and then off γ1. For this +reason, the diffraction effects due to these two rays cancel out. Incidentally, the same phenomenon can be expected whenever +the interior angle 2π − α is of the form π +n , with n ∈ {2, 3, . . .}. +15 + +D. Pradovera, M. Nonino, and I. Perugia +Geometry-based approximation of waves in complex domains +�u +uFEM +�u − uFEM +t = 0 +t = 3 +t = 6 +t = 9 +−1 +−0.5 +0 +0.5 +1 +−0.1 +0 +0.1 +−5 +0 +5 +·10−4 +−0.1 +0 +0.1 +−5 +0 +5 +·10−2 +−0.1 +0 +0.1 +−5 +0 +5 +·10−2 +Figure 10: Results for the cavity domain. Each row corresponds to a different time instant t ∈ {0, 3, 6, 9}, +from top to bottom. In each row, from left to right: surrogate solution �u(·, t), FE solution uFEM(·, t), and +error �u(·, t) − uFEM(·, t). The color scales for the first two columns are the same. +16 + +D. Pradovera, M. Nonino, and I. Perugia +Geometry-based approximation of waves in complex domains +5.2 +A tall room +We consider a simplified sound propagation problem in a room. For simplicity, we consider a 2-dimensional +problem, thus assuming an infinitely tall room, and modeling line sources (in the z-direction) as point sources. +The complicated domain Ω ⊂ R2 is depicted in Fig. 11. It is composed of two communicating “rooms” +with sound-hard walls, as well as of a third large room (above), which is modeled as infinitely large. In the +main room, three sound-soft triangular obstacles are also present. +Setting once more u1 = f = 0, we are interested in modeling the propagation of an initial condition +u0 modeled as a Ricker wavelet centered at 0, see Fig. 11 (top left), over the time horizon t ∈ [0, T], with +T = 20. To this aim, we employ our proposed method from Section 2. +As in the previous example, we start by computing an approximation of the free-space solution Ψ = Ψ(ρ, t) +for (ρ, t) ∈ [0, T + R] × [0, T], see (4), with R being the radius of the support of the initial condition u0. +Again, we use P1-FEM with leapfrog timestepping for this. +Since many reflective surfaces face each other, the domain Ω is trapping. Accordingly, we expect the +number N of waves in the approximation �u to be rather large. In the interest of reducing the number of +such terms, we can employ the on-the-fly parsimonious strategy described in Remark 2.2, removing all wave +terms �un whose magnitude is smaller than tol = 10−2. After this, N ≈ 1.4 · 103 terms are left. Although +this value of N may seem large, the evaluation of the corresponding surrogate �u is rather quick, due to the +explicit nature of each wave contribution (and to the fact that their supports are smaller than the whole Ω). +We show the resulting u(·, t) for the four times t ∈ {0, 7.5, 15, 20} in Fig. 11. There, we can see why so +many terms are necessary for the approximation of u: we must model many reflection and diffraction effects. +Since energy escapes the system only through the top “door”, the wave will persist for quite a long time. +Accordingly, a larger T will make a larger N necessary. +In order to better inspect this effect, we show the trace of the solution at the arbitrarily chosen point +xtrace = (−1, −2) in Fig. 12. We notice that oscillations persist for t > 10. We use this last plot also to +validate our results. To this aim, we compare three results: +• The surrogate �u obtained as described above, with tol = 10−2. +• The surrogate �u obtained with our strategy, but with tol = 10−3. This leads to an increased number +of rays N ≈ 7.3 · 103. +• The reference solution uFEM obtained by the P1-FEM with leapfrog timestepping, as described at the +beginning of Section 5. The mesh size must be chosen small enough to resolve both the initial condition +and the domain Ω well. In our case, we have a mesh with approximately 1.4 · 106 elements. To satisfy +the CFL condition on this mesh, we choose a time step ∆t ≈ 7 · 10−3. +We can observe that the two surrogates obtained with our approach give very similar results. Indeed, +the cutoff tolerance tol affects the results only for large t > 15, due to the accumulation of “small” waves +that are excluded from the coarser surrogate but included in the finer one. +Moreover, taking the FE solution as reference, we see that most of the peaks of the surrogates are +aligned with the FE ones (i.e., the “phase” of the wave is well approximated), but there are some noticeable +discrepancies in their amplitudes. This is due to the fact that, in our approach, reflection is modeled exactly, +whereas the magnitudes of the diffraction waves are only roughly approximated. For this reason, we should +not expect the amplitude error to get smaller if we reduce tol. +The only “real” way of improving the +approximation is using a better diffraction modeling. +As a final result, we also report: +• The so-called “offline” time, i.e., the time required to compute the numerical solution. For �u, this +means executing Algorithm 1. For uFEM, this means building the mesh, assembling the FE stiffness +and (lumped) mass matrices, and carrying out the timestepping. +• The so-called “online” time, i.e., the time required to evaluate the numerical solution (�u or uFEM) at +a single (x, t)-point. +They can be found in Table 2. +17 + +D. Pradovera, M. Nonino, and I. Perugia +Geometry-based approximation of waves in complex domains +−1 +0 +1 +t = 0 +−0.1 +0 +0.1 +t = 7.5 +−0.1 +0 +0.1 +t = 15 +−0.1 +0 +0.1 +t = 20 +Figure 11: 2-dimensional domain Ω modeling a room. Top left plot: initial condition �u(·, 0) = u(·, 0) = u0, +a Ricker wavelet; we also show the point xtrace as a cross. Top right plot: intermediate solution �u(·, 7.5). +Bottom left plot: intermediate solution �u(·, 15). Bottom right plot: final solution �u(·, 20). +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +−0.1 +0 +0.1 +t +u(xtrace, t) +�u (tol = 10−2) +�u (tol = 10−3) +uFEM +Figure 12: Value of solution at point xtrace = (−1, −2). +Method +�u (tol = 10−2) +�u (tol = 10−3) +uFEM +Offline +46.66 [s] +252.9 [s] +188.9 [s] +Online +2.04 [ms] +8.48 [ms] +27.67 [µs] +Table 2: Timings for the room test case. To obtain more statistically significant results, each displayed time +is the average over 3 (resp. 103) runs of the offline (resp. online) phase with identical parameters. +18 + +D. Pradovera, M. Nonino, and I. Perugia +Geometry-based approximation of waves in complex domains +We can observe the increased training and evaluation time that results from decreasing tol. Moreover, we +see that, in this example, our proposed approach is more competitive offline, but less so online. Somewhat +surprisingly, we have observed that the most expensive step in evaluating �u (taking about half of the online +time) is determining whether an evaluation point is in the light cones Ωn. The reason for this is that they +can have rather complicated shapes, cf. Section 3. +Evaluating the FE solution at a space-time point is an extremely cheap operation (essentially corre- +sponding to a vector dot product) while evaluating �u is more expensive, requiring the evaluation of O(N) +nonlinear functions. However, the FE solution comes with the serious drawback of memory usage. Indeed, in +our example, storing uFEM as a (∼ 1.5 · 106) × (∼ 2.9 · 103) array of double-precision floating-point numbers +requires approximately 34 GB. +Concerning the timing results, we also wish to mention that the online times in Table 2 should be +interpreted carefully. Indeed, the online time for the FE solution is artificially deflated by the fact that +xtrace is a vertex of the FE mesh: each evaluation of uFEM corresponds to extracting a vector entry. If xtrace +had not been a vertex of the mesh, the online time could have been larger by at least one order of magnitude, +if not more, depending on the FE implementation. Moreover, we note that accessing point-evaluations of +uFEM at arbitrary times a posteriori, namely, after the timestepping has been carried out, is feasible only if +enough memory is available to store the whole “timestepping history”. Considering the numbers mentioned +in the previous paragraph, this might not be possible in practice, especially for more complex and/or larger +domains. +5.2.1 +A time-harmonic source +One of the advantages of our approach is that it allows changing the source terms of the problem in a +seamless way. Notably, under minor technical constraints (e.g., the support of the new source term should +not be larger than the old one), this kind of change does not require training a new surrogate. +To showcase this, we approximate the wave propagating from a time-harmonic point source at x = 0 +with angular frequency ω > 0. In our tests, we pick ω ∈ {2π, 10π}. To this aim, we define u as the solution +of the following (ω-dependent) problem: +� +� +� +� +� +� +� +� +� +∂ttu(x, t) = ∆u(x, t) − ω2 sin(ωt)δ0(x) +for (x, t) ∈ Ω × (0, T), +u(x, 0) = 0 +for x ∈ Ω, +∂tu(x, 0) = 0 +for x ∈ Ω, +∂νu(x, t) = 0 +for (x, t) ∈ ∂Ω × (0, T], +(23) +where δ0 denotes the usual 2-dimensional “delta function” centered at x = 0. +As usual, we define Ψ = Ψ(ρ, t) as the (ω-dependent) solution of the free-space version of (23) in radial- +temporal coordinates. Note that, in free space, i.e., without boundary effects3, the forcing term in (23) is +equivalent to the following non-homogeneous Dirichlet-like condition at ρ = 0: +U(0, t) = Ψ(0, t) = +� t +0 +� t′ +0 +−ω2 sin(ωt′′)dt′′dt′ = sin(ωt) +∀t > 0. +(24) +Accordingly, the free-space solution Ψ has space-time support {(ρ, t) ∈ [0, ∞)2, ρ ≤ t}, which is a subset of +the free-space solution Ψ from the previous section, namely, {(ρ, t) ∈ [0, ∞)2, ρ ≤ t + R}. As such, to obtain +an approximation for the wave u generated by the time-harmonic source for an arbitrary ω, it suffices to +plug the corresponding Ψ in each term of the surrogate �u from the previous section! We show the results of +our approximation in Figs. 13 and 14. +We note that, if we had chosen to apply the FEM to approximate the wave u generated by the time- +harmonic source, we would have been forced to carry out a new simulation from scratch for every frequency +to be studied. To this end, we would have needed to choose a mesh with ω-dependent resolution: the mesh +size should be small enough for the well-known pollution effect (see, e.g., [22, 2]) to be absent. +3If Ω is bounded, reflected or diffracted waves will generally bounce back to x = 0. As such, the value of u(0, t), u being +the solution of (23), will be different from the source signal sin(ωt). For this reason, we cannot turn the forcing term in (23) +into a condition like (24), except in free space. +19 + +D. Pradovera, M. Nonino, and I. Perugia +Geometry-based approximation of waves in complex domains +−0.4 +−0.2 +0 +0.2 +0.4 +−0.2 +0 +0.2 +Figure 13: Surrogate solution found with the proposed approach. Left plot: ω = 2π. Right plot: ω = 10π. +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +−0.2 +−0.1 +0 +0.1 +0.2 +t +u(xtrace, t) +ω = 2π +ω = 10π +Figure 14: Value of solution at point xtrace = (−1, −2) for different excitation frequencies. +In our proposed approach, we also have a constraint on the mesh resolution. However, it only applies +to the problem defining the free-space solution Ψ, which is 1-dimensional in space. Hence, having to refine +the mesh represents a much smaller obstacle to efficiency. In particular, for a fixed time horizon T, the +computation of �u becomes more and more efficient, when compared to the computation of uFEM, as the +frequency ω increases. +6 +Conclusions +We have presented a method for approximating waves propagating through complex 2-dimensional domains +with polygonal boundaries. Our method relies on the automatic identification of reflection and diffraction +effects caused by the domain geometry. Each effect is modeled through a relatively simple nonlinear expres- +sion. In our numerical tests, we have observed rather a good approximation quality, with the main features +of the target wave being well identified. As a way to improve the approximation accuracy, we recall that any +diffraction model could replace the current one. +In terms of complexity, our method requires the solution of a simplified 1D-in-space problem, much +simpler than the original 2D-in-space one. We expect such improved accuracy to increase even further if +one were to consider 3D instead of 2D problems. However, in order to generalize our method to 3 space +dimensions, a suitable diffraction model would be necessary. This is one of our ongoing research directions. +Another favorable aspect of our algorithm is its potential to be run on parallel architectures, since +the computation of different rays can be carried out independently. +This is not the case for standard +timestepping-based discretizations, due to their intrinsically sequential nature. +Other envisioned extensions of our technique involve the cases of domains with curvilinear boundaries +and of propagation media with non-uniform properties (e.g., density and refraction index). Specifically, this +latter case would effectively result in a non-uniform wave speed, with reflections and diffractions happening +20 + +D. Pradovera, M. Nonino, and I. Perugia +Geometry-based approximation of waves in complex domains +also within the domain Ω. +Finally, we recall that, in many applications, the ultimate target is understanding how the wave u solving +(1) depends on underlying parameters µ, e.g., the forcing term f, the shape of the domain Ω, etc. In this +setting, MOR methods try to construct a surrogate model of the form �u = �u(x, t; µ), providing a good +approximation of u over a whole range of parameter values. Even though our technique was presented here +in the non-parametric setting, we believe that it potentially allows incorporating the parameter dependence +in a natural and efficient way. In our opinion, this might be achievable by leveraging the simple and in- +terpretable structure of the rays (free-space solution, light cone, and angular modulation). As a simple +preliminary example, we showcased this in Section 5.2.1 for a parametric source term, with the parame- +ter being the frequency. 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SIAM +Journal on Scientific Computing, 39(4):A1225–A1250, 2017. +22 + diff --git a/8dFRT4oBgHgl3EQfpTfl/content/tmp_files/load_file.txt b/8dFRT4oBgHgl3EQfpTfl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b4d6a9c57e5837b9090c4bf14e13d293ab6d7adc --- /dev/null +++ b/8dFRT4oBgHgl3EQfpTfl/content/tmp_files/load_file.txt @@ -0,0 +1,1066 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf,len=1065 +page_content='Geometry-based approximation of waves propagating through complex domains∗ Davide Pradovera† Monica Nonino† Ilaria Perugia† February 1, 2023 Abstract We consider wave propagation problems over 2-dimensional domains with piecewise-linear bound- aries, possibly including scatterers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Under the assumption that the initial conditions and forcing terms are radially symmetric and compactly supported (which is common in applications), we propose an ap- proximation of the propagating wave as the sum of some special nonlinear space-time functions: each term in this sum identifies a particular ray, modeling the result of a single reflection or diffraction ef- fect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We describe an algorithm for identifying such rays automatically, based on the domain geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' To showcase our proposed method, we present several numerical examples, such as waves scattering off wedges and waves propagating through a room in presence of obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Keywords: wave propagation, model reduction, scattering, geometrical optics, diffraction AMS subject classifications: 35L05, 35Q60, 65M25, 78A45, 78M34 1 Introduction The discretization of numerical models for the simulation of complex phenomena results in high-dimensional systems to be solved, usually at an extremely high cost in terms of computational time and storage memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Among these models, wave propagation problems represent an extremely interesting topic: relevant applica- tions can be found, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', in the field of array imaging, where acoustic, electromagnetic, and elastic waves in scattering media are modeled by the reflectivity coefficient, which is often unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Some examples in this direction can be found in [5, 6, 7, 30], where inverse scattering problems are used to infer the reflectivity of one or more scatterers embedded either in a known and smooth medium, or in a randomly inhomogeneous medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Another example of application of wave propagation problems is numerical acoustics, where the goal is to simulate the propagation of sound in a room, in presence of obstacles and walls with different absorbing and/or reflecting properties, see [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Wave propagation problems in the time-harmonic setting (the Helmholtz problem, cast in the frequency domain) have been widely studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' See, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', [4, 13, 19, 24, 25, 27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' However, our focus here are problems in the time domain, whose numerical simulation is expensive, mainly because one needs to use both a fine spatial mesh and a carefully chosen time step in order to satisfy the CFL condition [11, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In the interest of making these simulations feasible, model order reduction (MOR) [3, 9, 14, 17] represents a promising framework, whose goal is to reduce the computational cost of solving the problem of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In this context, it is well known [12, 15] that wave propagation problems are characterized by a slowly decaying Kolmogorov n-width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Because of this, classical linear-subspace MOR methods are not able to reproduce the behavior of the wave propagation without relying on a very high-dimensional linear manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' This makes linear surrogate models unappealing, since they do not yield significant speed-ups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In recent years, many approaches have been proposed to overcome the intrinsic “difficulty” of problems with slowly ∗M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Nonino and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Perugia have been funded by the Austrian Science Fund (FWF) through project F 65 “Taming Complexity in Partial Differential Systems” and project P 33477.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' †Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria (da- vide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='pradovera@univie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='at, monica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='nonino@univie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='at, ilaria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='perugia@univie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='at).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='13613v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='NA] 31 Jan 2023 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Pradovera, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Nonino, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Perugia Geometry-based approximation of waves in complex domains decaying Kolmogorov n-width, with the target of making MOR more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' To this end, such methods rely on nonlinear and/or hybrid space-time approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' For more details, we refer to [8, 10, 18, 21, 26, 29, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In this work, we focus on wave propagation over 2-dimensional spatial domains, possibly including ob- stacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We limit our investigation to domains with piecewise-linear boundaries and a constant wave speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The initial conditions and forcing terms are assumed to be compactly supported and radially symmetric around a “source point”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' This situation arises in many of the above-mentioned applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Under these assumptions, we propose to approximate the solution of the problem of interest with the sum of some special nonlinear space-time functions, which we call “rays”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Each ray models a reflection or diffraction effect, and is composed of different parts: the free-space radially symmetric solution of the wave equation, modeling the space-time propagation of the ray;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' a spatial indicator function, determining the light cone of each ray;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' a nonlinear spatial term encoding the angular modulation of the ray, which is crucial when modeling diffraction effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The number of terms appearing in the sum is determined by the number of reflection and diffraction effects that are required to faithfully approximate the target wave, which ultimately depends on the geometry of the physical domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Among the advantages of the proposed approach, we mention the fact that each ray is separable into time-radial and angular components (in the “polar coordinates” sense).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' As we will see, we can leverage this to reduce drastically the computational cost and the storage memory that are required by our approximation, with respect to competitor methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The rest of the paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1 we present the problem of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In Section 2 we introduce the main ingredients of our method, and we describe the “training phase” of the algorithm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', the construction of the approximated wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In Sections 3 and 4 we detail how we model reflection and diffraction effects, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The latter section is rather extensive, since diffraction is much harder to model than reflection, and requires special care.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In Section 5 we present some numerical results to showcase our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Both simple benchmarks (wedges) and more complicated tests (2D room model with scatterers) are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Some final considerations follow in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1 Target problem We are interested in the numerical approximation of the solution of the wave equation in complex domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In this work, we consider 2-dimensional domains only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' However, most of our discussion generalizes to 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We defer a discussion on this till Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We denote by Ω ⊂ R2 the physical domain in which the wave equation is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We assume that Ω is either a closed polygon or a set-subtraction of polygons (to allow for multiply connected domains).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We denote by ne and nv the number of edges and vertices of ∂Ω, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We study the propagation of waves in Ω over a given time interval of interest [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The model problem is the wave equation with constant (unit) wave speed: � � � � � � � � � ∂ttu(x, t) = ∆u(x, t) + f(x, t) for (x, t) ∈ Ω × (0, T), u(x, 0) = u0(x) for x ∈ Ω, ∂tu(x, 0) = u1(x) for x ∈ Ω, ∂νu(x, t) = 0 for (x, t) ∈ ∂Ω × (0, T], (1) with ∆ the Laplacian operator, defined, in 2 dimensions, as ∆ = �2 j=1 ∂xjxj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The homogeneous Neumann condition (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', the last equation above) models the whole boundary ∂Ω as sound-hard [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' More generally, all or parts of ∂Ω may be modeled as sound-soft via a Dirichlet-type condition: u(x, t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We assume that the initial conditions u0 and u1, as well as the forcing term f, have radial symmetry around a given point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Without loss of generality, we will take such point to be the origin of R2: u0(x) = η0(∥x∥), u1(x) = η1(∥x∥), f(x, t) = η2(∥x∥ , t) ∀(x, t) ∈ Ω × (0, T), (2) 2 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Pradovera, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Nonino, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Perugia Geometry-based approximation of waves in complex domains with ∥x∥2 = �2 j=1 x2 j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We further assume that the functions ηj have compact support, namely, that there exist R > 0 such that ηj(ρ) = 0 for all ρ > R and j = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Moreover, to avoid incompatibilities with the boundary conditions, for simplicity we will only consider the situation where the supports of the functions ηj are fully contained in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 2 Approximation framework Before we can model boundary effects (reflection and diffraction), we need to understand how the solution u would behave if no boundary were present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' To this aim, we consider the wave equation in free space � � � � � ∂ttU(x, t) = ∆U(x, t) + f(x, t) for (x, t) ∈ R2 × (0, ∞), U(x, 0) = u0(x) for x ∈ R2, ∂tU(x, 0) = u1(x) for x ∈ R2, (3) which we have obtained from (1) by replacing Ω with the whole plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Due to radial symmetry (of the initial conditions and of the forcing term), we can recast the problem in polar coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' This allows us to define the free-space solution in the radial coordinate Ψ, as the solution of � � � � � � � � � ∂ttΨ(ρ, t) = �∆Ψ(ρ, t) + η2(ρ, t) for (ρ, t) ∈ (0, ∞) × (0, ∞), Ψ(ρ, 0) = η0(ρ) for ρ ∈ [0, ∞), ∂tΨ(ρ, 0) = η1(ρ) for ρ ∈ [0, ∞), ∂ρΨ(0, t) = 0 for t ∈ (0, ∞), (4) where �∆ is the Laplace operator in polar coordinates (under radial symmetry), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' �∆ = ∂ρρ + 1 ρ∂ρ, and U(x, t) = Ψ(∥x∥ , t) for all x ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Note that, by the compact support of the initial conditions and of the forcing term, and by the finite (unit) speed of propagation of the wave equation, we have Ψ(ρ, t) = 0 whenever ρ > t + R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Generally, the free-space solution Ψ is not available analytically, except for very simple choices of initial conditions and forcing term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Accordingly, in most applications, the function Ψ will need to be replaced with a suitable approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' To this effect, one could discretize (4), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', with a finite element approximation (in space) and some timestepping scheme (in time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' See Section 5 for more details on how this can be carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Our goal is to approximate, for all (x, t) ∈ Ω × [0, T], the solution u(x, t) of the wave equation problem (1) with the following sum of special functions: u(x, t) ≈ �u(x, t) = N � n=1 Ψ(∥x − ξn∥ + rn, t)1Ωn(x)ζn(x − ξn) � �� � �un(x,t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' (5) Each term �un is what we will call a “ray”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Therein, Ψ is the above-mentioned free-space radially symmetric solution of (4), and 1A denotes the indicator function with support A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', 1A(y) = � 1 if y ∈ A, 0 if y /∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' (6) Moreover, in (5), we have introduced the following quantities: N is the number of rays used in the approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' ξn is the location of the new source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' rn ≥ 0 is a spatial delay, which will be used for the synchronization of diffraction effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Ωn ⊂ Ω is the light cone (the spatial support) of a term of the sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 3 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Pradovera, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Nonino, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Perugia Geometry-based approximation of waves in complex domains ζn : R2 \\ {0} → R is a weight function describing the angular modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We require that ζn be a positive-homogeneous functions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', ζn(y) = ζn(τy) for all τ > 0 and y ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Note that, due to the finite speed of propagation of the free-space solution Ψ, we have that a generic term �un(x, t) is zero whenever t < ∥x − ξn∥ + rn − R, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', for t small enough, depending on x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The number of rays N in the sum (5) will be determined based on how many boundary effects (reflections and diffractions) need to be included in �u in order to have a good approximation of the target wave u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We describe a strategy for automatically identifying a good N in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' See, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1 Building the low-rank skeleton Recalling that u solves the wave equation (1) in the domain Ω, we use the first term in (5), namely, �u1, to approximate the outgoing component of u, ignoring any effect due to the boundary ∂Ω, except for shadows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Then, given such �u1, we use the other terms �u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' , �uN to correct this first approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Each extra term models a single effect due to a certain portion of the boundary, specifically, an edge (reflection off that edge) or a vertex (diffraction about that vertex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Going back to the first ray �u1, let us define it, by providing its “ingredients” ξ1, r1, Ω1, and ζ1, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We set ξ1 = 0, the center of the initial condition, as well as r1 = 0, since no delay is necessary for this first term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Then, leveraging symmetry, we set ζ1 ≡ 1, which corresponds to the (physical) assumption that the propagation of �u1 is purely radial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Finally, we set Ω1 (the light cone around 0) as the set of points that can be reached from 0 via a straight line without going outside ∂Ω, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', Ω1 = {x ∈ Ω : τx ∈ Ω ∀0 ≤ τ ≤ 1} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' (7) In summary, the first term of �u is �u1(x, t) = Ψ(∥x∥ , t)1Ω1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' (8) Then we can move to the subsequent terms �un, n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Their expressions depend on our choice of reflection and diffraction modeling, and will be provided in the upcoming sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Instead, in the rest of the present section we focus on understanding how large N should be, in order for �u to provide a faithful approximation of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Equivalently, we want to count the number of times the wave gets reflected or diffracted at the boundary ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' This is done incrementally, starting from the initial value N = 1 (no boundary effects) and then updating this guess as more and more boundary effects get “discovered”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' To help us in this endeavor, we employ what we call a timetable, which, in this work, is simply a list of vectors, each with size ne + nv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The timetable is built incrementally starting from an empty list, appending one new vector every time a new term is added in the sum (5), starting from �u1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The entries of the n-th timetable vector are the waiting times before �un comes in contact with an edge or a vertex of ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' If it is impossible for �un to “cast light” (along a straight path) onto a certain edge or vertex, then the corresponding entry in the timetable is set to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' After this, it suffices to look for the smallest not-yet-explored entry of the timetable to identify what the next term of the approximation �u should be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Once the entry in the timetable has been explored, its value is set to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We start by describing how the first vector a1 ∈ Rne+nv of the timetable (corresponding to �u1) is computed, and how a1 allows us to identify the (geometric) features of �u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The vector a1 can be partitioned into edges-related part (the first ne entries) and vertices-related part (the last nv entries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Edge times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Given a generic edge γj ⊂ ∂Ω (j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' , ne) belonging to the domain boundary, we define the corresponding entry of a1 as (a1)j = � r1 + inf � ∥x − ξ1∥ : x ∈ γj ∩ Ω1 � if the set is non-empty, ∞ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' (9) Note that we have taken the shortest path from ξ1 to γj, and that we have denoted the closure of the light cone Ω1 as Ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Vertex times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Given a generic vertex yj ⊂ ∂Ω (j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' , nv) of the domain boundary, we set (a1)ne+j = � r1 + ∥yj − ξ1∥ if yj ∈ Ω1, ∞ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' (10) 4 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Pradovera, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Nonino, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Perugia Geometry-based approximation of waves in complex domains Note that we have included the delay r1 (which is actually zero here) as a way to streamline Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' (9) and (10) for the upcoming section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 1 for a diagram showcasing these formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' (a1)1 (a1)2 (a1)5 (a1)17 Ω1 Ω \\ Ω1 γ1 γ2 γ3 γ4 γ5 y3 y4 y6 ξ1 (a1)3 = ∞ (a1)4 = ∞ (a1)14 = ∞ (a1)15 = ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Figure 1: Computation of some timetable entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The boundary ∂Ω has 11 sides, so that, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', (a1)14 is related to y3 and (a1)17 is related to y6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The shadowed area Ω \\ Ω1 is in darker grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The smallest entry of a1 is the time at which the first “boundary event” (reflection or diffraction) can happen1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The index of the smallest entry tells us whether the event is a reflection (index 1 ≤ j ≤ ne) or a diffraction (index ne + 1 ≤ j ≤ ne + nv), and also what edge/vertex causes the event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' From here, we use the models described in Sections 3 and 4 to build �u2, by computing ξ2, r2, Ω2, and ζ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Then, the second timetable vector a2 can be computed by replacing all subscripts “1” by “2” in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' (9) and (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' This is followed by the construction of �u3, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The process continues until all not-yet- explored entries of the timetable are larger than T +R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Indeed, starting from this time instant, the would-be next terms of �u do not affect the approximation anymore, since, due to the finite speed of wave propagation, they only act (on Ω) after the end of the time horizon, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', for t > T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The total number of rays N is simply the number of vectors in the timetable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We summarize the overall procedure for the construction of the rays �un in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' For ease of presentation, once an entry of the timetable has been explored, it is set to ∞ as a way for the algorithm to ignore it from that point forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Algorithm 1 Step-by-step construction of the surrogate model Set N ← 1, find Ω1 as in (7), and define �u1 as in (8) Define a1 ∈ Rne+nv using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' (9) and (10) Set i ← 1 and j ← arg minj=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=',ne+nv (a1)j while (ai)j ≤ T + R do Set (ai)j ← ∞ and N ← N + 1 if j ≤ ne then Find ξN, rN, ΩN, and ζN as in Section 3 ← Reflection from edge j else Find vertex index j′ ← j − ne Find ξN, rN, ΩN, and ζN as in Section 4 ← Diffraction from vertex j′ end if Define �uN from ξN, rN, ΩN, and ζN, as in (5) Define aN ∈ Rne+nv using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' (9) and (10), with “N” replacing “1” in subscripts Set (i, j) ← arg mini=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=',N,j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=',ne+nv (ai)j end while 1We say “can happen” since not all vertices can cause diffraction, when hit from a certain point source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' This issue is discussed in Section 4, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 5 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Pradovera, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Nonino, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Perugia Geometry-based approximation of waves in complex domains γ ξi ξn y(x) x θr θi φi(y(x)) φn(x) βγ ξn ξi y(x) x Figure 2: Graphical representation of a reflection off edge γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' On the left, the law of reflection prescribes θr = θi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We show the straight line �γ supporting γ with a dotted stroke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' For a given observation point x, y(x) denotes the point of incidence of the reflected ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' On the right, computation of the light cone Ωn (light-grey area) and its complementary shadow zone Ω \\ Ωn (dark-grey area) for the reflected ray, in the presence of a rectangular obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The dashed portion of edge γ denotes the shadow γ \\ γ(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The shadow zone consists of two connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In trapping domains, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='2, the number of terms N might be rather large due to waves repeatedly “bouncing back and forth” between two or more edges/vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' A large N, although necessary for a good approximation of all wavefronts, is undesirable since it increases the computational cost of both the construction of the surrogate �u and its evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' As a compromise, one could remove all terms �un that are smaller than a certain tolerance tol, uniformly over x and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' This can be done as a post-processing step (thus speeding up the evaluation of �u but not its construction) or while building the surrogate itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' This can be achieved with a simple modification of Algorithm 1, by introducing a test on the magnitude of each soon-to-be-added wave contribution �un, discarding terms that are too small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 3 Modeling reflection We now present the strategy for modeling reflection due to an edge γ of the domain boundary ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We rely on the well-known geometrical optics model, which describes wave propagation in terms of rays, not accounting for any diffraction [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We assume that we are adding a new ray �un to the surrogate model (5), due to a reflection phenomenon caused by ray �ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Specifically, we assume that a ray coming from source point ξi hits the edge γ ⊂ ∂Ω, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', that γ ∩ Ωi ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We need to prescribe several ingredients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Spatial correction rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We just transfer rn over from the incoming wave: rn = ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Indeed, as we will see in Section 4, we require the term rn only when modeling diffraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Source point ξn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We use the method of images, which gives the position of ξn as the reflection of ξi with respect to the edge γ: ξn = 2 arg min z∈�γ ∥z − ξi∥ − ξi, (11) where �γ ⊂ R2 is the straight line on which edge γ lies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 2 (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Weight function ζn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Let x − ξn be a generic point where we wish to evaluate the weight function ζn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We define the incidence point y(x) as the intersection (if any) between edge γ and the segment from ξn to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 2 (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' According to the method of images, the amplitude of the reflected wave is equal (up to sign) to the amplitude of the incoming wave: ζn(x − ξn) = (2σγ − 1)ζi(y(x) − ξi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' (12) 6 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Pradovera, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Nonino, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Perugia Geometry-based approximation of waves in complex domains + = Figure 3: Example of reflection off an edge in the presence of an obstacle, from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Neumann conditions are imposed on all edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Source wave (left), reflected wave (middle), and superimposition of the two (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Note how the obstacle creates a shadow zone for source and reflected waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' For simplicity, in this plot we are not showing any reflection or diffraction effects due to the rectangular obstacle, since they would be modeled at different stages of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In the equation above, the quantity σγ is related to the kind of boundary conditions that are imposed on γ: if γ is an edge with Neumann boundary conditions, we set σγ = 1 (ζn and ζi have the same sign), whereas we set σγ = 0 if we have Dirichlet boundary conditions on γ (ζn and ζi have opposite signs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Now, recall that we are assuming all weight functions to be positive-homogeneous: ζi(x − ξi) = ζi(τ(x − ξi)), for all τ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Accordingly, as we are in 2D, ζi(x − ξi) is only a function of the direction (with sign) υi(x) = (x − ξi)/ ∥x − ξi∥, or, equivalently, of the angle φi(x) between υi(x) and the positive x1-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 2 (left) for a graphical depiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Specifically, with an abuse of notation, let ζi(x − ξi) = ζi(φi(x)) and ζn(x − ξn) = ζn(φn(x)), where the “new” angle-dependent functions ζi and ζn are 2π-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' By (12), we deduce the property ζn(φn(x)) = (2σγ − 1)ζi(φi(y(x))) = (2σγ − 1)ζi(2βγ − φn(y(x))) = (2σγ − 1)ζi(2βγ − φn(x)), (13) where βγ is the angle between edge γ and the positive x1-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' This uniquely identifies ζn given ζi and βγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Light cone Ωn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We first identify what portion of γ is actually “lit” by �ui: γ(i) = γ ∩ Ωi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Note that we may have γ ̸= γ(i), for instance when obstacles are present between ξi and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 2 (right) for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Then, roughly speaking, we define the new light cone Ωn as the union of all rays from ξn that pass through γ(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' To be more precise, given x ∈ Ω, let y(x) be the intersection (if any) between γ and the line segment from ξn to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Also, if y(x) exists, we define τ0(x) = ∥y(x) − ξn∥ / ∥x − ξn∥ ∈ (0, 1), which satisfies y(x) = ξn + τ0(x)(x − ξn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The new light cone is defined as Ωn = � x ∈ Ω : y(x) ∈ γ(i) and ξn + τ(x − ξn) ∈ Ω ∀τ0(x) < τ ≤ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' (14) Figure 3 represents a possible output of the numerical algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In this case, we simulate only the reflections, thus discarding, for the time being, any effect due to diffraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' It is clear that, by modeling reflection effects only, we may obtain a discontinuous approximation of the solution of our target problem, where the discontinuity happens exactly at the shadow boundaries (the boundaries of light cones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' As we will see in the next section, introducing diffraction in our approximation will allow us to obtain a continuous approximation �u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 4 Modeling diffraction Here, we describe a strategy for modeling waves diffracted by a vertex of the domain boundary ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' This is required in building a new ray �un whenever the smallest unexplored entry of the timetable is related to a vertex, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', j > ne in Algorithm 1, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We need to identify several ingredients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Source point ξn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We employ the (standard, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', [23]) assumption that diffraction emerges as a wave outgoing from a point source located at the diffraction point yj′ = yj−ne (we are employing the notation of Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' This motivates the choice of the center ξn = yj′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 7 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Pradovera, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Nonino, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Perugia Geometry-based approximation of waves in complex domains ∂Ω γ′ γ ξ α φ π − θ θ 3π − 2α − θ ∂Ω γ′ γ ξ α θ − π 3π − 2α − θ θ Figure 4: Diagrams for the two cases of scattering for concave corners (0 < α < π): without (left plot) and with shadow zone (right plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The dashed lines are reflection boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The dash-dotted line is a shadow boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Shadow regions are absent if and only if π − α ≤ θ ≤ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The angular coordinate 0 < φ < 2π − α is measured starting from one of the two adjacent edges of ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' ∂Ω γ′ γ ξ α φ π + θ − α θ Figure 5: Diagrams for the scattering at convex corners (π < α < 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The source point ξ is virtual, being used to model reflection off of edge γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The dash-dotted line is the shadow boundary due to edge γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The shadow region is present if and only if α − π < θ < π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The angular coordinate 0 < φ < 2π − α is measured starting from one of the two adjacent edges of ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Light cone Ωn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Since the diffracted wave propagates in all geometrically allowed directions, we define the support Ωn as the set of all points that are visible (along straight-line paths) from ξn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', Ωn = {x ∈ Ω : ξn + τ(x − ξn) ∈ Ω ∀0 < τ ≤ 1} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' (15) Modeling diffraction is substantially more complicated than modeling reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' For this reason, before we can describe how the remaining unknown quantities rn and ζn are defined, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' (5), we need to introduce several assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1 (Separability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Diffracted waves are separable into radial-temporal and angular compo- nents around the diffraction point ξn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Otherwise stated, �un(x, t) can be expressed (at least locally) as ψn(∥x − ξn∥ , t)ζn(x − ξn), where ζn is positive-homogeneous, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', ζn(z) is independent of ∥z∥ (as long as z ̸= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Using an abuse of notation, we will express ζn as a function of φ only, with φ defined as the angular coordinate around ξn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' See Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' This, together with the following assumption on the angular component ζn, will allow us to recover the approximation structure presented in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='2 (Piecewise-linear angular component).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The angular component ζn is a piecewise-linear function of the angular coordinate φ, with discontinuities at all reflection and shadow boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Using the geometrical optics approximation, we can explicitly compute the locations of such discontinuities: at concave corners (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 4), φ1 = |π − θ| = max{π−θ, θ−π} and φ2 = 2π−α−|π − α − θ| = min {θ + π, 3π − 2α − θ};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' at convex corners (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 5), φ3 = π + θ − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Now we are ready to describe our full diffraction model, which satisfies Assumptions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='2, as well as the following three standard requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 8 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Pradovera, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Nonino, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Perugia Geometry-based approximation of waves in complex domains Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='3 (Characterization of diffracting vertices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' A vertex ξn emits a diffraction wave in “re- sponse” to �ui only if both following conditions are met: ξn is visible from ξi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', ξn ∈ Ωi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' one of the following is true: – the domain Ω is locally concave near ξn, with ξi being located on the “concave side” of ξn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', 0 < α < π and 0 ≤ θ ≤ 2π − α in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' or – the domain Ω is locally convex near ξn and a “shadow zone” is present, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', π < α < 2π and π − α < θ < π in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='4 (Continuity of the full approximation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The full wave approximation �u is continuous, in particular across reflection and shadow boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='5 (Conservation of mass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Diffracted waves have zero “net mass”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', � R2 �un(x, t)dx = 0, leading to mass conservation of the full wave approximation �u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' (Note that we are stating mass conservation in free space to ignore further reflections and diffractions of �un, which are also assumed to conserve mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=') Spatial correction rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' As in Algorithm 1, let i be the index of the term �ui that causes the diffraction �un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' With the objective of satisfying (5) and Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='4, we define the radial component ψn as ψn(∥x − ξn∥ , t) = Ψ(∥x − ξn∥ + ∥ξn − ξi∥ + ri � �� � =:rn , t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' (16) By direct inspection of this definition, we can see that, by our choice of rn, we are “aligning” the wavefronts of the diffracted waves with the wavefronts of the reflected wave at the reflection boundaries (the shadow boundary of the reflected waves, if any) and with the wavefronts of the incoming wave �ui at its shadow boundary (if any).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' For instance, it is easy to see that, using (16), a point close to the diffraction point (x ≈ ξn) is within the support of the diffracted wave �un only for t ≥ rn − R, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', only when the wave �ui has crossed the distance from ξi to ξn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Weight function ζn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' According to Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='2, we define the discontinuous piecewise-linear function ζn : [0, 2π − α] → R as ζn(φ) = � � � � � z1(φ1−φ)+z2φ φ1 for 0 < φ < φ1 := |π − θ| , z3(φ2−φ)+z4(φ−φ1) φ2−φ1 for φ1 < φ < φ2 := 2π − α − |π − α − θ| , z5(2π−α−φ)+z6(φ−φ2) 2π−α−φ2 for φ2 < φ < 2π − α, (17) for concave corners, and ζn(φ) = � z1(φ3−φ)+z7φ φ3 for 0 < φ < φ3 := π + θ − α, z8(2π−α−φ)+z6(φ−φ3) 2π−α−φ3 for φ3 < φ < 2π − α, (18) for convex corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The scalars z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' , z8 are nodal values of ζn: ζn(0) = z1, ζn(φ+ 1 ) = z3 for concave corners, ζn(φ− 3 ) = z7 for convex corners, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' These values are chosen so as to satisfy: The boundary conditions at the edges ending at ξn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', γ and γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='4 at the discontinuity angles φ1, φ2, and φ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' To this aim, we prescribe values for the jumps (z3 − z2), (z5 − z4), and (z8 − z7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Given the radial-angular decomposition of �un from Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1, this is equivalent to the condition � 2π−α 0 ζn(φ)dφ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 9 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Pradovera, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Nonino, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Perugia Geometry-based approximation of waves in complex domains + = + = Figure 6: Examples of diffraction at the concave (top) and convex (bottom) corners from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 4 (right) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Neumann conditions are imposed on all edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In each row of plots, we have: discontinuous wave without diffraction (left), diffraction wave (middle), and continuous wave with diffraction (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Note that, in the convex case, we are not showing the wave �ui that causes the reflection off edge γ nor the reflection and scattering of �ui off edge γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In the case of a convex corner, this set of condition uniquely identifies the four degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' See Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1 for the formulas and for their derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' However, in the case concave case, an additional condition is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In this work, we set this last condition as described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 6 the results obtained with our diffraction modeling in two simple illustrative cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Before proceeding further, we deem it important to make the following remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Our proposed strategy is able to deliver only a fairly crude approximation of diffraction effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' (We refer to Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1 for a validation of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=') However, it has the great advantage of being extremely simple to build and to evaluate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Thanks to the modularity of our approach, it would be surely possible to replace our diffraction model with more sophisticated ones (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', removing Assumptions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='2), in the interest of achieving a better approximation of the exact solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' To this aim, we mention that a wide body of works has been dedicated to the modeling of diffraction in the time-harmonic (Helmholtz) setting: among others, we name the geometrical [20] and uniform [23] theories of diffraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' However, the authors have not been able to find any satisfactory all-purpose time-domain diffraction modeling in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1 Convex diffraction coefficients Consider the situation depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 5 and the notation introduced therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Also, we rely on the quantities ξn, rn, Ωn, and i introduced in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' For diffraction to happen, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='3, �ui must be a wave reflected off either edge γ or γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Indeed, a convex vertex ξn cannot be “hit” from outside the domain Ω by the source wave �u1, nor by any wave reflected off a different edge, nor by any diffracted wave centered at some vertex of ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' For this reason, the shadow boundary {φ = φ3} must belong to ∂Ωi (the boundary of the light cone Ωi), at least locally around ξn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Without loss of generality, we assume that �ui is a wave reflected off edge γ, so that Ωi consists (locally) of point whose angular coordinate is 0 < φ < φ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' This means that φ3 < φ < 2π −α is a shadow zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The alternative case (of reflection off edge γ′) can be obtained by symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Let σγ = 0 if γ is a Dirichlet edge and σγ = 1 if it is a Neumann edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Define σγ′ similarly for edge γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' To satisfy the conditions described in Section 4, the quantities z1, z7, z8, z6 appearing in the angular weight ζn, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' (18), must satisfy the conditions: Boundary condition at γ: z1 = σγz7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Boundary condition at γ′: z6 = σγ′z8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' For �u to be continuous at φ = φ3, there must be a jump to account for the fact that �ui is nonzero for φ → φ− 3 but zero for φ → φ+ 3 : given the angular component of �ui at the shadow boundary, namely, 10 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Pradovera, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Nonino, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Perugia Geometry-based approximation of waves in complex domains hi := ζi(ξn − ξi), we impose z8 − z7 = hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Conservation of mass: for all t > 0, 0 = � R2 �un(x, t)dx = � ∞ 0 � 2π−α 0 ψn(ρ, t)ζn(φ)ρdφdρ = �� 2π−α 0 ζn(φ)dφ � �� ∞ 0 ψn(ρ, t)ρdρ � = �z1 + z7 2 φ3 + z8 + z6 2 (2π − α − φ3) � �� ∞ 0 ψn(ρ, t)ρdρ � , which leads to the condition z1+z7 2 φ3 + z8+z6 2 (2π − α − φ3) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' With simple algebra, we now obtain z7 = (σγ′ + 1)hi(φ3 + α − 2π) (σγ′ + 1)(2π − α) + (σγ − σγ′)φ3 , z8 = (σγ + 1)hiφ3 (σγ′ + 1)(2π − α) + (σγ − σγ′)φ3 , (19) as well as z1 = σγz7 and z6 = σγ′z8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 6 (bottom) for an example of the resulting diffraction wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='2 Concave diffraction coefficients Consider the setup depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 4 and the notation introduced therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Also, we rely on the quantities ξn, rn, Ωn, and i introduced in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Let σγ = 0 if γ is a Dirichlet edge and σγ = 1 if it is a Neumann edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Define σγ′ similarly for edge γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Without loss of generality, we assume that 0 < θ ≤ π − α 2 , since the other case can be easily obtained by symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In this setting, γ is in the light cone of �ui (at least locally around ξn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Accordingly, let �ui′ be the wave component obtained by reflection of �ui off edge γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' If 0 < θ ≤ π − α, γ′ is not in the light cone of �ui, so that: for 0 < φ < φ1 = π − θ, both �ui and �ui′ are present;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' for φ1 < φ < φ2 = π + θ, only �ui is present, since the light cone Ωi′ ends at {φ = φ1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' for φ2 < φ < 2π − α, neither �ui nor �ui′ is present, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', we have a shadow zone, since the light cone Ωi ends at {φ = φ2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Otherwise, assume that π − α < θ ≤ π − α 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In this case, γ′ is also in the light cone of �ui (at least locally around ξn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We denote the wave component obtained by reflection of �ui off edge γ′ by �ui′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Then: for 0 < φ < φ1 = π − θ, both �ui and �ui′ are present;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' for φ1 < φ < φ2 = 3π − 2α − θ, only �ui is present, since the light cones Ωi′ and Ωi′′ end at {φ = φ1} and at {φ = φ2}, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' for φ2 < φ < 2π − α, both �ui and �ui′′ are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' To satisfy the conditions described in Section 4, the quantities z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' , z6 appearing in the angular weight ζn, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' (17), must satisfy the conditions: Boundary condition at γ: z1 = σγz2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Boundary condition at γ′: z6 = σγ′z5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' For �u to be continuous at φ = φ1, there must be a jump whose height is the angular component of �ui′ at φ = φ1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', hi′ := ζi′(ξn −ξi′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' we impose z3 −z2 = hi′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Note that, by the law of reflection, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' (11), hi′ = τhi, with τ = 2σγ − 1 and hi := ζi(ξn − ξi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' For �u to be continuous at φ = φ2, there must be a jump whose height depends on whether θ ≤ π − α or not: 11 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Pradovera, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Nonino, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Perugia Geometry-based approximation of waves in complex domains – If 0 < θ ≤ π − α, the jump equals the angular component of �ui at φ = φ2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', hi := ζi(ξn − ξi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' we impose z5 − z4 = hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' – If π − α < θ ≤ π − α 2 , the jump equals minus the angular component of �ui′′ at φ = φ2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', hi′′ := ζi′′(ξn − ξi′′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' we impose z5 − z4 = −hi′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Note that, by the law of reflection, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' (11), hi′′ = (2σγ′ − 1)hi, with hi := ζi(ξn − ξi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In summary, z5 − z4 = τ ′hi, with τ ′ = 1 if 0 < θ ≤ π − α and τ ′ = 1 − 2σγ′ if π − α < θ ≤ π − α 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Conservation of mass: for all t > 0, 0 = � R2 �un(x, t)dx = � ∞ 0 � 2π−α 0 ψn(ρ, t)ζn(φ)ρdφdρ = �� 2π−α 0 ζn(φ)dφ � �� ∞ 0 ψn(ρ, t)ρdρ � � �� � =:C(t) = �z1 + z2 2 φ1 + z3 + z4 2 (φ2 − φ1) + z5 + z6 2 (2π − α − φ2) � C(t), which leads to the condition z1+z2 2 φ1 + z3+z4 2 (φ2 − φ1) + z5+z6 2 (2π − α − φ2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' One constraint is missing for the values z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' , z6 to be uniquely determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Specifically, some simple algebra shows that, for any δ ∈ R, the following set of values satisfies all the above conditions: z2 = τhi + δ (σγφ1 + φ2)/(φ1 − φ2), z5 = τ ′hi + δ (φ1 + σγ′φ2 − (σγ′ + 1)(2π − α))/(φ1 − φ2), (20) together with z1 = σγz2, z3 = z2 + τhi, z4 = z5 − τ ′hi, and z6 = σγ′z5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Note that the jump heights τhi and τ ′hi appear in the numerators of z2 and z5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Our diffraction model is simply the one given by δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Intuitively, this corresponds to a “balanced” partitioning of the mass of the diffracted wave into the components related to the two (reflection and/or shadow) boundaries φ1 and φ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 6 (top) for an example of the resulting diffraction wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' To further highlight the (mostly beneficial) effects of the choice δ = 0, we note that: in the symmetric case θ = π − α 2 , δ = 0 leads to a symmetric ζn: (1 − 2σγ′)z2 = (1 − 2σγ)z5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' in the case θ = π − α, the second transition happens at γ′, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', φ2 = 2π − α, and the choice δ = 0 yields z4 = 0, so that the diffraction wave vanishes at γ′ (which is physically sound);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' in the “grazing incidence” case θ = 0, the two transition coalesce into one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', φ1 = φ2, and the choice δ = 0 leads to ζn(φ) = 0 for all φ, which is unphysical;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' see the following remark for a possible solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In the “grazing incidence” case θ = 0 (which corresponds to φ1 = φ2 = π), the diffraction wave �un does not cure the discontinuity of �u at the boundary {φ = π}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' This is because, in some sense, the two discontinuities of �un at φ1 and φ2 cancel each other out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' For a similar reason, a small θ ≈ 0 will result in a continuous total wave, but a sharp gradient will be present for φ1 < φ < φ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' By tweaking the value of δ, we can obtain an alternative diffraction model, which guarantees Assump- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='4 even in the case of grazing incidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' To this aim, we can set δ = � (σγ + 1) σγφ1 + φ2 − (σγ′ + 1)(2π − α) (σγ − σγ′)φ1 + (σγ′ + 1)(2π − α)φ1τ+ +(σγ′ + 1) φ1 + σγφ2 (σγ − σγ′)φ2 + (σγ′ + 1)(2π − α)(φ2 − 2π + α)τ ′ � hi φ1 − φ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' (21) Roughly speaking, this corresponds to a different “balancing” of the mass of the diffracted wave into the components related to the two boundaries φ1 and φ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' This being said, in our numerical tests, such alternative model, albeit recovering a continuous total wave, resulted in a reduced accuracy of approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Specifically, using the value of δ above, we have observed an exaggerated magnitude of the diffraction wave, especially in the shadow zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 12 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Pradovera, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Nonino, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Perugia Geometry-based approximation of waves in complex domains Example exterior incidence ∥�u(·, T)∥L2(Ω) ∥�u(·, T) − uFEM(·, T)∥L2(Ω) index angle α angle ω #1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='712 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='984 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='50 · 10−1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='65 · 10−4 #2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='093 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='603 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='02 · 10−3 #3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='761 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='363 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='05 · 10−2 #4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='761 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='277 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='00 · 10−2 Table 1: Setup for the four wedge examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The angle ω is as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' ω ω ω ω −1 0 1 Figure 7: Initial conditions for the wedge examples, indexed #1 through #4 from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The (dashed) distance between the center of the Gaussian and the boundary vertex is 4 units in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 5 Numerical results In our experiments, we require a “reference” solution of (1) to validate our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' To this effect, we use the solution uFEM obtained by discretizing (1) with: the P1-finite element method (P1-FEM) with mass-lumping, over a regular triangulation (mesh) of the physical domain Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' explicit leapfrog timestepping with a uniform time step that satisfies the CFL condition on the chosen mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' See [11, 16] for more details on this discretization strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' If the domain Ω is unbounded, we first need to truncate it in such a way that reflections from the non- physical truncation boundary do not affect the solution in the region of interest for t < T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Recalling that the problem data are supported in a ball of radius R and center 0, this can be done, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', by truncating Ω at the sphere with radius R + T and center 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In our tests, we rely on FEniCS [1] to carry out the P1-FEM discretization on 2-dimensional domains Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Instead, note that with our proposed approach, modeling unbounded domains is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Indeed, we can simply ignore any reflection or diffraction from its “infinitely far” vertices/edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' All our tests are performed in Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='8 on a machine with an 8-core 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='60 GHz Intel® processor and 64 GB of RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' For reproducibility, our code is made available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='com/pradovera/ ray-wave-2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1 Some simple wedges As a way to assess our proposed method in simple settings, we consider four different “wedge” domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Similarly to the diagrams in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 4 and 5, we define Ω to be one of the portions of the plane R2 delimited by straight lines intersecting at a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Locally around such point, Ω “looks” like either Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 4 or Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 5, with α being the outer angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The specific choices of wedge angles α are reported in Table 1 for the four cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We set up a problem of the form (1), with u0 an isotropic Gaussian with standard deviation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The center of u0 is at a point located at a 4-unit distance from the wedge vertex, in the direction determined by the “incidence angle” ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 7 for a representation of the initial conditions in the four cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We set u1 = f = 0, we enforce Neumann boundary conditions on the whole ∂Ω, and we seek the solution at the final time T = 5, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', 1 time unit after the wave crest has reached the wedge vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 13 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Pradovera, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Nonino, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Perugia Geometry-based approximation of waves in complex domains 0 1 2 3 4 5 0 2 4 6 t ρ −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='5 1 Figure 8: Free-space solution Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The dashed line denotes the upper bound of the “causality cone” of Ψ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', ρ = t + R, with R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' �u uFEM �u − uFEM Example #1 Example #2 Example #3 Example #4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1 −2 · 10−4 0 2 · 10−4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1 −5 · 10−3 0 5 · 10−3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1 −1 · 10−2 0 1 · 10−2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1 −5 · 10−2 0 5 · 10−2 Figure 9: Results for the four wedge examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Each row pertains to a different example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In each row, from left to right: surrogate solution, FE solution, and error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The color scales for the first two columns are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' All results are shown at the final time t = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 14 AD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Pradovera, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Nonino, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Perugia Geometry-based approximation of waves in complex domains To this aim, we employ our proposed approach, see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' First, we compute an approximation of the free-space solution Ψ, which solves (4), by employing P1-FEM with explicit leapfrog timestepping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Note that, since (4) is cast in polar coordinates, we only need to discretize a 1D interval with P1-FEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Since the initial condition u0 is supported within the unit disk, we have R = 1, and it suffices to approximate Ψ(ρ, t) for (ρ, t) ∈ [0, T + R] × [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Since this space-time domain is only 2-dimensional, we can afford even a very fine discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In our experiments, we employ a 1001 × 2001 uniform Cartesian space-time grid, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', the mesh size is δx = T +R 1000 and the time step is δt = T 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' This satisfies the CFL condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We show the resulting Ψ (which, in fact, we should denote by ΨFEM) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' After this preliminary step, we use the timetable-based strategy from Section 2 to identify reflection and scattering effects, which are then added up to give the final approximation �u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We show the resulting �u(·, T) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In this figure, we also display a reference solution uFEM(·, T), which we obtain by direct discretization of (1) by P1-FEM and leapfrog timestepping, as described at the beginning of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In all four examples, we see that �u and the reference uFEM seem qualitatively close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Notably, we can observe a good representation of the most prominent wavefronts, which are due to propagation of either the main “free-space” wave or to its reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Indeed, those wave contributions are reconstructed exactly: the only errors are the ones due to FE approximation and timestepping, which affect both uFEM and �u (the latter through the approximation of Ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Instead, some differences are present when comparing diffraction effects, which arise as circular waves about the wedge vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We can quantitatively observe this in the last column of both Table 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In example #1, we observe a very small error, which, in fact, is simply the (FEM and timestepping) discretization error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' This is related to the fact that the wedge has exterior angle α = 3 2π, which makes diffraction unnecessary in approximating the wave u: reflections are enough2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In the other examples, diffraction effects are necessary to correctly identify u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' While a good qualitative behavior can be observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 9, we can see in Table 1 that a modest error is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Specifically, we report the L2(Ω)-norm of �u and of the error �u − uFEM at the final time t = T, defined as ∥v∥L2(Ω) = �� Ω v(x)2dx �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' (22) We see the largest error in example #4, where the relative L2(Ω)-approximation error amounts to 8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' This was to be expected, since this last example is rather close to the setting of grazing incidence (α+ω ≈ 2π), which, as discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='2, is approximated rather poorly by our diffraction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Qualitatively, the bad approximation quality is apparent in the form of a rather sharp gradient in the corresponding plot of �u in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 9 (bottom left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1 Building a cavity out of wedges As a slightly more complicated example, we now combine the four wedges from the previous section to obtain the open cavity represented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In this case, more reflection and diffraction effects will arise, due to the trapping nature of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Our initial conditions and forcing term are the same as before, but now all edges are sound-soft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Accordingly, we model them using Dirichlet boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The time horizon is T = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Using our strategy from Section 2, we build the approximation �u, which contains 47 wave terms (1 source wave, 32 reflected waves, and 14 diffraction waves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We compare the approximation �u with the reference solution uFEM, obtained as described at the beginning of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We show the results of the comparison in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 10, at 4 time instants t ∈ {0, 3, 6, 9}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Once more, we see a good qualitative agreement between �u and uFEM, with the most important features of u being identified well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Up to t = 3, only reflections have happened, so that the error �u − uFEM consists only of FEM and timestepping errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' On the other hand, for larger times, we see “error waves” of small-to-moderate amplitude propagating from the 3 vertices of Ω that generate diffraction effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' These correspond to errors in diffraction modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 2To intuitively understand why, let γ1 and γ2 be the two sides forming ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The domain Ω is partitioned exactly into (i) the light cone of the reflection off γ1 and then off γ2 and (ii) the light cone of the reflection off γ2 and then off γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' For this reason, the diffraction effects due to these two rays cancel out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Incidentally, the same phenomenon can be expected whenever the interior angle 2π − α is of the form π n , with n ∈ {2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 15 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Pradovera, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Nonino, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Perugia Geometry-based approximation of waves in complex domains �u uFEM �u − uFEM t = 0 t = 3 t = 6 t = 9 −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='5 1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1 −5 0 5 10−4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1 −5 0 5 10−2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1 −5 0 5 10−2 Figure 10: Results for the cavity domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Each row corresponds to a different time instant t ∈ {0, 3, 6, 9}, from top to bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In each row, from left to right: surrogate solution �u(·, t), FE solution uFEM(·, t), and error �u(·, t) − uFEM(·, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The color scales for the first two columns are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 16 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Pradovera, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Nonino, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Perugia Geometry-based approximation of waves in complex domains 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='2 A tall room We consider a simplified sound propagation problem in a room.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' For simplicity, we consider a 2-dimensional problem, thus assuming an infinitely tall room, and modeling line sources (in the z-direction) as point sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The complicated domain Ω ⊂ R2 is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' It is composed of two communicating “rooms” with sound-hard walls, as well as of a third large room (above), which is modeled as infinitely large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In the main room, three sound-soft triangular obstacles are also present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Setting once more u1 = f = 0, we are interested in modeling the propagation of an initial condition u0 modeled as a Ricker wavelet centered at 0, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 11 (top left), over the time horizon t ∈ [0, T], with T = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' To this aim, we employ our proposed method from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' As in the previous example, we start by computing an approximation of the free-space solution Ψ = Ψ(ρ, t) for (ρ, t) ∈ [0, T + R] × [0, T], see (4), with R being the radius of the support of the initial condition u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Again, we use P1-FEM with leapfrog timestepping for this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Since many reflective surfaces face each other, the domain Ω is trapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Accordingly, we expect the number N of waves in the approximation �u to be rather large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In the interest of reducing the number of such terms, we can employ the on-the-fly parsimonious strategy described in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='2, removing all wave terms �un whose magnitude is smaller than tol = 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' After this, N ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='4 · 103 terms are left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Although this value of N may seem large, the evaluation of the corresponding surrogate �u is rather quick, due to the explicit nature of each wave contribution (and to the fact that their supports are smaller than the whole Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We show the resulting u(·, t) for the four times t ∈ {0, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='5, 15, 20} in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' There, we can see why so many terms are necessary for the approximation of u: we must model many reflection and diffraction effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Since energy escapes the system only through the top “door”, the wave will persist for quite a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Accordingly, a larger T will make a larger N necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In order to better inspect this effect, we show the trace of the solution at the arbitrarily chosen point xtrace = (−1, −2) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We notice that oscillations persist for t > 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We use this last plot also to validate our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' To this aim, we compare three results: The surrogate �u obtained as described above, with tol = 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The surrogate �u obtained with our strategy, but with tol = 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' This leads to an increased number of rays N ≈ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='3 · 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The reference solution uFEM obtained by the P1-FEM with leapfrog timestepping, as described at the beginning of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The mesh size must be chosen small enough to resolve both the initial condition and the domain Ω well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In our case, we have a mesh with approximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='4 · 106 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' To satisfy the CFL condition on this mesh, we choose a time step ∆t ≈ 7 · 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We can observe that the two surrogates obtained with our approach give very similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Indeed, the cutoff tolerance tol affects the results only for large t > 15, due to the accumulation of “small” waves that are excluded from the coarser surrogate but included in the finer one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Moreover, taking the FE solution as reference, we see that most of the peaks of the surrogates are aligned with the FE ones (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', the “phase” of the wave is well approximated), but there are some noticeable discrepancies in their amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' This is due to the fact that, in our approach, reflection is modeled exactly, whereas the magnitudes of the diffraction waves are only roughly approximated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' For this reason, we should not expect the amplitude error to get smaller if we reduce tol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The only “real” way of improving the approximation is using a better diffraction modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' As a final result, we also report: The so-called “offline” time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', the time required to compute the numerical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' For �u, this means executing Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' For uFEM, this means building the mesh, assembling the FE stiffness and (lumped) mass matrices, and carrying out the timestepping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The so-called “online” time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', the time required to evaluate the numerical solution (�u or uFEM) at a single (x, t)-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' They can be found in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 17 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Pradovera, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Nonino, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Perugia Geometry-based approximation of waves in complex domains −1 0 1 t = 0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1 t = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1 t = 15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1 t = 20 Figure 11: 2-dimensional domain Ω modeling a room.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Top left plot: initial condition �u(·, 0) = u(·, 0) = u0, a Ricker wavelet;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' we also show the point xtrace as a cross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Top right plot: intermediate solution �u(·, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Bottom left plot: intermediate solution �u(·, 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Bottom right plot: final solution �u(·, 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 0 2 4 6 8 10 12 14 16 18 20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1 t u(xtrace, t) �u (tol = 10−2) �u (tol = 10−3) uFEM Figure 12: Value of solution at point xtrace = (−1, −2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Method �u (tol = 10−2) �u (tol = 10−3) uFEM Offline 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='66 [s] 252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='9 [s] 188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='9 [s] Online 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='04 [ms] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='48 [ms] 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='67 [µs] Table 2: Timings for the room test case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' To obtain more statistically significant results, each displayed time is the average over 3 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 103) runs of the offline (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' online) phase with identical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 18 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Pradovera, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Nonino, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Perugia Geometry-based approximation of waves in complex domains We can observe the increased training and evaluation time that results from decreasing tol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Moreover, we see that, in this example, our proposed approach is more competitive offline, but less so online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Somewhat surprisingly, we have observed that the most expensive step in evaluating �u (taking about half of the online time) is determining whether an evaluation point is in the light cones Ωn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The reason for this is that they can have rather complicated shapes, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Evaluating the FE solution at a space-time point is an extremely cheap operation (essentially corre- sponding to a vector dot product) while evaluating �u is more expensive, requiring the evaluation of O(N) nonlinear functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' However, the FE solution comes with the serious drawback of memory usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Indeed, in our example, storing uFEM as a (∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='5 · 106) × (∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='9 · 103) array of double-precision floating-point numbers requires approximately 34 GB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Concerning the timing results, we also wish to mention that the online times in Table 2 should be interpreted carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Indeed, the online time for the FE solution is artificially deflated by the fact that xtrace is a vertex of the FE mesh: each evaluation of uFEM corresponds to extracting a vector entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' If xtrace had not been a vertex of the mesh, the online time could have been larger by at least one order of magnitude, if not more, depending on the FE implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Moreover, we note that accessing point-evaluations of uFEM at arbitrary times a posteriori, namely, after the timestepping has been carried out, is feasible only if enough memory is available to store the whole “timestepping history”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Considering the numbers mentioned in the previous paragraph, this might not be possible in practice, especially for more complex and/or larger domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1 A time-harmonic source One of the advantages of our approach is that it allows changing the source terms of the problem in a seamless way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Notably, under minor technical constraints (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', the support of the new source term should not be larger than the old one), this kind of change does not require training a new surrogate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' To showcase this, we approximate the wave propagating from a time-harmonic point source at x = 0 with angular frequency ω > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In our tests, we pick ω ∈ {2π, 10π}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' To this aim, we define u as the solution of the following (ω-dependent) problem: � � � � � � � � � ∂ttu(x, t) = ∆u(x, t) − ω2 sin(ωt)δ0(x) for (x, t) ∈ Ω × (0, T), u(x, 0) = 0 for x ∈ Ω, ∂tu(x, 0) = 0 for x ∈ Ω, ∂νu(x, t) = 0 for (x, t) ∈ ∂Ω × (0, T], (23) where δ0 denotes the usual 2-dimensional “delta function” centered at x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' As usual, we define Ψ = Ψ(ρ, t) as the (ω-dependent) solution of the free-space version of (23) in radial- temporal coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Note that, in free space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', without boundary effects3, the forcing term in (23) is equivalent to the following non-homogeneous Dirichlet-like condition at ρ = 0: U(0, t) = Ψ(0, t) = � t 0 � t′ 0 −ω2 sin(ωt′′)dt′′dt′ = sin(ωt) ∀t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' (24) Accordingly, the free-space solution Ψ has space-time support {(ρ, t) ∈ [0, ∞)2, ρ ≤ t}, which is a subset of the free-space solution Ψ from the previous section, namely, {(ρ, t) ∈ [0, ∞)2, ρ ≤ t + R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' As such, to obtain an approximation for the wave u generated by the time-harmonic source for an arbitrary ω, it suffices to plug the corresponding Ψ in each term of the surrogate �u from the previous section!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We show the results of our approximation in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 13 and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We note that, if we had chosen to apply the FEM to approximate the wave u generated by the time- harmonic source, we would have been forced to carry out a new simulation from scratch for every frequency to be studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' To this end, we would have needed to choose a mesh with ω-dependent resolution: the mesh size should be small enough for the well-known pollution effect (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', [22, 2]) to be absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 3If Ω is bounded, reflected or diffracted waves will generally bounce back to x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' As such, the value of u(0, t), u being the solution of (23), will be different from the source signal sin(ωt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' For this reason, we cannot turn the forcing term in (23) into a condition like (24), except in free space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 19 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Pradovera, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Nonino, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Perugia Geometry-based approximation of waves in complex domains −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='2 Figure 13: Surrogate solution found with the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Left plot: ω = 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Right plot: ω = 10π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 0 2 4 6 8 10 12 14 16 18 20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='2 t u(xtrace, t) ω = 2π ω = 10π Figure 14: Value of solution at point xtrace = (−1, −2) for different excitation frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In our proposed approach, we also have a constraint on the mesh resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' However, it only applies to the problem defining the free-space solution Ψ, which is 1-dimensional in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Hence, having to refine the mesh represents a much smaller obstacle to efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In particular, for a fixed time horizon T, the computation of �u becomes more and more efficient, when compared to the computation of uFEM, as the frequency ω increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' 6 Conclusions We have presented a method for approximating waves propagating through complex 2-dimensional domains with polygonal boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Our method relies on the automatic identification of reflection and diffraction effects caused by the domain geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Each effect is modeled through a relatively simple nonlinear expres- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In our numerical tests, we have observed rather a good approximation quality, with the main features of the target wave being well identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' As a way to improve the approximation accuracy, we recall that any diffraction model could replace the current one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In terms of complexity, our method requires the solution of a simplified 1D-in-space problem, much simpler than the original 2D-in-space one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We expect such improved accuracy to increase even further if one were to consider 3D instead of 2D problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' However, in order to generalize our method to 3 space dimensions, a suitable diffraction model would be necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' This is one of our ongoing research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Another favorable aspect of our algorithm is its potential to be run on parallel architectures, since the computation of different rays can be carried out independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' This is not the case for standard timestepping-based discretizations, due to their intrinsically sequential nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Other envisioned extensions of our technique involve the cases of domains with curvilinear boundaries and of propagation media with non-uniform properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', density and refraction index).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Specifically, this latter case would effectively result in a non-uniform wave speed, with reflections and diffractions happening 20 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Pradovera, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Nonino, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Perugia Geometry-based approximation of waves in complex domains also within the domain Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Finally, we recall that, in many applications, the ultimate target is understanding how the wave u solving (1) depends on underlying parameters µ, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=', the forcing term f, the shape of the domain Ω, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In this setting, MOR methods try to construct a surrogate model of the form �u = �u(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' µ), providing a good approximation of u over a whole range of parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Even though our technique was presented here in the non-parametric setting, we believe that it potentially allows incorporating the parameter dependence in a natural and efficient way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' In our opinion, this might be achievable by leveraging the simple and in- terpretable structure of the rays (free-space solution, light cone, and angular modulation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' As a simple preliminary example, we showcased this in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='1 for a parametric source term, with the parame- ter being the frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' We are currently investigating how to extend our method to more complicated parametric problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' References [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Alnæs, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Blechta, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Hake, and Others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' The FEniCS Project version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFRT4oBgHgl3EQfpTfl/content/2301.13613v1.pdf'} +page_content=' Archive of Numerical Software, 3(100), 2015.' metadata={'source': 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chaos and dielectric microresonators +L. Ponzellini Marinelli[1,2] +luciano@fceia.unr.edu.ar +L. Raviola[1] +raviola@fceia.unr.edu.ar +[1] Faculty of Exact Sciences, Engineering and Surveying, +National University of Rosario, Rosario, Argentina. +[2] Rosario Physics Institute, UNR-CONICET, +Rosario, Argentina. +January 3, 2023 +Abstract +Most problems in electrodynamics do not have an analytical so- +lution so much effort has been put in the development of numerical +schemes, such as the finite-difference method, volume element meth- +ods, boundary element methods, and related methods based on bound- +ary integral equations. +In this paper we introduce a local integral +boundary domain method with a stable calculation based on Radial +Basis Functions (RBF) approximations, in the context of wave chaos +in acoustics and dielectric microresonator problems. RBFs have been +gaining popularity recently for solving partial differential equations +numerically, becoming an extremely effective tool for interpolation on +scattered node sets in several dimensions with high-order accuracy +and flexibility for nontrivial geometries. One key issue with infinitely +1 +arXiv:2301.00069v1 [math.NA] 30 Dec 2022 + +smooth RBFs is the choice of a suitable value for the shape param- +eter which controls the flatness of the function. It is observed that +best accuracy is often achieved when the shape parameter tends to +zero. However, the system of discrete equations obtained from the +interpolation matrices becomes ill-conditioned, which imposes severe +limits to the attainable accuracy. A few numerical algorithms have +been presented that are able to stably compute an interpolant, even +in the increasingly flat basis function limit. We present the recently +developed Stabilized Local Boundary Domain Integral Method in the +context of boundary integral methods that improves the solution of +the Helmholtz equation with RBFs. Numerical results for small shape +parameters that stabilize the error are shown. Accuracy and compar- +ison with other methods are also discussed for various case studies. +Applications in wave chaos, acoustics and dielectric microresonators +are discussed to showcase the virtues of the method, which is com- +putationally efficient and well suited to the kind of geometries with +arbitrary shape domains. +1 +Introduction and motivation +Dielectric microresonators, also known as dielectric microcavities, have at- +tracted interest in the last decades due to technological applications like +microlasers and and as systems with intrinsic theoretical interest for its con- +nections with quantum billiards and wave chaos [2, 9, 20]. +A quantum billiard is a system in which a free particle is confined within +a 2D domain and whose dynamics is governed by the Schr¨odinger equation +iψt(x, t) = −∆ψ(x, t), +x ∈ Ω ⊂ R2, t > 0. +(1) +where ψ(x, t) = 0 for x ∈ Γ being Γ the boundary of the domain Ω. +When searching the time harmonic solutions of this system in the form +ψ(x, t) = ˜ψ(x)eikt, the spatial dependence, ˜ψ(x), satisfies the well-known +Helmholtz Equation (HE) +� +∆ + k2� ˜ψ(x) = 0, +x ∈ Ω ⊂ R2, t > 0. +(2) +In this case, the eigenvalues to equation (2) are related to the energy of the +particle. +On the other hand, a similar situation arises when trying to solve the +problem of light waves propagating inside a dielectric medium satisfying the +2 + +Maxwell equations. Also in this case, the search for time harmonic solutions +leads to the Helmholtz equation for the spatial dependence of the electro- +magnetic field [2]. +For generic domains, the equation (2) cannot be solved analytically to find +stationary states. So we must resort to finding efficient and reliable numerical +methods to solve this equation. There are many numerical techniques to +address this equation such as the finite element method (FEM), the finite +volume method (FVM), the Boundary Element Method (BEM) or spectral +methods (PS) [19]. However, several of these require the construction of a +specific mesh or refinement to efficiently address certain numerical problems +on non-trivial geometries. +The BEM transforms the formulated Partial Differential Equations (PDE) +into integral equations, that is, into an integral form over the boundary +[1, 13]. In BEM the PDE that describes the physical problem is transformed +into a Boundary Integral Equation (BIE), which is achieved by using Green’s +identities to then apply this integral formulation over points distributed in +the domain. Many local integral methods are based on an integral formula- +tion on small, strongly overlapping stencils with local interpolations. +In recent decades, methods involving the Radial Basis Functions (RBF) +have become an extremely effective tool in non-trivial geometries for inter- +polation in sets of scattered nodes and for numerically approximating PDE. +There are many modern books dealing with theory, implementations and ap- +plications [3, 4, 6]. One advantage is that when the distribution nodes are +created, it is possible to achieve local refinement in critical areas depending +on the specific problem [5]. Particularly, this is interesting to resolve local- +ized structures like the scarred states observed in quantum chaos phenomena +[18]. +Using infinitely differential RBFs like Gaussians, exponential convergence +can be shown. A practical obstacle is the ill-conditioning of the interpolation +matrix when the shape parameter ε that defines the Gaussian RBF tends +to zero. It is known that when this parameter is reduced, the interpolation +accuracy of the method improves considerably but the numerical conditioning +of the problem worsens if it is solved with a direct type numerical method. +That is, there is a conflict between accuracy and the constraint known as the +uncertainty principle [17]. +In this paper we present the Stabilized Localized Boundary-Domain Inte- +gral Method (SLBDIM) [16] in the context of Helmholtz type equations. This +is a new stable integral local numerical method for approximating elliptic- +3 + +type PDE solutions to solve Boundary Value Problems (BVP) in 2D that +uses local interpolations with RBF for low values ε > 0. This technique is +a combination of meshless methods, local integral formulations and bound- +ary elements in multi-domains independent of a structured mesh and that +only requires an unstructured distribution of nodes of the domain Ω and its +boundary Γ = ∂Ω that allows to deal with complex geometries. For local +interpolations, the Gaussian RBFs ϕ(r) = e−(εr)2 are used when ε → 0 in +local interpolations in stable form. +Numerical results are shown for a small shape parameter that stabilizes +the error. Comparisons with other methods in several cases are also dis- +cussed. It is shown that the method is computationally efficient and suit- +able for geometries that come from applications of wave chaos and dielectric +microresonators. In particular, we solve differential problems with Dirichlet- +type boundary conditions over square domains with quasi-uniform point dis- +tributions. +2 +The Stabilized Localized Boundary Domain +Integral Method for Helmholtz equations +2.1 +Problem description and local integral method +We consider the following Boundary Value Problem (BVP) on an open, +bounded and simply connected domain Ω ⊂ R2 +(BV P) +� L [u] (x) = f(x), +x ∈ Ω, +(3a) +B [u] (x) = g(x), +x ∈ Γ = ∂Ω, +(3b) +where L[ . ] = ∆ + λ is an elliptic differential Helmholtz-type operator, +∆ = +∂ +∂x2 + +∂ +∂y2 is tha Laplacian, λ ∈ R (when λ = k2 > 0, k is the wave- +number) and f(x) is the smooth source term. B[ . ] is the boundary operator +with the boundary conditions (BC). +The BC are Dirichlet, Neumann or mixed over Γ = Γ1∪Γ2 and Γ1∩Γ2 = ∅ +� +� +� +u(x) = g1(x), +x ∈ Γ1, +(4a) +∂u(x) +∂n += g2(x), +x ∈ Γ2, +(4b) +with g1 and g2 known data, and ∂u(x) +∂n +the outward normal derivative of the +unknown field u. +4 + +We propose that PDE (3a) can be written as +∆u (x) = f(x) − λu (x) = b (x, u (x)) , +(5) +where u (x) is the potential in the point x ∈ Ω. +We consider x ∈ Ω ⊂ R2 +∆u∗ = δ(x − ξ), +(6) +where δ(x − ξ) is Delta’s delta centered at ξ ∈ Ω with fundamental solution +u∗(x, ξ) = 1 +2πln(r), +r = ∥x − ξ∥. +(7) +From equation (5) +∆u (x) = b ⇔ +� +Ω +u∗ (x, ξ) ∆u (x) dΩx = +� +Ω +u∗(x, ξ)b dΩx. +(8) +Applying Green’s second identity for u that satisfies (5) and u∗ that +satisfies (6) +� +Ω +(u∗∆u − u∆u∗) dΩx = +� +Γ +� +u∗ ∂u +∂n − u∂u∗ +∂n +� +dΓx, +(9) +we obtain +u(ξ) = +� +Ω +u∗ (x, ξ) b dΩx − +� +Γ +� +u∗ (x, ξ) ∂u(x) +∂n +− u(x)u∗(x, ξ) +∂n +� +dΓx. +(10) +From equation (10) we have a formula for the integral representation of +the PDE over a subregion Ωi with boundary Γi. The interior collocation +point xi is obtained as before from the fundamental solution and Green’s +second identity +u(ξ) = +� +Γi +q∗ (x, ξ) u (x) dΓx − +� +Γi +u∗ (x, ξ) q (x) dΓx + +� +Ωi +b u∗ (x, ξ) dΩx, +(11) +where q = ∂u +∂n is the normal derivative of the unknown field, u∗ is the fun- +damental Laplacian solution and q∗ = +∂u∗ +∂n is the normal derivative of the +fundamental solution. +5 + +Using the well-known Green-Dirichlet function (FGD), G (x, ξ), and its +normal derivative Q (x, ξ) [8] in (11) we obtain a new integral formulation of +the form +u(ξ) = +� +Γi +Q (x, ξ) u (x) dΓx + +� +Ωi +b G (x, ξ) dΩx. +(12) +since the integral over Γi involving G in (11) vanishes since its value is zero. +In addition, if the non-homogeneous term b of the PDE can be split +b (x, u (x)) = f (x) − λu (x) , +(13) +where the funcion source f is data. +The integral representation (12) in each subregion of integration Ωi is +u(ξ) = +� +Γi +Q(x, ξ)u(x) dΓx+ +� +Ωi +G(x, ξ)f(x) dΩx+ +� +Ωi +−λu (x) G(x, ξ) dΩx, +(14) +where ξ is the interior source point. The collocation technique is done only +at interior points of the domain. +2.2 +Local interpolations with RBF +A function ϕ : Rd → R is an RBF if there exists φ : [0, ∞) → R such that +ϕ (x) = φ(r), +r = ∥x − xj∥, +(15) +where ∥.∥ is the Euclidean norm on Rd and depends on the distance to a +center xj ∈ Rd. If it depends on the shape parameter ε > 0, then ϕε +j (x) = +φ(r, ε) is often noted. +In the LBDIM the field u is locally interpolated with RBF {ϕj}n +j=1 with +centers of the stencil Θx = {xj}n +j=1 +u (x) ≈ +n +� +j=1 +αjϕj(x), +(16) +where the interpolation matrix Ai is such that +(Ai)jk = ϕk(xj) = φ(∥xj − xk∥), +j, k = 1, . . . , n +(17) +6 + +The term b of (13) is interpolated with RBF {χj}m +j=1 with centers of the +stencil Θy = {yj}m +j=1 +�b (u (x) , ∇u (x)) ≈ +m +� +j=1 +βjχj (x) , +(18) +where the interpolation matrix �Ai is such that +(�Ai)jk = χk(yj) = χ(∥yj − yk∥), j, k = 1, . . . , m +(19) +The RBFs are eventually of the same type and with the same centers. If +we take the same RBF bases with the same centers, the result is {ϕj}n +j=1 +and {χj}m +j=1 for m = n although they could be different depending on the +application problem or numerical experience. +The local integral formulation of (14) is of the form +u(ξ) +≈ +n +� +j=1 +αj +�� +Γi +Q(x, ξ)ϕj(x) dΓx +� ++ +m +� +j=1 +βj +�� +Ωi +G(x, ξ)χj (x) dΩx +� ++ +� +Ωi +G(x, ξ)f(x) dΩx. +(20) +If Θ = {x1, . . . , xN} is the discretization of domain Ω and ξ = xi ∈ Θ is +the collocation point, the discretized formulae of the unknown field is +ui = u (xi) = +n +� +j=1 +αj�hij + +m +� +j=1 +βj�gij + �fi, +(21) +where αj and βj come from equations (16) and (18). The coefficients �hij, �gij +and �fi are of the form +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +�hij = +� +Γi +Q (x, xi) ϕj (x) dΓx, +(22a) +�gij = +� +Ωi +G (x, xi) χj (x) dΩx, +(22b) +�fi = +� +Ωi +G (x, xi) f (x) dΩx, +(22c) +which are calculated by Gauss-Legendre quadratures. +7 + +Defining the vectors α = [α1, . . . , αn]T and β = [β1, . . . , βm]T as interpo- +lation coefficients, the discretized form (21) of u can be expressed as +ui = �hT +i α + �gT +i β + �fi, +(23) +where �hi = [�hi1, . . . ,�hin]T and �gi = [�gi1, . . . , �gim]T are the influence coeffi- +cients, and �fi ∈ R is data. +The vector α arises from the local system by interpolating with the RBF +basis {ϕj}n +j=1 +Aiα = di ⇔ α = A−1 +i di +(24) +and the vector β arises from the local system by interpolating with the RBF +basis {χj}m +j=1 +�Aiβ = �bi ⇔ β = �A−1 +i �bi = �A−1 +i +� +A�biα +� += �A−1 +i +� +A�biA−1 +i di +� +, +(25) +where A�bi is the calculation matrix of the vector �bi with known coefficients +(A�bi)jk = �b (ϕk (yj) , ∇ϕk (yj)) , +j = 1, . . . , m, k = 1, . . . , n. +(26) +Substituting (24) and (25) in the discretized form (23), we obtain the +discretized matrix form for ui in terms of di +ui = +� +�hT +i A−1 +i ++ �gT +i �A−1 +i A�biA−1 +i +� +di + ˜fi. +(27) +Rewriting (27) we obtain an algorithmic procedure to avoid the compu- +tation of inverses A−1 +i +and �A−1 +i +(see [14]) +ui = zTdi + �fi +donde zT = �hT +i A−1 +i ++ �gT +i �A−1 +i A�biA−1 +i +(28) +which are assembled into a global sparse-like system and numerically resolved +with Generalized Minimal Residual (GMRES). +2.3 +Stability with Gaussian RBFs +Convergence in global interpolations with ε-dependent RBFs can be studied +in a stationary way (n = cte. and ε → 0) or non-stationary (ε = cte. and +increasesn). In the case of Gaussian RBFs, they produce convergence of +order O(e +− const +(εh)2 ) (superspectral). +8 + +The RBF interpolation matrix is +A(ε) = +� +���� +φ(∥x1 − x1∥, ε) +φ(∥x1 − x2∥, ε) +. . . +φ(∥x1 − xn∥, ε) +φ(∥x2 − x1∥, ε) +φ(∥x2 − x2∥, ε) +. . . +φ(∥x2 − xn∥, ε) +... +... +... +... +φ(∥xn − x1∥, ε) +φ(∥xn − x2∥, ε) +. . . +φ(∥xn − xn∥, ε) +� +���� . +When ε is small, the RBFs become almost linearly dependent (’flat’) +forming a bad basis of functions and generating ill-conditioned interpolation +matrices A(ε) in a good interpolation space. To avoid this problem in [7, 10] +numerical techniques were developed that stabilize the solutions of linear +systems where the RBFs that form the matrix of the system take arbitrarily +small shape parameters. The RBF-QR method developed for global inter- +polations of scattered nodes using Gaussian RBFs is numerically stable for +nearly zero parameters. The idea of the RBF-QR algorithm is to change +the base {φj} to a new base {ψj} using combinations of polynomial powers, +Chebyshev polynomials and trigonometric functions. +3 +Implementation of the SLBDIM +The new matrix form for u of (27) at each node is +ui = +� +lT +i Bi +−1 + �l +T +i � +Bi +−1B˜biBi +−1� +di + ˜fi, +(29) +where li = [. . . , lik, . . .]T and �li = [. . . ,�lik, . . .]T are the column vectors. +For internal stencils, the local interpolation matrix is +Bi +ψ = V +� +In +� +R +T +� +, +(30) +where (Bi +ψ)jk = ψk(xj) and Vjk = Vk(xj) for j, k = 1, . . . , n ([7] for details). +For boundary stencils, the local matrix interpolation matrix is Bi has +two blocks, +Bi = +� Bi +ψ +Bi +Bψ +� +, +(31) +where the first matrix block is +(Bi +ψ)jk = ψk(xj), +(32) +9 + +for j = 1, . . . , nint (interior nodes) and k = 1, . . . , n (boundary nodes), and +the second matrix block is +(Bi +Bψ)jk = Bψk(xj) +(33) +for j = nint + 1, . . . , n and k = 1, . . . , n. +To avoid calculating Bi +−1 and � +Bi +−1 when ε → 0 we follow an algorithmic +procedure. The inclusion of this technique in the local integral method allows +to stabilize the numerical error of the approximation of the Helmholtz-type +equations. +This Stabilized Domain and Boundary Local Integral Method +(SLBDIM) was presented at [16] for Poisson problems, convection-diffusion +equations and elliptic PDEs. Another strategy of stability technique for local +integral methods that uses RBF interpolation functions was presented in [15]. +4 +Numerical examples on several billiars +In this section we report two numerical experiments to show the accuracy and +efficiency of the proposed numerical scheme to solve Helmholtz-type equa- +tions in two dimensions. Implementations and numerical experiments were +performed using MATLAB version R2017a numerical calculation software on +a PC with 7.5 GB of RAM and an Intel Core i7-7500U 7th Generation CPU. +running at 2.70GHz. +The reported errors are the standard error L2 (L2-Error) +L2-Error += +� +�N +i=1(uexac +i +−uapprox +i +) +2 +�N +i=1(uexac +i +) +2 +(34) +and the root mean square error (RMS): +RMS += +� +�N +i=1(uexac +i +−uapprox +i +) +2 +N +. +(35) +4.1 +Polygonal billiars: case 1 +This Helmholtz-type PDE is given over the rectangular domain Ω = [−1, 1]× +[−1, 1] +� ∆u(x) − k2u(x) += +f(x), +x = (x, y) ∈ Ω, +u(x) += +g(x), +(x, y) ∈ Γ = ∂Ω, +(36) +10 + +where f(x, y) = 2 cos(x2 + y) − (4x2 + 1 + k2)sin(x2 + y) and the parameter +k = 9. The BCs of this BVP are of the Dirichlet type, the analytical solution +being u(x, y) = sin(x2+y). In our case, we will use the local integral method +presented in its original form with Gaussian RBF kernels φ(r) = e−(εr)2 (we +will call it LBDIM) and in its stabilized form (SLBDIM). +There are several ways to discretize the Ω domain with distributions of +nodes. In our case we will use the algorithm for generating quasi-uniform +distributions developed in [5] for 2D. These distributions were created with +a fast-forward method that generates a set of nodes from a density function +starting from the Γ boundary towards the interior of the domain. +-1 +-0.5 +0 +0.5 +1 +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +-1 +1 +-0.5 +0.5 +1 +0 +0.5 +0.5 +0 +1 +0 +-0.5 +-0.5 +-1 +-1 +Figure 1: Quasi-uniform 2D node distribution for Nint = 916 internal col- +location points and Ncol = 124 boundary points with Dirichlet BC (left). +Analytical solution of BVP (right). +We compare the L2-Error of the formulation of the LBDIM and the SLB- +DIM using the Gaussian RBFs in the local interpolations varying the pa- +rameter in the form ε ∈ [1, 10]. +Figure 2 shows that as ε decreases, the +accuracy increases but the LBDIM is destabilized and the convergence is +interrupted all for cases N = 400, 916, 1610, 3604 quasi-uniform nodes. In +turn, we observe that as we increase the number of nodes on the domain +and the boundary, the errors decrease. This plot shows that for local in- +terpolation with Gaussian RBF lead to a loss in accuracy for small shape +parameters. +However, the best performance is obtained by the stabilized +local integral method to address this Helmholtz-type equation with known +analytical solutions. The error for N = 916, 1610, 3604 is of order 1 × 10−8. +The application of the RBF-QR kernel makes the system well-posed to solve +11 + +them with a direct method in the LBDIM. In this numerical experiment the +size of the stencil is n = 50. +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +10 -8 +10 -6 +10 -4 +10 -2 +10 0 +LBDIM (N=400) +SLBDIM (N=400) +LBDIM (N=916) +SLBDIM (N=916) +LBDIM (N=1610) +SLBDIM (N=1610) +LBDIM (N=3604) +SLBDIM (N=3604) +Figure 2: Comparison of the L2-Error between LBDIM and SLBDIM versus +the shape parameter ε. +In Figure 3 we show the isolines of the error log10(L2-Error) for the range +of the shape parameter [1, 10] and for different sizes of stencils n=10:10:100. +As n increases, the linear systems increase, worsening the conditioning of +the interpolation matrices. To understand the importance of local stability +technique, both graphs of this figure must be observed simultaneously. The +yellow region at the top left shows the region of error instability due to poor +numerical conditioning while in the isolines of the graphs on the right, the +region dark blue shows how 1 × 10−8 could be kept in order. As N increases +from 916 to 3604 this numerical behaviour is similar reading the figure row- +wise. +12 + +-6 +-6 +-6 +-5 +-5 +-5 +-5 +-4 +-4 +-4 +-4 +-3 +-3 +-3 +-3 +-3 +-2 +-2 +-2 +-2 +-2 +-1 +-1 +-1 +0 +0 +2 +4 +6 +8 +10 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +-6 +-5 +-4 +-3 +-2 +-1 +0 +-7 +-7 +-6 +-6 +-5 +-5 +-4 +-4 +-3 +-3 +-3 +-2 +-2 +-2 +-1 +2 +4 +6 +8 +10 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +-7 +-6 +-5 +-4 +-3 +-2 +-1 +-6 +-6 +-6 +-5 +-5 +-5 +-5 +-4 +-4 +-4 +-4 +-4 +-3 +-3 +-3 +-3 +-3 +-2 +-2 +-2 +-2 +-1 +-1 +-1 +0 +0 +2 +4 +6 +8 +10 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +-6 +-5 +-4 +-3 +-2 +-1 +0 +-7 +-7 +-6 +-6 +-5 +-5 +-4 +-4 +-4 +-3 +-3 +-3 +-2 +-2 +-1 +2 +4 +6 +8 +10 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +-7 +-6 +-5 +-4 +-3 +-2 +-1 +-6 +-5 +-5 +-5 +-5 +-5 +-4 +-4 +-4 +-4 +-4 +-3 +-3 +-3 +-3 +-2 +-2 +-2 +-1 +-1 +0 +0 +2 +4 +6 +8 +10 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +-6 +-5 +-4 +-3 +-2 +-1 +0 +-7 +-7 +-6 +-6 +-5 +-5 +-5 +-4 +-4 +-4 +-3 +-3 +-2 +2 +4 +6 +8 +10 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +-7 +-6 +-5 +-4 +-3 +-2 +-1 +Figure 3: Accuracy isolines (log10(L2-Error)) with Nint = 916, 1610, 3604 +interior points varying the shape parameter ε and the stencil size n. +In [12] this same Helmholtz type PDE is worked with mixed type BC. +In said work it can be seen that for N = 900 nodes the L2-Error 1 × 10−5 +is reached using the Radial Basis Function - Finite Difference (RBF-FD) +technique using a kernel hybrid of the Gaussian of type φ(r) = αe−(εr)2 +βr3. +13 + +4 +6 +6 +8 +8 +10 +10 +12 +12 +14 +14 +16 +16 +18 +18 +20 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +10 +20 +30 +40 +50 +60 +70 +80 +90 +4 +6 +8 +10 +12 +14 +16 +18 +20 +4 +4 +6 +6 +8 +8 +8 +10 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +10 +20 +30 +40 +50 +60 +70 +80 +90 +3 +4 +5 +6 +7 +8 +9 +10 +6 +6 +8 +8 +8 +10 +10 +12 +12 +14 +14 +16 +16 +18 +18 +20 +20 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +10 +20 +30 +40 +50 +60 +70 +80 +90 +4 +6 +8 +10 +12 +14 +16 +18 +20 +4 +4 +6 +6 +6 +8 +8 +8 +10 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +10 +20 +30 +40 +50 +60 +70 +80 +90 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +6 +8 +10 +10 +12 +12 +12 +14 +14 +16 +16 +18 +18 +20 +20 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +10 +20 +30 +40 +50 +60 +70 +80 +90 +6 +8 +10 +12 +14 +16 +18 +20 +4 +4 +6 +6 +8 +8 +10 +10 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +10 +20 +30 +40 +50 +60 +70 +80 +90 +3 +4 +5 +6 +7 +8 +9 +10 +Figure 4: Condition number isolines (log10(κ(Ai)) with Nint=916,1610,3604 +interior points varying the shape parameter ε and the stencil size n. +In Figure 4 the isolines condition number log10(κ(Ai) is shown. +The +range of the shape parameter is [1, 10] and the for different sizes of stencils +are n=10:10:100. As n increases, the conditioning of the local interpolation +matrices increases. The yellow region at the top left shows the region of +the condition number up to 1 × 1020. In the isolines of the graphs on the +right column, the region dark blue shows better conditioning up to 1 × 1010. +This ten order of magnitude are significant when when using linear solvers +14 + +numerically. Also we can observe thar as N increases from 916 to 3604 the +conditioning behaviour is similar reading the figure row-wise. +In Figure 3 it was observed that the error plots suggest smaller values +of ε0 for better accuracy, whereas in this figure the condition isolines plots +suggest the larger values of ε for better stability. This numerical results are +interpreted as the well-known uncertainty principle in RBF local interpola- +tions [17]. The idea behind this principle is that one cannot simultaneously +achieve good conditioning and high accuracy using RBF basis. The relation +between numerical stability and accuracy may be reviewed from different +perspectives as in our case to obtain a stable formulation our option was to +find a better basis in the same space of approximation using RBF-QR [7] in +the local boundary domain integral method. +4.2 +Polygonal billiars: case 2 +Consider the following two-dimensional Helmholtz equation +� ∆u(x, y) + k2u(x, y) += +f(x, y), +Ω = [0, 1] × [0, 1], +u(x, y) += +g(x, y), +Γ = ∂Ω, +(37) +where k2 = 2, f(x, y) = 2x − 4y and the exact solution is given by u(x, y) = +sin( +√ +3x)sinh(y) + cos( +√ +2y) + x − 2y, and g(x, y) is chosen to match the +exact one, thus giving BC of type Dirichlet. We use quasi-uniform nodes +within the domain and stencils of size n = 25 counting the collocation center +as shown in Figure 5. +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Figure 5: Quasi-uniform node distribution with N = 900 interior nodes (left). +Stencil node sets with n = 25 (right). +15 + +In Table 1 we show the accuracy of the SLBDIM for the shape parameter +ε = 1 and for a range of low values, ε ∈ {1 × 100, 1 × 10−1, 1 × 10−2, 1 × +10−3, 1 × 10−4, 1 × 10−5}. The number of quasi-uniform interior points of +the domain, N, varies from 121 to 900. It can be seen that for fixed ε = 1, +the ´orders of magnitude decrease from 1 × 10−6 to 1 × 10−8 starting at 441 +nodes. In turn, the convergence of the method is observed for low values of +the shape parameter, reaching RMS of the order 1 × 10−8 from 225 nodes. +The ε shown is where the best error is reached in that range. +N +SLBDIM +SLBDIM +ϵ +RMS +low ϵ +RMS +121 +1.0 +1.2028E-06 +0.1 +2.1405E-07 +225 +1.0 +5.8570E-07 +0.1 +5.0834E-08 +361 +1.0 +3.9338E-07 +0.01 +3.3821E-08 +441 +1.0 +7.8581e-08 +0.1 +3.3866E-08 +530 +1.0 +5.2907E-08 +0.00001 +3.5984E-08 +628 +1.0 +4.3843E-08 +0.00001 +3.6887E-08 +Table 1: RMS for low shape parameters ε ∈ {1 × 10−1, . . . , 1 × 10−5}. +In [11] this differential problem with mixed BC over the same domain +is investigated using Multiquadric RBF kernels ϕ(r, ε) = +� +1 + (εr)2 and a +new RBF with N ∈ [50, 350] placement points. The results obtained in said +reference reach errors of the order of 1 × 10−5 for ε ∈ [0.4]. +5 +Summary +In this work we have introduced a new local integral method to compute reso- +nances in dielectric cavities with different shapes. We have discussed numer- +ical solutions, the node quasi-uniform node distributions over the domains +and cavities with corners. Numerical results for Helmholtz-type equations +were obtained using a stabilized local integral method that uses interpola- +tions with RBF Gaussians. This method does not depend on a mesh, so it +can be easily adapted to problems with complex geometries from . The good +performance of the method has been shown with good results as shown in +numerical tests 1 and 2 comparing with other results in the literature. Test 1 +shows the advantage of using the SLBDIM to find regions of convergence of +the L2-Error of the order 1×10−8 when the shape parameter approaches zero. +16 + +In test 2, a low shape parameter range is studied reaching the same order of +the RMS. Having investigated the computational efficiency of the method, +the future work consists of approaching some applications in wave chaos and +dielectric microresonators, which is adequate to deal with geometries that +come from arbitrary domains without analytical solutions. +References +[1] C. Brebbia and D. Dominguez. Boundary Elements. An Introductory +Course. 2nd Ed. WIT Press, Computational Mechanics Publications, +Southampton and Boston, 1998. +[2] H. Cao and J. Wiersig. Dielectric microcavities: Model systems for wave +chaos and non-hermitian physics. Reviews of Modern Physics, 87:61–111, +2015. +[3] G. Fasshauer. Meshfree Approximation Methods with MATLAB. World +Scientific Publishing Co., Hackensack, NJ, USA, 2007. +[4] G. Fasshauer and M. McCourt. Kernel-based Approximation Methods +using MATLAB. World Scientific Publishing Co., Hackensack, NJ, USA, +2015. +[5] B. Fornberg and N. Flyer. Fast generation of 2-D node distributions +for mesh-free PDE discretizations. Computers and Mathematics with +Applications, 69:531–544, 2015. +[6] B. Fornberg and N. Flyer. A Primer on Radial Basis Functions with Ap- +plications to the Geosciences. Society for Industrial and Applied Math- +ematics, Philadelphia, PA, USA, 2015. +[7] B. Fornberg, E. Larsson, and N. Flyer. Stable Computations with Gaus- +sian Radial Basis Functions. SIAM Journal of Scientific Computing, +33:869–892, 2011. +[8] M. Greenberg. Applications of Green’s Functions in Science and Engi- +neering. Dover Publications, Mineola, New York, 2015. +17 + +[9] D. Kaufman, I. Kosztin, and K. Schulten. Expansion method for station- +ary states of quantum billiards. American Journal of Physics, 67:133– +141, 1999. +[10] E. Larsson, E. Lehto, A. Heryudono, and B. Fornberg. Stable Compu- +tation of Differentiation Matrices and Scattered Node Stencils on Gaus- +sian Radial Basis Functions. SIAM Journal of Scientific Computating, +35:A2096–A2119, 2013. +[11] J. Lin, W. Chen, and K. Sze. A new radial basis function for helmholtz +problems. Engineering Analysis with Boundary Elements, 36(12):1923– +1930, 2012. +[12] P. Mishra, G. Fasshauer, M. Sen, and L. Ling. A stabilized radial basis- +finite difference (RBF-FD) method with hybrid kernels. Computers & +Mathematics with Applications, 77(9):2354–2368, 2019. +[13] P. Partridge and C. B. andL.C. Wrobel. The Dual Reciprocity Boundary +Element Method. Computational Mechanics Publications co-published +with Elsevier Applied Science, Southampton Boston, 1992. +[14] L. Ponzellini Marinelli. Estabilidad num´erica de un m´etodo local inte- +gral basado en funciones de base radial para problemas de valores de +contorno. Universidad Nacional de Rosario, 2021:164 p´aginas, 2021. +[15] L. Ponzellini Marinelli. Stabilizing radial basis functions techniques for +a local boundary integral method. +Revista de la Uni´on Matem´atica +Argentina, 64:in press, 2021. +[16] L. Ponzellini Marinelli, N. Caruso, and M. Portapila. A stable com- +putation on local boundary-domain integral method for elliptic PDE. +Mathematics and Computers in Simulation, 180:379–400, 2021. +[17] R. Schaback. Error estimates and condition numbers for Radial Basis +Function interpolants. Advances in Computational Mathematics, 3:251– +264, 1995. +[18] H.-J. St¨ockmann. Quantum Chaos: An Introduction. Cambridge Uni- +versity Press, Cambridge, UK, 1999. +18 + +[19] L. Trefethen. Spectral Methods in Matlab. Society for Industrial and +Applied Mathematics, Philadelphia, PA, USA, 2000. +[20] J. Wiersig. Boundary element method for resonances in dielectric mi- +crocavities. +Journal of Optics A: Pure and Applied Optics, 5:53–60, +2003. +19 + diff --git a/ANAyT4oBgHgl3EQfRffl/content/tmp_files/load_file.txt b/ANAyT4oBgHgl3EQfRffl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1e982077f8e5a5bcd4972e6c3577774423fbbcbb --- /dev/null +++ b/ANAyT4oBgHgl3EQfRffl/content/tmp_files/load_file.txt @@ -0,0 +1,912 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf,len=911 +page_content='A stabilized local integral method using RBFs for the Helmholtz equation with applications to wave chaos and dielectric microresonators L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' Ponzellini Marinelli[1,2] luciano@fceia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='unr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='ar L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' Raviola[1] raviola@fceia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='unr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='ar [1] Faculty of Exact Sciences, Engineering and Surveying, National University of Rosario, Rosario, Argentina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' [2] Rosario Physics Institute, UNR-CONICET, Rosario, Argentina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' January 3, 2023 Abstract Most problems in electrodynamics do not have an analytical so- lution so much effort has been put in the development of numerical schemes, such as the finite-difference method, volume element meth- ods, boundary element methods, and related methods based on bound- ary integral equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' In this paper we introduce a local integral boundary domain method with a stable calculation based on Radial Basis Functions (RBF) approximations, in the context of wave chaos in acoustics and dielectric microresonator problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' RBFs have been gaining popularity recently for solving partial differential equations numerically, becoming an extremely effective tool for interpolation on scattered node sets in several dimensions with high-order accuracy and flexibility for nontrivial geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' One key issue with infinitely 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='00069v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='NA] 30 Dec 2022 smooth RBFs is the choice of a suitable value for the shape param- eter which controls the flatness of the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' It is observed that best accuracy is often achieved when the shape parameter tends to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' However, the system of discrete equations obtained from the interpolation matrices becomes ill-conditioned, which imposes severe limits to the attainable accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' A few numerical algorithms have been presented that are able to stably compute an interpolant, even in the increasingly flat basis function limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' We present the recently developed Stabilized Local Boundary Domain Integral Method in the context of boundary integral methods that improves the solution of the Helmholtz equation with RBFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' Numerical results for small shape parameters that stabilize the error are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' Accuracy and compar- ison with other methods are also discussed for various case studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' Applications in wave chaos, acoustics and dielectric microresonators are discussed to showcase the virtues of the method, which is com- putationally efficient and well suited to the kind of geometries with arbitrary shape domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' 1 Introduction and motivation Dielectric microresonators, also known as dielectric microcavities, have at- tracted interest in the last decades due to technological applications like microlasers and and as systems with intrinsic theoretical interest for its con- nections with quantum billiards and wave chaos [2, 9, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' A quantum billiard is a system in which a free particle is confined within a 2D domain and whose dynamics is governed by the Schr¨odinger equation iψt(x, t) = −∆ψ(x, t), x ∈ Ω ⊂ R2, t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' (1) where ψ(x, t) = 0 for x ∈ Γ being Γ the boundary of the domain Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' When searching the time harmonic solutions of this system in the form ψ(x, t) = ˜ψ(x)eikt, the spatial dependence, ˜ψ(x), satisfies the well-known Helmholtz Equation (HE) � ∆ + k2� ˜ψ(x) = 0, x ∈ Ω ⊂ R2, t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' (2) In this case, the eigenvalues to equation (2) are related to the energy of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' On the other hand, a similar situation arises when trying to solve the problem of light waves propagating inside a dielectric medium satisfying the 2 Maxwell equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' Also in this case, the search for time harmonic solutions leads to the Helmholtz equation for the spatial dependence of the electro- magnetic field [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' For generic domains, the equation (2) cannot be solved analytically to find stationary states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' So we must resort to finding efficient and reliable numerical methods to solve this equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' There are many numerical techniques to address this equation such as the finite element method (FEM), the finite volume method (FVM), the Boundary Element Method (BEM) or spectral methods (PS) [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' However, several of these require the construction of a specific mesh or refinement to efficiently address certain numerical problems on non-trivial geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' The BEM transforms the formulated Partial Differential Equations (PDE) into integral equations, that is, into an integral form over the boundary [1, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' In BEM the PDE that describes the physical problem is transformed into a Boundary Integral Equation (BIE), which is achieved by using Green’s identities to then apply this integral formulation over points distributed in the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' Many local integral methods are based on an integral formula- tion on small, strongly overlapping stencils with local interpolations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' In recent decades, methods involving the Radial Basis Functions (RBF) have become an extremely effective tool in non-trivial geometries for inter- polation in sets of scattered nodes and for numerically approximating PDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' There are many modern books dealing with theory, implementations and ap- plications [3, 4, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' One advantage is that when the distribution nodes are created, it is possible to achieve local refinement in critical areas depending on the specific problem [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' Particularly, this is interesting to resolve local- ized structures like the scarred states observed in quantum chaos phenomena [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' Using infinitely differential RBFs like Gaussians, exponential convergence can be shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' A practical obstacle is the ill-conditioning of the interpolation matrix when the shape parameter ε that defines the Gaussian RBF tends to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' It is known that when this parameter is reduced, the interpolation accuracy of the method improves considerably but the numerical conditioning of the problem worsens if it is solved with a direct type numerical method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' That is, there is a conflict between accuracy and the constraint known as the uncertainty principle [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' In this paper we present the Stabilized Localized Boundary-Domain Inte- gral Method (SLBDIM) [16] in the context of Helmholtz type equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' This is a new stable integral local numerical method for approximating elliptic- 3 type PDE solutions to solve Boundary Value Problems (BVP) in 2D that uses local interpolations with RBF for low values ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' This technique is a combination of meshless methods, local integral formulations and bound- ary elements in multi-domains independent of a structured mesh and that only requires an unstructured distribution of nodes of the domain Ω and its boundary Γ = ∂Ω that allows to deal with complex geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' For local interpolations, the Gaussian RBFs ϕ(r) = e−(εr)2 are used when ε → 0 in local interpolations in stable form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' Numerical results are shown for a small shape parameter that stabilizes the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' Comparisons with other methods in several cases are also dis- cussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' It is shown that the method is computationally efficient and suit- able for geometries that come from applications of wave chaos and dielectric microresonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' In particular, we solve differential problems with Dirichlet- type boundary conditions over square domains with quasi-uniform point dis- tributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' 2 The Stabilized Localized Boundary Domain Integral Method for Helmholtz equations 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='1 Problem description and local integral method We consider the following Boundary Value Problem (BVP) on an open, bounded and simply connected domain Ω ⊂ R2 (BV P) � L [u] (x) = f(x), x ∈ Ω, (3a) B [u] (x) = g(x), x ∈ Γ = ∂Ω, (3b) where L[ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' ] = ∆ + λ is an elliptic differential Helmholtz-type operator, ∆ = ∂ ∂x2 + ∂ ∂y2 is tha Laplacian, λ ∈ R (when λ = k2 > 0, k is the wave- number) and f(x) is the smooth source term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' B[ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' ] is the boundary operator with the boundary conditions (BC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' The BC are Dirichlet, Neumann or mixed over Γ = Γ1∪Γ2 and Γ1∩Γ2 = ∅ � � � u(x) = g1(x), x ∈ Γ1, (4a) ∂u(x) ∂n = g2(x), x ∈ Γ2, (4b) with g1 and g2 known data, and ∂u(x) ∂n the outward normal derivative of the unknown field u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' 4 We propose that PDE (3a) can be written as ∆u (x) = f(x) − λu (x) = b (x, u (x)) , (5) where u (x) is the potential in the point x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' We consider x ∈ Ω ⊂ R2 ∆u∗ = δ(x − ξ), (6) where δ(x − ξ) is Delta’s delta centered at ξ ∈ Ω with fundamental solution u∗(x, ξ) = 1 2πln(r), r = ∥x − ξ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' (7) From equation (5) ∆u (x) = b ⇔ � Ω u∗ (x, ξ) ∆u (x) dΩx = � Ω u∗(x, ξ)b dΩx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' (8) Applying Green’s second identity for u that satisfies (5) and u∗ that satisfies (6) � Ω (u∗∆u − u∆u∗) dΩx = � Γ � u∗ ∂u ∂n − u∂u∗ ∂n � dΓx, (9) we obtain u(ξ) = � Ω u∗ (x, ξ) b dΩx − � Γ � u∗ (x, ξ) ∂u(x) ∂n − u(x)u∗(x, ξ) ∂n � dΓx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' (10) From equation (10) we have a formula for the integral representation of the PDE over a subregion Ωi with boundary Γi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' The interior collocation point xi is obtained as before from the fundamental solution and Green’s second identity u(ξ) = � Γi q∗ (x, ξ) u (x) dΓx − � Γi u∗ (x, ξ) q (x) dΓx + � Ωi b u∗ (x, ξ) dΩx, (11) where q = ∂u ∂n is the normal derivative of the unknown field, u∗ is the fun- damental Laplacian solution and q∗ = ∂u∗ ∂n is the normal derivative of the fundamental solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' 5 Using the well-known Green-Dirichlet function (FGD), G (x, ξ), and its normal derivative Q (x, ξ) [8] in (11) we obtain a new integral formulation of the form u(ξ) = � Γi Q (x, ξ) u (x) dΓx + � Ωi b G (x, ξ) dΩx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' (12) since the integral over Γi involving G in (11) vanishes since its value is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' In addition, if the non-homogeneous term b of the PDE can be split b (x, u (x)) = f (x) − λu (x) , (13) where the funcion source f is data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' The integral representation (12) in each subregion of integration Ωi is u(ξ) = � Γi Q(x, ξ)u(x) dΓx+ � Ωi G(x, ξ)f(x) dΩx+ � Ωi −λu (x) G(x, ξ) dΩx, (14) where ξ is the interior source point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' The collocation technique is done only at interior points of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='2 Local interpolations with RBF A function ϕ : Rd → R is an RBF if there exists φ : [0, ∞) → R such that ϕ (x) = φ(r), r = ∥x − xj∥, (15) where ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='∥ is the Euclidean norm on Rd and depends on the distance to a center xj ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' If it depends on the shape parameter ε > 0, then ϕε j (x) = φ(r, ε) is often noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' In the LBDIM the field u is locally interpolated with RBF {ϕj}n j=1 with centers of the stencil Θx = {xj}n j=1 u (x) ≈ n � j=1 αjϕj(x), (16) where the interpolation matrix Ai is such that (Ai)jk = ϕk(xj) = φ(∥xj − xk∥), j, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' , n (17) 6 The term b of (13) is interpolated with RBF {χj}m j=1 with centers of the stencil Θy = {yj}m j=1 �b (u (x) , ∇u (x)) ≈ m � j=1 βjχj (x) , (18) where the interpolation matrix �Ai is such that (�Ai)jk = χk(yj) = χ(∥yj − yk∥), j, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' , m (19) The RBFs are eventually of the same type and with the same centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' If we take the same RBF bases with the same centers, the result is {ϕj}n j=1 and {χj}m j=1 for m = n although they could be different depending on the application problem or numerical experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' The local integral formulation of (14) is of the form u(ξ) ≈ n � j=1 αj �� Γi Q(x, ξ)ϕj(x) dΓx � + m � j=1 βj �� Ωi G(x, ξ)χj (x) dΩx � + � Ωi G(x, ξ)f(x) dΩx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' (20) If Θ = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' , xN} is the discretization of domain Ω and ξ = xi ∈ Θ is the collocation point, the discretized formulae of the unknown field is ui = u (xi) = n � j=1 αj�hij + m � j=1 βj�gij + �fi, (21) where αj and βj come from equations (16) and (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' The coefficients �hij, �gij and �fi are of the form � � � � � � � � � � � � � � � � � �hij = � Γi Q (x, xi) ϕj (x) dΓx, (22a) �gij = � Ωi G (x, xi) χj (x) dΩx, (22b) �fi = � Ωi G (x, xi) f (x) dΩx, (22c) which are calculated by Gauss-Legendre quadratures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' 7 Defining the vectors α = [α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' , αn]T and β = [β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' , βm]T as interpo- lation coefficients, the discretized form (21) of u can be expressed as ui = �hT i α + �gT i β + �fi, (23) where �hi = [�hi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' ,�hin]T and �gi = [�gi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' , �gim]T are the influence coeffi- cients, and �fi ∈ R is data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' The vector α arises from the local system by interpolating with the RBF basis {ϕj}n j=1 Aiα = di ⇔ α = A−1 i di (24) and the vector β arises from the local system by interpolating with the RBF basis {χj}m j=1 �Aiβ = �bi ⇔ β = �A−1 i �bi = �A−1 i � A�biα � = �A−1 i � A�biA−1 i di � , (25) where A�bi is the calculation matrix of the vector �bi with known coefficients (A�bi)jk = �b (ϕk (yj) , ∇ϕk (yj)) , j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' , m, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' (26) Substituting (24) and (25) in the discretized form (23), we obtain the discretized matrix form for ui in terms of di ui = � �hT i A−1 i + �gT i �A−1 i A�biA−1 i � di + ˜fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' (27) Rewriting (27) we obtain an algorithmic procedure to avoid the compu- tation of inverses A−1 i and �A−1 i (see [14]) ui = zTdi + �fi donde zT = �hT i A−1 i + �gT i �A−1 i A�biA−1 i (28) which are assembled into a global sparse-like system and numerically resolved with Generalized Minimal Residual (GMRES).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='3 Stability with Gaussian RBFs Convergence in global interpolations with ε-dependent RBFs can be studied in a stationary way (n = cte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' and ε → 0) or non-stationary (ε = cte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' and increasesn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' In the case of Gaussian RBFs, they produce convergence of order O(e − const (εh)2 ) (superspectral).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' 8 The RBF interpolation matrix is A(ε) = � ���� φ(∥x1 − x1∥, ε) φ(∥x1 − x2∥, ε) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' φ(∥x1 − xn∥, ε) φ(∥x2 − x1∥, ε) φ(∥x2 − x2∥, ε) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' φ(∥x2 − xn∥, ε) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' φ(∥xn − x1∥, ε) φ(∥xn − x2∥, ε) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' φ(∥xn − xn∥, ε) � ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' When ε is small, the RBFs become almost linearly dependent (’flat’) forming a bad basis of functions and generating ill-conditioned interpolation matrices A(ε) in a good interpolation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' To avoid this problem in [7, 10] numerical techniques were developed that stabilize the solutions of linear systems where the RBFs that form the matrix of the system take arbitrarily small shape parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' The RBF-QR method developed for global inter- polations of scattered nodes using Gaussian RBFs is numerically stable for nearly zero parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' The idea of the RBF-QR algorithm is to change the base {φj} to a new base {ψj} using combinations of polynomial powers, Chebyshev polynomials and trigonometric functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' 3 Implementation of the SLBDIM The new matrix form for u of (27) at each node is ui = � lT i Bi −1 + �l T i � Bi −1B˜biBi −1� di + ˜fi, (29) where li = [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' , lik, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' ]T and �li = [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' ,�lik, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' ]T are the column vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' For internal stencils, the local interpolation matrix is Bi ψ = V � In � R T � , (30) where (Bi ψ)jk = ψk(xj) and Vjk = Vk(xj) for j, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' , n ([7] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' For boundary stencils, the local matrix interpolation matrix is Bi has two blocks, Bi = � Bi ψ Bi Bψ � , (31) where the first matrix block is (Bi ψ)jk = ψk(xj), (32) 9 for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' , nint (interior nodes) and k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' , n (boundary nodes), and the second matrix block is (Bi Bψ)jk = Bψk(xj) (33) for j = nint + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' , n and k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' To avoid calculating Bi −1 and � Bi −1 when ε → 0 we follow an algorithmic procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' The inclusion of this technique in the local integral method allows to stabilize the numerical error of the approximation of the Helmholtz-type equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' This Stabilized Domain and Boundary Local Integral Method (SLBDIM) was presented at [16] for Poisson problems, convection-diffusion equations and elliptic PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' Another strategy of stability technique for local integral methods that uses RBF interpolation functions was presented in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' 4 Numerical examples on several billiars In this section we report two numerical experiments to show the accuracy and efficiency of the proposed numerical scheme to solve Helmholtz-type equa- tions in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' Implementations and numerical experiments were performed using MATLAB version R2017a numerical calculation software on a PC with 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='5 GB of RAM and an Intel Core i7-7500U 7th Generation CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' running at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='70GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' The reported errors are the standard error L2 (L2-Error) L2-Error = � �N i=1(uexac i −uapprox i ) 2 �N i=1(uexac i ) 2 (34) and the root mean square error (RMS): RMS = � �N i=1(uexac i −uapprox i ) 2 N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' (35) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='1 Polygonal billiars: case 1 This Helmholtz-type PDE is given over the rectangular domain Ω = [−1, 1]× [−1, 1] � ∆u(x) − k2u(x) = f(x), x = (x, y) ∈ Ω, u(x) = g(x), (x, y) ∈ Γ = ∂Ω, (36) 10 where f(x, y) = 2 cos(x2 + y) − (4x2 + 1 + k2)sin(x2 + y) and the parameter k = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' The BCs of this BVP are of the Dirichlet type, the analytical solution being u(x, y) = sin(x2+y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' In our case, we will use the local integral method presented in its original form with Gaussian RBF kernels φ(r) = e−(εr)2 (we will call it LBDIM) and in its stabilized form (SLBDIM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' There are several ways to discretize the Ω domain with distributions of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' In our case we will use the algorithm for generating quasi-uniform distributions developed in [5] for 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' These distributions were created with a fast-forward method that generates a set of nodes from a density function starting from the Γ boundary towards the interior of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='5 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='8 1 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='5 0 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='5 1 1 Figure 1: Quasi-uniform 2D node distribution for Nint = 916 internal col- location points and Ncol = 124 boundary points with Dirichlet BC (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' Analytical solution of BVP (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' We compare the L2-Error of the formulation of the LBDIM and the SLB- DIM using the Gaussian RBFs in the local interpolations varying the pa- rameter in the form ε ∈ [1, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' Figure 2 shows that as ε decreases, the accuracy increases but the LBDIM is destabilized and the convergence is interrupted all for cases N = 400, 916, 1610, 3604 quasi-uniform nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' In turn, we observe that as we increase the number of nodes on the domain and the boundary, the errors decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' This plot shows that for local in- terpolation with Gaussian RBF lead to a loss in accuracy for small shape parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' However, the best performance is obtained by the stabilized local integral method to address this Helmholtz-type equation with known analytical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' The error for N = 916, 1610, 3604 is of order 1 × 10−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' The application of the RBF-QR kernel makes the system well-posed to solve 11 them with a direct method in the LBDIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' In this numerical experiment the size of the stencil is n = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 10 -8 10 -6 10 -4 10 -2 10 0 LBDIM (N=400) SLBDIM (N=400) LBDIM (N=916) SLBDIM (N=916) LBDIM (N=1610) SLBDIM (N=1610) LBDIM (N=3604) SLBDIM (N=3604) Figure 2: Comparison of the L2-Error between LBDIM and SLBDIM versus the shape parameter ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' In Figure 3 we show the isolines of the error log10(L2-Error) for the range of the shape parameter [1, 10] and for different sizes of stencils n=10:10:100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' As n increases, the linear systems increase, worsening the conditioning of the interpolation matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' To understand the importance of local stability technique, both graphs of this figure must be observed simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' The yellow region at the top left shows the region of error instability due to poor numerical conditioning while in the isolines of the graphs on the right, the region dark blue shows how 1 × 10−8 could be kept in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' As N increases from 916 to 3604 this numerical behaviour is similar reading the figure row- wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='6 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='Figure 3: Accuracy isolines (log10(L2-Error)) with Nint = 916,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' 1610,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' 3604 interior points varying the shape parameter ε and the stencil size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' In [12] this same Helmholtz type PDE is worked with mixed type BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' In said work it can be seen that for N = 900 nodes the L2-Error 1 × 10−5 is reached using the Radial Basis Function - Finite Difference (RBF-FD) technique using a kernel hybrid of the Gaussian of type φ(r) = αe−(εr)2 +βr3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='4 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='Figure 4: Condition number isolines (log10(κ(Ai)) with Nint=916,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='1610,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='3604 interior points varying the shape parameter ε and the stencil size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' In Figure 4 the isolines condition number log10(κ(Ai) is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' The range of the shape parameter is [1, 10] and the for different sizes of stencils are n=10:10:100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' As n increases, the conditioning of the local interpolation matrices increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' The yellow region at the top left shows the region of the condition number up to 1 × 1020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' In the isolines of the graphs on the right column, the region dark blue shows better conditioning up to 1 × 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' This ten order of magnitude are significant when when using linear solvers 14 numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' Also we can observe thar as N increases from 916 to 3604 the conditioning behaviour is similar reading the figure row-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' In Figure 3 it was observed that the error plots suggest smaller values of ε0 for better accuracy, whereas in this figure the condition isolines plots suggest the larger values of ε for better stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' This numerical results are interpreted as the well-known uncertainty principle in RBF local interpola- tions [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' The idea behind this principle is that one cannot simultaneously achieve good conditioning and high accuracy using RBF basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' The relation between numerical stability and accuracy may be reviewed from different perspectives as in our case to obtain a stable formulation our option was to find a better basis in the same space of approximation using RBF-QR [7] in the local boundary domain integral method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='2 Polygonal billiars: case 2 Consider the following two-dimensional Helmholtz equation � ∆u(x, y) + k2u(x, y) = f(x, y), Ω = [0, 1] × [0, 1], u(x, y) = g(x, y), Γ = ∂Ω, (37) where k2 = 2, f(x, y) = 2x − 4y and the exact solution is given by u(x, y) = sin( √ 3x)sinh(y) + cos( √ 2y) + x − 2y, and g(x, y) is chosen to match the exact one, thus giving BC of type Dirichlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' We use quasi-uniform nodes within the domain and stencils of size n = 25 counting the collocation center as shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='9 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='9 1 Figure 5: Quasi-uniform node distribution with N = 900 interior nodes (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' Stencil node sets with n = 25 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' 15 In Table 1 we show the accuracy of the SLBDIM for the shape parameter ε = 1 and for a range of low values, ε ∈ {1 × 100, 1 × 10−1, 1 × 10−2, 1 × 10−3, 1 × 10−4, 1 × 10−5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' The number of quasi-uniform interior points of the domain, N, varies from 121 to 900.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' It can be seen that for fixed ε = 1, the ´orders of magnitude decrease from 1 × 10−6 to 1 × 10−8 starting at 441 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' In turn, the convergence of the method is observed for low values of the shape parameter, reaching RMS of the order 1 × 10−8 from 225 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' The ε shown is where the best error is reached in that range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' N SLBDIM SLBDIM ϵ RMS low ϵ RMS 121 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='2028E-06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='1405E-07 225 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='8570E-07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='0834E-08 361 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='9338E-07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='01 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='3821E-08 441 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='8581e-08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='3866E-08 530 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='2907E-08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='00001 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='5984E-08 628 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='3843E-08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='00001 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='6887E-08 Table 1: RMS for low shape parameters ε ∈ {1 × 10−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' , 1 × 10−5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' In [11] this differential problem with mixed BC over the same domain is investigated using Multiquadric RBF kernels ϕ(r, ε) = � 1 + (εr)2 and a new RBF with N ∈ [50, 350] placement points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' The results obtained in said reference reach errors of the order of 1 × 10−5 for ε ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' 5 Summary In this work we have introduced a new local integral method to compute reso- nances in dielectric cavities with different shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' We have discussed numer- ical solutions, the node quasi-uniform node distributions over the domains and cavities with corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' Numerical results for Helmholtz-type equations were obtained using a stabilized local integral method that uses interpola- tions with RBF Gaussians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' This method does not depend on a mesh, so it can be easily adapted to problems with complex geometries from .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' The good performance of the method has been shown with good results as shown in numerical tests 1 and 2 comparing with other results in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' Test 1 shows the advantage of using the SLBDIM to find regions of convergence of the L2-Error of the order 1×10−8 when the shape parameter approaches zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' 16 In test 2, a low shape parameter range is studied reaching the same order of the RMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' Having investigated the computational efficiency of the method, the future work consists of approaching some applications in wave chaos and dielectric microresonators, which is adequate to deal with geometries that come from arbitrary domains without analytical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' References [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' Brebbia and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' Dominguez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' Boundary Elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' An Introductory Course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' 2nd Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' WIT Press, Computational Mechanics Publications, Southampton and Boston, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' [2] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' Cao and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' Wiersig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' Dielectric microcavities: Model systems for wave chaos and non-hermitian physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfRffl/content/2301.00069v1.pdf'} +page_content=' Reviews of Modern Physics, 87:61–111, 2015.' metadata={'source': 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Misconfiguration Repair of Configurable +Cyber-Physical Systems with Search: an Industrial +Case Study on Elevator Dispatching Algorithms +Pablo Valle +Mondragon University +Mondragon, Spain +pablo.valle@alumni.mondragon.edu +Aitor Arrieta +Mondragon University +Mondragon, Spain +aarrieta@mondragon.edu +Maite Arratibel +Orona +Hernani, Spain +marratibel@orona-group.com +Abstract—Real-world Cyber-Physical Systems (CPSs) are usu- +ally configurable. Through parameters, it is possible to configure, +select or unselect different system functionalities. While this +provides high flexibility, it also becomes a source for failures +due to misconfigurations. The large number of parameters these +systems have and the long test execution time in this context +due to the use of simulation-based testing make the manual +repair process a cumbersome activity. Subsequently, in this +context, automated repairing methods are paramount. In this +paper, we propose an approach to automatically repair CPSs’ +misconfigurations. Our approach is evaluated with an industrial +CPS case study from the elevation domain. Experiments with a +real building and data obtained from operation suggests that our +approach outperforms a baseline algorithm as well as the state of +the practice (i.e., manual repair carried out by domain experts). +Index Terms—Cyber-Physical Systems, Repair, Debugging, +Configurable Systems. +I. INTRODUCTION +Cyber-Physical Systems combine digital cyber computa- +tions with parallel physical processes [1]–[3]. In such sys- +tems, digital technologies, such as computational units, low +and high-level software and communication protocols interact +among them to control a physical process through sensors and +actuators [1]. In practice, most CPSs deal with parameters. +For instance, a heavy duty lifting system involved more than +2,000 configuration parameters [4]. The behavior of CPSs can +significantly change depending on these parameters. This often +causes misconfigurations, even when selecting parameters that +are within the ranges provided by the manufacturer [5]. A +recent study showed that 19.6% of UAV-specific bugs were +caused by parameters [6]. Garcia et al. [7] found that 27.25% +of autonomous vehicle bugs were caused by incorrect con- +figurations. In our industrial case study, which involves the +traffic dispatching algorithm of a system of elevators, around +55% of the issues assigned to the traffic team are solved +through configuration changes. Therefore, it is paramount to +leverage automated and scalable techniques to automatically +repair CPS misconfigurations. However, this involves four core +challenges: +1) Challenge 1 – Expensive execution of the tests: It +is well-known that executing CPS tests is highly time- +consuming [8]–[15]. This is because, as the execution +of tests is carried out at system level, CPSs involve +compute-intensive models to simulate the physical part of +the system (e.g., models of electrical engines, dynamics +of a system). This makes the computation of the fitness +to assess how close the algorithm is from repairing the +misconfiguration expensive. For instance, in our industrial +case study, executing a test case takes around 5 minutes. +2) Challenge 2 – Large configuration space: Since con- +figurable CPSs involve many parameters, the amount of +possible configurations that a CPS can have is huge. +Subsequently, testing all of these configurations is compu- +tationally unfeasible [16]–[21]. Furthermore, it is usually +unknown which the reason (i.e., the parameters) that +causes the misconfiguration is. +3) Challenge 3 – Multiple requirements: Multiple fail- +ing requirements may exist. Some of them might be +independent from one-another [15], while others may be +conflicting (e.g., in our case study, better energy con- +sumption could lead to passengers needing to wait more). +Therefore, the repair algorithm shall be approached as a +many-objective optimization problem. +4) Challenge 4 – Prioritize severe failures: The repair +technique needs to resolve failures in their order of sever- +ity [15]. For instance a test case that shows a passengers’ +average waiting time (AWT) of 55 seconds is more +critical than one showing 35 seconds. Therefore, similar +to other CPS repairing techniques [15], our algorithm +shall give priority to more critical test cases over the less +critical ones. +On the one hand, there are approaches that target the prob- +lem of repairing misconfigurations [22], [23] of configurable +software. However, such approaches only cover the second +aforementioned challenge. On the other hand, Swarmbug [24] +focuses on repairing misconfigurations of swarm robots, which +can be considered CPSs. However, Swarmbug [24] solely +focuses on one specific objective (e.g., not crashing), therefore, +not tackling the third and fourth challenges that our industrial +case study requires. +arXiv:2301.01487v1 [cs.SE] 4 Jan 2023 + +In this paper we propose an automated repairing approach +specifically targeting CPSs’ misconfigurations. Specifically, +we tackle this by recasting the misconfiguration repair problem +to that of a many-objective search problem. To deal with +the aforementioned first challenge, we propose an algorithm +that follows a single population-based approach. Multiple +population-based algorithms, such as genetic algorithms, are +not appropriate for this context because the repair process +requires interaction with the simulator for executing test cases. +Such algorithms require a large population, and the large test +execution time would lead the algorithm to require too much +time to converge. This could eventually lead to scalability is- +sues in the context of CPSs. To deal with the second challenge, +our repairing approach implements a strategy that permits +measuring the suspiciousness of each parameter. This permits, +as the search process evolves, increasing the probability of +selecting suspicious parameters to provide a new patch. As a +result, in the beginning of the search, our approach focuses +on exploring which the critical parameters can be. As the +search evolves, the algorithm starts to focus on the exploitation +by targeting suspicious parameters. To deal with the third +challenge, our approach includes a Pareto-optimal archive- +based strategy to select and evolve potential misconfiguration +patches. This permits focusing on more than one requirement +at the same time when repairing the misconfiguration. To deal +with the last challenge, search objectives are prioritized based +on their severity level. +Our main contributions can be highlighted as follows: +1) We propose a scalable and automated approach to repair +misconfigurations in CPSs. +2) We integrate the approach with an industrial case study +from Orona, one of the largest elevator companies in +Europe. The case study involves the traffic dispatching +algorithm, a highly configurable software system. +3) We empirically evaluate our approach by using a real +scenario in which Orona’s engineers had to manually +intervene in the misconfiguration repair process. Our +repairing technique not only outperforms a baseline al- +gorithm, but also the manually derived repairing patches +by Orona’s domain experts. +4) We extract key lessons learned from the application of +our approach in an industrial case study, and provide ap- +plicability guidelines in order our approach to be adopted +by other CPS developers. +The rest of the paper is structured as follows: Section II +explains our industrial case study, how the testing is carried out +and why misconfigurations occur. In Section III we present our +approach to repair misconfigurations in our industrial context. +Section IV presents how we evaluated our approach. We ex- +tract key lessons learned and we explain the required changes +in our approach to be applied in other CPSs in Section V. +We position our work with relevant studies in Section VI. We +conclude and present future work in Section VII. +II. INDUSTRIAL CASE STUDY +Our repair algorithm is applied in an industrial case study +from the elevation domain. This section explains the different +details of the case study. +The Cyber-Physical System: Figure 1 shows an overview +of the CPS. A system of elevators is a complex CPS, whose +goal is to transport passengers from one floor to another safely +while trying to provide the highest comfort as possible. In this +system, a passenger registers a call in a floor by pushing a call +button. This information is transferred to the traffic master +through a Controller Area Network (CAN) bus. The traffic +master, after collecting other CPS information (e.g., position of +each elevator, elevator occupancy), assigns one of the available +elevators to each active call. This assignation can be carried +out through different objectives (e.g., reducing the passengers’ +waiting times, reducing energy consumption). When the call +is assigned, the elevator attends the passenger. +Elevator 1 +Elevator 2 +Elevator 3 +Floor 1 +Floor 2 +Floor N +Controller Area Network +Controller 1 +Controller 2 +Controller 3 +Traffic Master (SUT) +Fig. 1: Overview of our industrial case study +The System Under Test (SUT): Our SUT is the traffic +dispatching algorithm (i.e., dispatcher), which is an important +module inside the traffic master. To deal with different func- +tionalities and priorities, the dispatcher is highly configurable +through parameters. Different traffic dispatching algorithms +exist in Orona, and each of them encompasses one config- +uration file. The number of potential configurations of each +dispatcher is over trillions. +Test Executions: Three different phases are undertaken +when testing the dispatching algorithm [25], [26]: the +Software-in-the-Loop (SiL), the Hardware-in-the-Loop (HiL) +and Operation. Our algorithm is designed for the first phase, +i.e., the SiL test level. At this stage, a domain-specific +simulator, i.e., Elevate1, takes as input (1) the dispatching +algorithm’s executable, (2) the building installation, (3) the +configuration file and (4) the passenger file. The passenger +file is considered the test input, and it involves a set of +1https://peters-research.com/index.php/elevate/ + +passengers traveling through different floors in a building. +Each passenger has different attributes, such as, its arrival +time (i.e., time at which the passenger arrives to the floor +and pushes the button), arrival floor (i.e., floor at which the +passenger arrives), destination floor (i.e., floor at which the +passenger is traveling to), passenger weight, etc. When a test +is executed, Elevate returns a file with the results of the +simulation (e.g., waiting time required by each passenger, their +traveling time, energy consumption, distance traveled by each +elevator). This information is parsed and the necessary test +oracles are employed to assess the quality of the execution of +the test. +Functional performance requirements: When executing +test cases, besides considering certain functional require- +ments, we focus on “functional performance requirements”. +Functional performance is defined as “the properties derived +indirectly from the output of the system, rather than the +system’s efficient usage of the computational resources” [26]. +These properties are directly employed for evaluating the +functional performance requirements of Orona’s dispatching +algorithms. The properties involve metrics from the elevator +traffic domain, such as the Average Waiting Time (AWT) of +passengers, the Average Transit Time (ATT) of passengers, +Longest Waiting Time (LWT), Longest Transit Time (LTT), +number of engine starts, traveled distance by each elevator +or consumed energy. Note that configuration changes affect +functional performance requirements, whereas functional re- +quirements (e.g., ensuring that reverse journeys do not take +place) are, in principle, not affected by such changes. +Why misconfigurations occur and how they are handled: +The dispatcher has different parameters to accommodate dif- +ferent functionalities that have a direct impact on the CPS per- +formance. However, it is noteworthy that a configuration may +perform well in one installation of elevators, while not well +in another one, causing a misconfiguration. This is because +the performance of a system of elevators largely depends on +(1) the type of building and its composition and (2) how its +traffic flow is. Regarding the former, the performance can vary +depending on aspects like number of elevators in a building, +the number of floors the building has, whether all elevators +attend all floors or not, etc. For some types of buildings, some +configurations are more appropriate than others. As for the +latter, the traffic is also different depending on the type of +buildings. For instance, the traffic flow is completely different +in a hospital and in a residential building. While in a hospital +inter-floor travels are common, in a residential building most +of the calls are from the base floor to the floor where the +apartment is and vice-versa. When a system of elevators shows +a poor performance, its traffic flow is reproduced at the SiL +test level to debug and try to improve its performance through +changing parameters. If a new set of parameters improves +the system performance, then, the original configuration is +considered a misconfiguration. It is important to note that +in our industrial case study, a misconfiguration might not be +detected nor foreseen before the system is in operation due to +the CPS exposition to uncertainty [27], [28]. +III. CPS MISCONFIGURATION REPAIR METHOD +Algorithm 1 shows an overview of our repairing algo- +rithm. The algorithm takes as input (1) a faulty configu- +ration file C, composed of N number of parameters, i.e., +C = {p1, p2, ..., pN}; and (2) a test suite, composed of M +failing test cases, i.e., TS = {tc1, tc2, ..., tcM}. The first step +of the algorithm consists on assessing the failing configuration +file, where all the parameter values are parsed (Line 1) and +all test cases are executed (Line 2). When the failing test +suite is executed, the values returned by the oracle are used to +initialize the Archive (Line 3) and the suspiciousness scores +of parameters initialized (Line 4). After that, the algorithm +enters into a while loop (Lines 5-11) that ends when the +termination criteria are met. These criteria involve (1) fixing +the misconfiguration or (2) exceeding the running time. +Algorithm 1: Overview of our search-based repairing +algorithm +Input: C //Faulty Configuration file +TS //Test Suite +Output: Archive //Archive containing improved +configurations +1 Patch0 ← getValues(C); +2 InitialScore← executeTestSuite(Patch0, TS); +3 Archive ← saveToArchive(Patch0, InitialScore); +4 Susp ← initSusp(); +5 while terminationCriteriaNotMet do +6 +Parent ← selectAParentArchive(Archive); +7 +Patch1 ← generatePatch(Parent,Susp); +8 +Score ← executeTestSuite(Patch1, TS); +9 +Susp ← updateSusp(Patch1, Parent, Score, +ScoreParent); +10 +Archive ← saveToArchive(Patch1, Score); +11 end +12 return Archive; +Inside this while loop, the first step consists in selecting a +solution from the Archive (Line 6), which will be the parent. +The solution is selected pure randomly. With the selected +solution, a potential patch is proposed (Line 7), which consists +of changing one or more parameters from the parent solution +(Section III-A). This patch is assessed by executing the failing +test suite (Line 8), and the test execution results are obtained +and stored as Scores (Section III-B). In a fourth step, the +suspiciousness score of each parameter is recalculated (Line +9, Section III-C). Lastly, the Archive is updated (Line 10, +Section III-D). +A. Patch generation +A patch in our context refers to a mutation of at least one +parameter. Algorithm 2 shows our algorithm for proposing a +potential patch. As input, it receives (1) a parent configuration, +which corresponds to one configuration in the archive of the +algorithm and (2) the suspiciousness ranking of all parameters. +First, a parameter to be mutated is selected (Line 4) based on + +the suspiciousness of each parameter (see Section +III-C for +more details on how to compute the suspiciousness score). +The higher the suspiciousness, the higher the probability of +being selected. The parameter to be mutated is obtained by +employing Algorithm 3. The selected parameter is mutated +(Line 5) by giving a random value within its ranges. After +this, it is decided whether a new parameter is mutated (Line +8). The probability of mutating a new parameter decreases as +the number of mutated parameters in the new patch increases. +We ensure that one parameter is not mutated more than once. +Algorithm 2: Patch generation algorithm +Input: Parent //Faulty Configuration +SuspRanking //Suspiciousness Ranking +Output: Patch //Mutated Configuration +1 numOfMutParams ← 0; +2 Patch ← Parent; +3 do +4 +varToMutate ← selectParam(SuspRanking); +5 +Patch ← mutate(Patch,varToMutate); +6 +numOfMutParams ← numOfMutParams +1; +7 +p ← rand(); //returns random value 0 to 1 +8 while p < 0.5numOfMutatedP arams; +9 return Patch; +Algorithm 3: Suspiciousness-based parameter selec- +tion algorithm +Input: SuspScore = {ss1, ss2, ..., ssN} +Output: selected //Index of the selected parameter +1 total ← �N +i=1(ssi); +2 iterativeSum←0; +3 prob ← []; +4 for i ← 1 to nPop do +5 +prob[i] ← iterativeSum + SuspScore[i]/total; +6 +iterativeSum←prob[i]; +7 end +8 prob←orderAscending(prob); +9 r←rand();//Returns random number 0 to 1 +10 j←0; +11 selected=N; +12 while j= 0.474, where d = 2| ˆA12 − 0.5|. According to this +categorization, the difference was negligible during the first +execution hour, small between the second and third execution +hours and medium during the fourth execution hour. In these +first four execution hours, there was no statistical significance +between the repair algorithm and the baseline. Conversely, +after the fifth hour, there was statistical significance (i.e., p- +value < 0.05) with large effect sizes based on the related +categorization [47], all of them in favor of our approach. +TABLE I: RQ1 – Summary of the statistical tests when +comparing the repair algorithm with its unguided version for +the HV quality indicator over the execution of 12 hours. An +ˆA12 value higher than 0.5 means that the results are in favor +of the repair algorithm. Statistical significance is set as p- +val<0.05 +Hour +ˆA12 +p-val +1 +0.51 +0.9698 +2 +0.61 +0.4273 +3 +0.65 +0.2730 +4 +0.71 +0.1212 +5 +0.80 +0.0256 +6 +0.86 +0.0081 +7 +0.89 +0.0040 +8 +0.82 +0.0172 +9 +0.85 +0.0090 +10 +0.85 +0.0090 +11 +0.90 +0.0028 +12 +0.92 +0.0017 +Besides the HV, we also analyzed the individual patches +provided by the decision maker (DM). In this case, the aim +of the algorithm was to reduce such metrics. Therefore, an +ˆA12 lower than 0.5 means that the repair algorithm performed +better. Table II summarizes the statistical tests for the ten runs +and each individual objective function. There was statistical +significance in half of the objective functions (i.e., LWT, ATT +and LTT). For such cases, the effect sizes were large (i.e., +ˆA12 between 0.18 to 0.2). For the remaining objectives, where +there was no statistical significance, in the case of the AWT +and %WT>55, the effect sizes showed a negligible difference, +whereas for the case of %TT>70, the difference was small. +TABLE II: Summary of the statistical test results when com- +paring the patches provided by the DM when employing repair +algorithm against the baseline and manual repair approaches +vs. Baseline +vs. Manual +ˆA12 +p-val +ˆA12 +p-val +AWT +0.52 +0.9097 +0.10 +0.0014 +LWT +0.18 +0.0165 +0.20 +0.0161 +%WT>55s +0.47 +0.8788 +0.00 +<0.0001 +ATT +0.20 +0.0312 +0.40 +0.4429 +LTT +0.20 +0.010 +0.00 +<0.0001 +%TT>70s +0.37 +0.3438 +0.00 +<0.0001 +Table III show the average value of each of the functional +performance metrics used by the oracles for the 10 runs and +the patches provided by the DMs. These results were somehow +consistent with those from Table II. As it can be appreciated, +the most striking difference relates to the LWT and the LTT +functional performance metrics. On the contrary, for the AWT, +%WT>55, ATT and %TT>70, the differences were not that +large. This could be due to the nature of the DM. Note +that for those metrics, the DM accepts values that are below +certain thresholds (e.g., AWT < 25 seconds), whereas for LWT +and LTT, the DM selects those patches with lowest values. +However, in all metrics except the AWT, our algorithm showed +lower average values. +TABLE III: Comparison between the misconfigured configu- +ration file, the patch provided by the DM with the manual +repair, the average values of the patches returned by the DM +for the baseline algorithm and the average values of the patches +returned by the DM for the repair algorithm +Misconf +Manual +Baseline DM +Repair DM +AWT +25.99 +23.10 +22.66 +22.77 +LWT +435.70 +223.00 +241.55 +213.72 +%WT >55s +12.78 +11.99 +9.93 +9.92 +ATT +42.01 +41.60 +41.77 +41.58 +LTT +209.80 +220.60 +206.24 +195.56 +%TT>70s +10.24 +10.02 +9.64 +9.45 +In conclusion, the first RQ can be answered as follows: +Answer to the first RQ: The repair algorithm outper- +formed the baseline algorithm. The average improvement +extent of the repair algorithm with respect to the baseline +was around 29% when considering the HV quality indi- +cator. Furthermore, there was statistical significance with +large effect sizes when comparing individual patches pro- +posed by the DM for half of the objective functions, all +of them in favor of the repair algorithm. All this suggests +that the problem of repairing CPSs misconfigurations is +non-trivial, and therefore, automated and scalable repair +techniques are necessary. +2) RQ2 – Comparison with manual repair: With the second +RQ, we aimed at comparing the proposed repairing algorithm + +Hypervolume +0.014 +0.012 +0.01 +C +0.008 + o- Repair +0.006 + Unguided +-- Manual +0.004 +-0 +G中 +0.002 +0d +2 +4 +6 +8 +10 +0 +12 +Execution hourswith the manual process of repairing the misconfiguration by +domain experts. Specifically, these domain experts provided a +total of 6 patches. With those patches, and by applying the six +oracles in our algorithm, we derived the HV metric. As can +be seen in Figure 2, the HV was quite low. This was because +only four patches were non-dominated, whereas our archive is +capable of handling up to twelve patches. Therefore, those four +patches were not able to cover a large volume in the objective +space. Furthermore, it is important to note that the time was not +considered here, because we do not have such information. In +terms of the HV, the average improvement extent of our repair +algorithm over the manually derived patches was up to 77.5%. +For this case, we also employed the DM to select one of +the non-dominated patches. Table II shows the statistical tests +carried out when comparing the patches provided by the DM +after executing the repair algorithm with the patch proposed +by the DM after processing the four non-dominated solutions. +As it can be appreciated, in five out of six metrics there +was statistical significance, where the effect size showed a +large difference according to the categorization proposed by +Romano et al. [47]. All these effect sizes were in favor of the +repair algorithm. On the other hand, for the case where there +was no statistical significance, i.e., for the case of the ATT +metric, the difference was small in terms of the ˆA12 value, +but in favor of the repair algorithm. +The improvement extent for each functional performance +metric obtained by the patches provided by the DM (over 10 +runs) with respect to the manual approach can be appreciated +in Table III. These results are consistent with the statistical +tests, where it can be appreciated a similar average value in the +case of the ATT. In this case, the improvement extent is higher +in the cases of the AWT, % WT > 55, LTT and %WT>70 +when compared to the baseline algorithm. However, in relation +to the LWT, the improvement was only of 10 seconds on +average, unlike with the baseline, where the improvement was +of nearly 29 seconds on average. +In summary, the second RQ can be answered as follows: +Answer to the second RQ: The repair algorithm +outperformed the manual repair process. The average +improvement extent of the repair algorithm with re- +spect to the patches provided by the domain experts +was around 77.5% when considering the HV quality +indicator. Furthermore, there was statistical significance +with large effect sizes when comparing individual patches +proposed by the DM in five out of six objective functions. +In addition, our approach provides a fully automated +approach, which can therefore increase the productivity +of engineers from Orona when dealing with misconfigu- +rations of the traffic dispatching algorithm. +C. Threats to Validity +We now summarize the threats to validity of our study and +the measures taken to mitigate them. +An internal validity threat in our evaluation could be related +to the parameters used in the algorithms, which were not +changed. Three main parameters need to be configured (1) +the time budget, which was set to 12 hours; (2) the number of +time a parameter needs to be selected to start computing its +suspiciousness score (i.e., Nsusp), which is set to 5; and (3) the +number of solutions in the archive. The first two parameters +were selected based on preliminary evaluations. Coversely, the +maximum number of solutions in the archive was the same as +other repair approaches targeting CPSs [15]. +As in any search-based software engineering problem, a +conclusion validity threat involves the stochastic nature of the +algorithms used. To mitigate such issue, we run each algorithm +10 times. It is important to note that our technique needs a long +time to converge because the simulations employed to assess +potential patches are exhaustive, therefore, we could not afford +a large number of runs. Furthermore, we applied statistical +tests to analyze the results, as recommended by Arcuri and +Briand [48]. +As in any study involving humans, our evaluation is also +subject to external validity threats. One such threats refers +to the patches proposed by engineers from Orona. It is note- +worthy, however, that these engineers have broad experience +and domain expertise, and that the patches they proposed +were the ones that were later deployed in the real CPS. The +generalizability of the results is also another external validity +threat of our study; note, however, that we used an industrial +case study with a real installation and data obtained from +operation. We plan to mitigate such threat in the future by +(1) using other case studies from a different domain and (2) +using other real installations where misconfigurations occured. +Lastly, construct validity threats arise when the measures +used are not comparable across algorithms. This was mitigated +by giving the same search budget to both algorithms (i.e., the +repair and the unguided algorithm). +V. LESSONS LEARNED AND APPLICABILITY +In this section, we describe the lessons we have learned +thorough the whole process of developing and evaluating the +repairing algorithm. In addition, we explain the main changes +our method would require when applying it to other CPS +domains. +A. Lessons Learned +Lesson 1 – Reduction of personnel cost: The current +state of the practice when repairing misconfigurations is purely +manual. This requires significant personnel cost since domain +experts are required in the process. Our fully automated +repairing approach not only outperforms the state of the +practice in terms of providing a better patch to repair the +misconfiguration, but also reduces significantly the personnel +costs that are required behind a manual repair process. +Lesson 2 – Scalable technique: Scalability is one of the +main concerns when testing and debugging CPSs, mainly +due to the need of considering properties involving physical +devices with continuous dynamics and complex concurrent +interactions between the system and its environment (e.g., +people) [49]. We saw that our search-based repair algorithm + +converges after around 10 hours, which is affordable for +our industrial partner as engineers can launch the automated +misconfiguration repair tool nightly. +Lesson 3 – Surrogate models are, in principle, not +appropriate: Despite we did not carefully assess this, while +we developed the algorithm, we intended to integrate surrogate +models to accelerate the repair process. However, we saw +that this technique required too much time to build reliable +surrogate models. This time was similar to the time budget +that our repair algorithm required to converge. Although we +assessed different types of surrogate models, we still need to +more carefully analyze this, which remains a future work. +Lesson 4 – Challenging conflicting installation: After +applying our experiments and showing the results to Orona’s +engineers, we noted that the conflicting installation we selected +was challenging. Indeed, the traffic was abnormal, with many +unforeseen situations (e.g., having too many calls in a short +time window) and therefore, repairing the misconfiguration +in such installation was, according to domain experts, more +difficult than other installations. +B. Applicability +The context at which we have applied our repairing ap- +proach is the elevator dispatching algorithm of Orona. How- +ever, we believe that the three key challenges that we tackle +(i.e., expensive execution of tests, large configuration space +and multiple functional performance requirements) are com- +mon in all types of configurable CPSs. As we involved domain +experts when developing the repair approach, several domain- +specific design choices were considered, which would require +adaptions when applying our approach in another domain. Be- +low we explain different alternatives and the changes required +for the adoption of our method in another domain. +Test execution process: One of the first changes our method +would require is the test execution. As explained in Section +III-B, we use a domain-specific simulator to execute test cases +and measure how close the algorithm is from repairing the +misconfiguration. This process would need to be substituted +by the simulator being used to execute the tests within +other CPSs. In addition, we employ a parallel test execution, +which was possible in our context. However, other simulators +(e.g., autonomous vehicles) could require more computing +resources. For instance, testing autonomous vehicles often +requires rendering driving scenes in virtual scenarios using +high-fidelity simulators [13], which may require the execution +of test cases to be sequential. Lastly, test oracles would need +to be defined. When using Simulink models to execute the +tests, which is a predominant CPS testing tool [50], an option +could be to use SOCRaTEs [11], a DSL-based test oracle +specification and generation tool for Simulink. Specifically, +SOCRaTEs [11] provides a quantitative measure of the degree +of violation of a requirement, similar to what we need in our +algorithm to guide the misconfiguration repair process. +Removing solutions from the archive: As explained in +Section III-D, the archive may increase in size, which may +have a direct implication in the convergence of the repairing +algorithm. Therefore, when the archive exceeds a predefined +number of solutions, one of the solutions needs to be removed. +Our algorithm removes the solution with longest AWT, given +that this is the most widely employed metric when testing dis- +patching algorithms [30]. In another domain, two alternatives +can be considered. The first one, employing one of the most +important metrics. If all metrics have a similar importance, +the second alternative could be to randomly remove one of +the solutions from the archive or use a crowding distance to +remove solutions that are too close from each other. +Decision maker: The decision maker is another component +that we developed ad-hoc for the traffic dispatching algorithm +by following the advise of domain experts. We recommend +to analyze priorities of the specific CPS to make a decision. +In case there are no clear priorities, a solution could be to +employ a weighted approach giving the same importance to +all objectives. +Patch confirmation: We only employed a failing test suite +to guide the repair process. The core reason was the high test +execution time. Eventually, it could happen that a proposed +patch makes a test case from the passing test suite fail. Because +of this, we implemented a patch confirmation process by +following a traditionally employed regression test method [26] +combined with a newly incorporated metamorphic testing +approach by Orona [37], [38]. The patch confirmation module +should follow the internally standardized testing approach, +which can vary from a company to another. +VI. RELATED WORK +The related work in automated program repair is huge. +Monperrus mantains a living review on such techniques [51]. +Table IV shows a summarized classification of the related work +analyzing four key characteristics covered by our approach. +The first characteristic (C1) analyzes the possibility of repair- +ing computationally expensive systems. The second one (C2), +whether the approach is intended to repair misconfigurations. +The third one (C3), analyzes if the approach is able to deal +with many requirements (i.e., more than 3). And the last one +(C4), whether the approach prioritizes critical faults over the +less critical ones. +TABLE IV: Related work comparison with different charac- +teristics required by our repairing technique +C1 +C2 +C3 +C4 +[15] ++ +- ++ ++ +[22], [23] +- ++ +- +- +[24] ++ ++ +- +- +[52]–[56] +- +- +- +- +[57]–[64] ++ +- +- +- +We found that, in the field of CPSs, repairing approaches +are still in their infancy. Indeed, to the best of our knowledge, +only two approaches tackle the problem of repairing CPSs. +On the one hand, Swarmbug [24] focuses on repairing mis- +configurations of swarm robotics. Specifically, they make use +of a mechanism called the “degree of causal contribution” +to abstract impacts of configurations to the swarm drones + +via behavior causal analysis. The evaluation is carried out +in four swarm algorithms, and the repair objectives are in- +dividual for each of them. These involve aspects like avoiding +obstacles or unsafe zones in order the drones not to crash. +The approach, however, does not cover C3 and C4. On the +other hand, Ariel [15] focuses on repairing feature interaction +failures in automated driving systems. Similar to our approach, +ARIEL [15] uses a many-objective and a single population- +based approach, and also employs an archive to keep track of +partially repaired solutions. However, unlike this paper, which +focuses on repairing misconfigurations, ARIEL [15] repairs +feature interaction bugs by applying modify and swift mutation +operators to integration rules that resolve conflicts between +automated driving system features. Therefore, ARIEL does not +cover C2. +CADET [22] does cover C2 as it is intended to debug +and fix misconfigurations that cause non-functional faults. +Xiong et al. [23] focus on repairing misconfigurations in +software product lines by generating a list of range fixes +to help satisfy a constraint. However, both approaches do +not consider systems that take high computation resources to +execute the tests. In addition, CADET [22] only covers two +non-functional properties (i.e., latency and energy), whereas +Xiong et al. [23] focus on satisfying individual constraints. +Lastly, the approaches do not prioritize fixing more critical +faults over the less critical ones. Subsequently, both techniques +do not cover C1, C3 and C4. +Besides these three studies, which are the most closely +related to our approach, other studies exist in the field +of automated program repair [52]–[64]. Similar to this ap- +proach, some consider search techniques, such as genetic +programming [55], [56]. GenProg [55] is one of the first +approaches that proposed the use of meta-heuristic search to +repair software programs. Specifically they leveraged genetic +programming to repair C programs. However, all these ap- +proaches focus on repairing bugs in the code. Conversely, our +approach focuses on repairing misconfigurations in the field +of configurable CPSs. +Another line of research related to our approach is that +of unified debugging [65], [66]. Such technique uses patch +execution results to improve localizing the fault [65], [66]. +Therefore, even if the repair process is unable to repair the bug, +unified debugging helps improving the fault localization for +latter manual repair. Our approach follows a similar strategy, +where we aim at localizing suspicious parameters that will +eventually help repair the misconfiguration. However, besides +the fact that unified debugging [65], [66] is not aimed at +debugging misconfigurations, but bugs at the code level, it +assumes that there is an initial suspiciousness score (i.e., at +statement level). Conversely, our approach begins with all +parameters having the same suspiciousness because there is +no information about which parameters have influence in the +system performance. +VII. CONCLUSION AND FUTURE WORK +Real-world CPSs, such as elevators, involve many param- +eters. The performance of CPSs is tightly linked to such +parameters, and therefore, misconfigurations may occur. On +the one hand, manually dealing with such misconfigurations +might not always be feasible. On the other hand, automated +solutions require dealing with certain challenges, such as, +expensive simulations to execute test cases. In this paper we +propose an automated and scalable solution based on meta- +heuristic search to repair misconfigurations in CPSs. Our +approach was integrated with an industrial case study provided +by Orona, one of the largest elevator manufacturers in Europe. +The evaluation was carried out with a real installation in which +domain experts from Orona had to manually intervene in +repairing a misconfiguration. The results suggest that, besides +automating a process that before was purely manual, our algo- +rithm provides better patches than those provided by domain +experts. Specifically, in five out of the six quality indicators +employed by domain experts to assess the quality of a patch, +our algorithm outperformed with statistical significance the +patch provided by domain experts. +In the future, we would like to extend our approach +from different perspectives. In terms of the applicability, we +would like to integrate our algorithm with other CPSs in +which configurations have been found to be problematic (e.g., +unmanned aerial vehicles [5]). Furthermore, we would like +to explore solutions to prevent potential overfitting issues +before proposing a plausible patch. This has been one of the +core challenges identified in automated program repair [32]– +[35], and therefore, we should be aware of it. In terms of +internal applicability within Orona, we would like to evaluate +our approach in other installations where misconfigurations +occurred. Furthermore, we would also like to transfer the +repair algorithm beyond the traffic team and within other +departments. Lastly, we would like to further study whether +other strategies exist to better train and integrate surrogate +models in the repair process. +ACKNOWLEDGMENT +Project supported by a 2021 Leonardo Grant for Researchers +and Cultural Creators, BBVA Foundation. The BBVA Founda- +tion is not responsible for the opinions, comments and contents +included in the project and/or the results derived from it, +which are the total and absolute responsibility of their authors. +Aitor Arrieta is part of the Software and Systems Engineer- +ing research group of Mondragon Unibertsitatea (IT1519-22), +supported by the Department of Education, Universities and +Research of the Basque Country. +REFERENCES +[1] P. Derler, E. A. Lee, and A. 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Lou, and L. Zhang, “Evaluating and improving +unified debugging,” IEEE Transactions on Software Engineering, 2021. + diff --git a/AdAzT4oBgHgl3EQfhf3-/content/tmp_files/load_file.txt b/AdAzT4oBgHgl3EQfhf3-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ebb2babaa212e262ab9297af064696097c6daf7d --- /dev/null +++ b/AdAzT4oBgHgl3EQfhf3-/content/tmp_files/load_file.txt @@ -0,0 +1,1202 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf,len=1201 +page_content='Automated Misconfiguration Repair of Configurable Cyber-Physical Systems with Search: an Industrial Case Study on Elevator Dispatching Algorithms Pablo Valle Mondragon University Mondragon, Spain pablo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content='valle@alumni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content='mondragon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content='edu Aitor Arrieta Mondragon University Mondragon, Spain aarrieta@mondragon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content='edu Maite Arratibel Orona Hernani, Spain marratibel@orona-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content='com Abstract—Real-world Cyber-Physical Systems (CPSs) are usu- ally configurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' Through parameters, it is possible to configure, select or unselect different system functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' While this provides high flexibility, it also becomes a source for failures due to misconfigurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' The large number of parameters these systems have and the long test execution time in this context due to the use of simulation-based testing make the manual repair process a cumbersome activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' Subsequently, in this context, automated repairing methods are paramount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' In this paper, we propose an approach to automatically repair CPSs’ misconfigurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' Our approach is evaluated with an industrial CPS case study from the elevation domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' Experiments with a real building and data obtained from operation suggests that our approach outperforms a baseline algorithm as well as the state of the practice (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=', manual repair carried out by domain experts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' Index Terms—Cyber-Physical Systems, Repair, Debugging, Configurable Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' INTRODUCTION Cyber-Physical Systems combine digital cyber computa- tions with parallel physical processes [1]–[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' In such sys- tems, digital technologies, such as computational units, low and high-level software and communication protocols interact among them to control a physical process through sensors and actuators [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' In practice, most CPSs deal with parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' For instance, a heavy duty lifting system involved more than 2,000 configuration parameters [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' The behavior of CPSs can significantly change depending on these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' This often causes misconfigurations, even when selecting parameters that are within the ranges provided by the manufacturer [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' A recent study showed that 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content='6% of UAV-specific bugs were caused by parameters [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' Garcia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' [7] found that 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content='25% of autonomous vehicle bugs were caused by incorrect con- figurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' In our industrial case study, which involves the traffic dispatching algorithm of a system of elevators, around 55% of the issues assigned to the traffic team are solved through configuration changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' Therefore, it is paramount to leverage automated and scalable techniques to automatically repair CPS misconfigurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' However, this involves four core challenges: 1) Challenge 1 – Expensive execution of the tests: It is well-known that executing CPS tests is highly time- consuming [8]–[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' This is because, as the execution of tests is carried out at system level, CPSs involve compute-intensive models to simulate the physical part of the system (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=', models of electrical engines, dynamics of a system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' This makes the computation of the fitness to assess how close the algorithm is from repairing the misconfiguration expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' For instance, in our industrial case study, executing a test case takes around 5 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' 2) Challenge 2 – Large configuration space: Since con- figurable CPSs involve many parameters, the amount of possible configurations that a CPS can have is huge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' Subsequently, testing all of these configurations is compu- tationally unfeasible [16]–[21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' Furthermore, it is usually unknown which the reason (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=', the parameters) that causes the misconfiguration is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' 3) Challenge 3 – Multiple requirements: Multiple fail- ing requirements may exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' Some of them might be independent from one-another [15], while others may be conflicting (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=', in our case study, better energy con- sumption could lead to passengers needing to wait more).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' Therefore, the repair algorithm shall be approached as a many-objective optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' 4) Challenge 4 – Prioritize severe failures: The repair technique needs to resolve failures in their order of sever- ity [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' For instance a test case that shows a passengers’ average waiting time (AWT) of 55 seconds is more critical than one showing 35 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' Therefore, similar to other CPS repairing techniques [15], our algorithm shall give priority to more critical test cases over the less critical ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' On the one hand, there are approaches that target the prob- lem of repairing misconfigurations [22], [23] of configurable software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' However, such approaches only cover the second aforementioned challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' On the other hand, Swarmbug [24] focuses on repairing misconfigurations of swarm robots, which can be considered CPSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' However, Swarmbug [24] solely focuses on one specific objective (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=', not crashing), therefore, not tackling the third and fourth challenges that our industrial case study requires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content='01487v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content='SE] 4 Jan 2023 In this paper we propose an automated repairing approach specifically targeting CPSs’ misconfigurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' Specifically, we tackle this by recasting the misconfiguration repair problem to that of a many-objective search problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' To deal with the aforementioned first challenge, we propose an algorithm that follows a single population-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' Multiple population-based algorithms, such as genetic algorithms, are not appropriate for this context because the repair process requires interaction with the simulator for executing test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' Such algorithms require a large population, and the large test execution time would lead the algorithm to require too much time to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' This could eventually lead to scalability is- sues in the context of CPSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' To deal with the second challenge, our repairing approach implements a strategy that permits measuring the suspiciousness of each parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' This permits, as the search process evolves, increasing the probability of selecting suspicious parameters to provide a new patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' As a result, in the beginning of the search, our approach focuses on exploring which the critical parameters can be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' As the search evolves, the algorithm starts to focus on the exploitation by targeting suspicious parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' To deal with the third challenge, our approach includes a Pareto-optimal archive- based strategy to select and evolve potential misconfiguration patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' This permits focusing on more than one requirement at the same time when repairing the misconfiguration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' To deal with the last challenge, search objectives are prioritized based on their severity level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' Our main contributions can be highlighted as follows: 1) We propose a scalable and automated approach to repair misconfigurations in CPSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' 2) We integrate the approach with an industrial case study from Orona, one of the largest elevator companies in Europe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' The case study involves the traffic dispatching algorithm, a highly configurable software system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' 3) We empirically evaluate our approach by using a real scenario in which Orona’s engineers had to manually intervene in the misconfiguration repair process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' Our repairing technique not only outperforms a baseline al- gorithm, but also the manually derived repairing patches by Orona’s domain experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' 4) We extract key lessons learned from the application of our approach in an industrial case study, and provide ap- plicability guidelines in order our approach to be adopted by other CPS developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' The rest of the paper is structured as follows: Section II explains our industrial case study, how the testing is carried out and why misconfigurations occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' In Section III we present our approach to repair misconfigurations in our industrial context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' Section IV presents how we evaluated our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' We ex- tract key lessons learned and we explain the required changes in our approach to be applied in other CPSs in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' We position our work with relevant studies in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' We conclude and present future work in Section VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' INDUSTRIAL CASE STUDY Our repair algorithm is applied in an industrial case study from the elevation domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' This section explains the different details of the case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' The Cyber-Physical System: Figure 1 shows an overview of the CPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' A system of elevators is a complex CPS, whose goal is to transport passengers from one floor to another safely while trying to provide the highest comfort as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' In this system, a passenger registers a call in a floor by pushing a call button.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' This information is transferred to the traffic master through a Controller Area Network (CAN) bus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' The traffic master, after collecting other CPS information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=', position of each elevator, elevator occupancy), assigns one of the available elevators to each active call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' This assignation can be carried out through different objectives (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=', reducing the passengers’ waiting times, reducing energy consumption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' When the call is assigned, the elevator attends the passenger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' Elevator 1 Elevator 2 Elevator 3 Floor 1 Floor 2 Floor N Controller Area Network Controller 1 Controller 2 Controller 3 Traffic Master (SUT) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' 1: Overview of our industrial case study The System Under Test (SUT): Our SUT is the traffic dispatching algorithm (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=', dispatcher), which is an important module inside the traffic master.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' To deal with different func- tionalities and priorities, the dispatcher is highly configurable through parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' Different traffic dispatching algorithms exist in Orona, and each of them encompasses one config- uration file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' The number of potential configurations of each dispatcher is over trillions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' Test Executions: Three different phases are undertaken when testing the dispatching algorithm [25], [26]: the Software-in-the-Loop (SiL), the Hardware-in-the-Loop (HiL) and Operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' Our algorithm is designed for the first phase, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=', the SiL test level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' At this stage, a domain-specific simulator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=', Elevate1, takes as input (1) the dispatching algorithm’s executable, (2) the building installation, (3) the configuration file and (4) the passenger file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' The passenger file is considered the test input, and it involves a set of 1https://peters-research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content='com/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content='php/elevate/ passengers traveling through different floors in a building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' Each passenger has different attributes, such as, its arrival time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=', time at which the passenger arrives to the floor and pushes the button), arrival floor (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=', floor at which the passenger arrives), destination floor (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=', floor at which the passenger is traveling to), passenger weight, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' When a test is executed, Elevate returns a file with the results of the simulation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=', waiting time required by each passenger, their traveling time, energy consumption, distance traveled by each elevator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' This information is parsed and the necessary test oracles are employed to assess the quality of the execution of the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' Functional performance requirements: When executing test cases, besides considering certain functional require- ments, we focus on “functional performance requirements”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' Functional performance is defined as “the properties derived indirectly from the output of the system, rather than the system’s efficient usage of the computational resources” [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' These properties are directly employed for evaluating the functional performance requirements of Orona’s dispatching algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' The properties involve metrics from the elevator traffic domain, such as the Average Waiting Time (AWT) of passengers, the Average Transit Time (ATT) of passengers, Longest Waiting Time (LWT), Longest Transit Time (LTT), number of engine starts, traveled distance by each elevator or consumed energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' Note that configuration changes affect functional performance requirements, whereas functional re- quirements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=', ensuring that reverse journeys do not take place) are, in principle, not affected by such changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' Why misconfigurations occur and how they are handled: The dispatcher has different parameters to accommodate dif- ferent functionalities that have a direct impact on the CPS per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' However, it is noteworthy that a configuration may perform well in one installation of elevators, while not well in another one, causing a misconfiguration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' This is because the performance of a system of elevators largely depends on (1) the type of building and its composition and (2) how its traffic flow is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' Regarding the former, the performance can vary depending on aspects like number of elevators in a building, the number of floors the building has, whether all elevators attend all floors or not, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' For some types of buildings, some configurations are more appropriate than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' As for the latter, the traffic is also different depending on the type of buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' For instance, the traffic flow is completely different in a hospital and in a residential building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' While in a hospital inter-floor travels are common, in a residential building most of the calls are from the base floor to the floor where the apartment is and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' When a system of elevators shows a poor performance, its traffic flow is reproduced at the SiL test level to debug and try to improve its performance through changing parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' If a new set of parameters improves the system performance, then, the original configuration is considered a misconfiguration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' It is important to note that in our industrial case study, a misconfiguration might not be detected nor foreseen before the system is in operation due to the CPS exposition to uncertainty [27], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' CPS MISCONFIGURATION REPAIR METHOD Algorithm 1 shows an overview of our repairing algo- rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' The algorithm takes as input (1) a faulty configu- ration file C, composed of N number of parameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=', C = {p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=', pN};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' and (2) a test suite, composed of M failing test cases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=', TS = {tc1, tc2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=', tcM}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' The first step of the algorithm consists on assessing the failing configuration file, where all the parameter values are parsed (Line 1) and all test cases are executed (Line 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' When the failing test suite is executed, the values returned by the oracle are used to initialize the Archive (Line 3) and the suspiciousness scores of parameters initialized (Line 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' After that, the algorithm enters into a while loop (Lines 5-11) that ends when the termination criteria are met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' These criteria involve (1) fixing the misconfiguration or (2) exceeding the running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' Algorithm 1: Overview of our search-based repairing algorithm Input: C //Faulty Configuration file TS //Test Suite Output: Archive //Archive containing improved configurations 1 Patch0 ← getValues(C);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' 2 InitialScore← executeTestSuite(Patch0, TS);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' 3 Archive ← saveToArchive(Patch0, InitialScore);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' 4 Susp ← initSusp();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' 5 while terminationCriteriaNotMet do 6 Parent ← selectAParentArchive(Archive);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' 7 Patch1 ← generatePatch(Parent,Susp);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' 8 Score ← executeTestSuite(Patch1, TS);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' 9 Susp ← updateSusp(Patch1, Parent, Score, ScoreParent);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' 10 Archive ← saveToArchive(Patch1, Score);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' 11 end 12 return Archive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' Inside this while loop, the first step consists in selecting a solution from the Archive (Line 6), which will be the parent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' The solution is selected pure randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' With the selected solution, a potential patch is proposed (Line 7), which consists of changing one or more parameters from the parent solution (Section III-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' This patch is assessed by executing the failing test suite (Line 8), and the test execution results are obtained and stored as Scores (Section III-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' In a fourth step, the suspiciousness score of each parameter is recalculated (Line 9, Section III-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' Lastly, the Archive is updated (Line 10, Section III-D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' Patch generation A patch in our context refers to a mutation of at least one parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' Algorithm 2 shows our algorithm for proposing a potential patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' As input, it receives (1) a parent configuration, which corresponds to one configuration in the archive of the algorithm and (2) the suspiciousness ranking of all parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' First, a parameter to be mutated is selected (Line 4) based on the suspiciousness of each parameter (see Section III-C for more details on how to compute the suspiciousness score).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' The higher the suspiciousness, the higher the probability of being selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' The parameter to be mutated is obtained by employing Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' The selected parameter is mutated (Line 5) by giving a random value within its ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' After this, it is decided whether a new parameter is mutated (Line 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' The probability of mutating a new parameter decreases as the number of mutated parameters in the new patch increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' We ensure that one parameter is not mutated more than once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' Algorithm 2: Patch generation algorithm Input: Parent //Faulty Configuration SuspRanking //Suspiciousness Ranking Output: Patch //Mutated Configuration 1 numOfMutParams ← 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' 2 Patch ← Parent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' 3 do 4 varToMutate ← selectParam(SuspRanking);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' 5 Patch ← mutate(Patch,varToMutate);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' 6 numOfMutParams ← numOfMutParams +1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' 7 p ← rand();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' //returns random value 0 to 1 8 while p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content='5numOfMutatedP arams;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' 9 return Patch;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' Algorithm 3: Suspiciousness-based parameter selec- tion algorithm Input: SuspScore = {ss1, ss2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=', ssN} Output: selected //Index of the selected parameter 1 total ← �N i=1(ssi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' 2 iterativeSum←0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' 3 prob ← [];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' 4 for i ← 1 to nPop do 5 prob[i] ← iterativeSum + SuspScore[i]/total;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' 6 iterativeSum←prob[i];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' 7 end 8 prob←orderAscending(prob);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' 9 r←rand();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content='//Returns random number 0 to 1 10 j←0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' 11 selected=N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfhf3-/content/2301.01487v1.pdf'} +page_content=' 12 while j 1036 erg s−1) coinciding or remaining in a very close spatial +proximity of UHE gamma-ray sources, may be a universal feature +★ E-mail: agnibha@rri.res.in +(Albert et al. 2021), further theoretical analyses of these LHAASO +detected UHE gamma-ray sources seem to tell a different story. +Recent studies have modeled a few of the significantly detected +LHAASO sources in detail. For example, De Sarkar & Gupta (2022) +found that the UHE gamma-ray emission observed from LHAASO +J1908+0621 is most likely hadronic in origin, emanated from the +interaction between SNR G40.5-0.5 and the associated MCs. On the +other hand, in De Sarkar et al. (2022), another significantly detected +source, LHAASO J2226+6057, was extensively modeled assuming +that the UHE gamma-ray emission is coming from the PWN associ- +ated with PSR J2229+6114. As caveats of the model, it was found +that the PWN interpretation of LHAASO J2226+6057 leads to a very +high radius of PWN, as well as a very small value of magnetic field. +Naturally, these results are in contrast with the observational results +(Ge et al. 2021; Liang et al. 2022), thus indicating that the PWN may +not be the contributing source to power the UHE gamma-ray source +detected. This indicates interaction between SNRs and associated +MCs may be the primary reason behind particle acceleration to PeV +energies in Galactic sources. +With this factors in mind, we focus on the emission of the recent +LHAASO detected unidentified UHE gamma-ray sources: LHAASO +J2108+5157 (Cao et al. 2021c) and LHAASO J0341+5258 (Cao +et al. 2021b). Both of these sources were found to be associated with +MCs, but no apparent association with energetic pulsars or SNRs +were established. Scenarios including leptonic emission from TeV +halo (Abe et al. 2022), injection of particles from past explosions +(Kar & Gupta 2022), hadronic interaction between SNR and MCs +(Cao et al. 2021c) were discussed in previous literatures. But most of +these models do not explain the HE-VHE-UHE gamma-ray spectral +© 2022 The Authors +arXiv:2301.13451v1 [astro-ph.HE] 31 Jan 2023 + +2 +A. De Sarkar +features entirely. Moreover, recent reveal of VHE gamma-ray upper +limits observed by the Large-Sized Telescope - Cherenkov Telescope +Array (LST-CTA) (Abe et al. 2022) has overruled some of these +models for the case of LHAASO J2108+5157. The absence of a +powerful pulsar or supernova remnant adds to the mystery as well, +leaving one asking what is the possible emission mechanism at play +in case of these unidentified UHE gamma-ray sources. +To that end, in this letter, we discuss and apply a phenomeno- +logical model, in which accelerated particles, escaped from an old, +shell-type SNR (now invisible), interact with the associated MCs to +produce the observed HE-VHE-UHE gamma-ray data for the case +of LHAASO J2108+5157. We also provide the possible age of the +old SNR, and account for the disappearance of the SNR in question. +Our simple model is also consistent with the X-ray 2𝜎 upper limits +given by Abe et al. (2022). We also discuss the applicability of the +model in other unidentified Galactic UHE gamma-ray source such +as LHAASO J0341+5258. Furthermore, we report that the neutrino +flux produced from the hadronic interaction considered in this model, +will be non-detectable, even by the next generation observatory such +as ICECUBE-Gen2 (Aartsen et al. 2021). +2 THE MODEL +In this section, we discuss the essentials of the model used to calculate +the hadronic and leptonic components produced from the interaction +between an old, now invisible SNR and the associated MCs. A more +detailed discussion of the model can be found in De Sarkar & Gupta +(2022), where we developed and applied our model to explain the +peculiar HE-VHE-UHE gamma-ray SED of LHAASO J1908+0621. +Our simple model assumes that the supernova had exploded at the +center of the cavity of a shell-like structure, which is surrounded by +dense MCs. After this explosion, the SNR shock front expands inside +the shell cavity, and finally hits the surrounding MCs. During the +collision between the shock front and associated MCs, the accelerated +particles get injected into the MCs to produce further emissions. +After the explosion, the supernova (SN) shock front expands freely +during its free expansion phase. When the amount of swept-up inter- +stellar medium (ISM) material becomes equal to that of the ejected +material at t = t𝑆𝑒𝑑𝑜𝑣, the SN enters the adiabatic Sedov phase. Fi- +nally after t = t𝑟𝑎𝑑, the SN enters the radiative phase, in which the +cooling timescales is less than the dynamic timescales. During its +evolution, the time dependence of the shock velocity and radius can +be given by the following simple relations (Fujita et al. 2009; Ohira +et al. 2012; De Sarkar & Gupta 2022), +𝑣𝑠ℎ(𝑡) = +� +𝑣𝑖 +(𝑡 < 𝑡𝑆𝑒𝑑𝑜𝑣) +𝑣𝑖(𝑡/𝑡𝑆𝑒𝑑𝑜𝑣)−3/5 +(𝑡𝑆𝑒𝑑𝑜𝑣 < 𝑡) +(1) +and, +𝑅𝑠ℎ(𝑡) ∝ +� +(𝑡/𝑡𝑆𝑒𝑑𝑜𝑣) +(𝑡 < 𝑡𝑆𝑒𝑑𝑜𝑣) +(𝑡/𝑡𝑆𝑒𝑑𝑜𝑣)2/5 +(𝑡𝑆𝑒𝑑𝑜𝑣 < 𝑡) +(2) +We note that for the entirety of this work, we have assumed the +following values: initial velocity of the shock v𝑖 = 109 cm s−1 (Fujita +et al. 2009), radius of the shock and time at the beginning of the Sedov +phase, R𝑆𝑒𝑑𝑜𝑣 and t𝑆𝑒𝑑𝑜𝑣, to be 2.1 pc and 210 yr, respectively +(Ohira et al. 2011; Makino et al. 2019). +The CR protons are accelerated through Diffusive Shock Acceler- +ation (DSA) mechanism when the SN is in the Sedov phase, where +the CR protons accelerate by scattering back and forth across the +shock front, while the shock is expanding towards the surround- +ing MCs. We assume an escape-limited acceleration scenario (Ohira +et al. 2010), in which the CR protons need to escape a geometrical +confinement region around the SN shock front produced by strong +magnetic turbulence, in order to get injected into the MCs and take +part in further interactions. The radius of the outermost boundary +of this confinement region (i.e., the escape boundary) is called the +escape radius, and it can be denoted by, +𝑅𝑒𝑠𝑐(𝑡) = (1 + 𝜅)𝑅𝑠ℎ(𝑡), +(3) +where 𝜅 ≈ 0.04 (Ohira et al. 2010; Makino et al. 2019), and is +defined by the geometrical confinement condition D𝑠ℎ/v𝑠ℎ ∼ l𝑒𝑠𝑐 = +𝜅R𝑠ℎ, where l𝑒𝑠𝑐 is the distance of the escape boundary from the +shock front and D𝑠ℎ is the diffusion coefficient around the shock +(Ohira et al. 2010). +After the explosion, the escape boundary in front of the shock +front eventually hits the surrounding MCs after traversing a distance +of R𝑀𝐶, the distance of MC surface from the cavity center. This +essentially means that at the time of collision t𝑐𝑜𝑙𝑙, the escape ra- +dius is equal to the MC surface distance, i.e. R𝑒𝑠𝑐 (t𝑐𝑜𝑙𝑙) = R𝑀𝐶 +≈ R𝑠ℎ (t𝑐𝑜𝑙𝑙), and at the time of collision, the velocity of the shock +is denoted by v𝑠ℎ(t𝑐𝑜𝑙𝑙). We assume that the particle acceleration +stops at t = t𝑐𝑜𝑙𝑙 (Fujita et al. 2009). Consequently, protons accel- +erated at t ≤ t𝑐𝑜𝑙𝑙 take part in further interactions inside the MCs. +Moreover, only the protons with sufficiently high energies will es- +cape the confinement region around the SNR shock front, whereas +the low energy protons will remain confined around the SNR. So a +suppression of fluxes in the lower energies, as well as a dominant con- +tribution of fluxes in the highest energies should be expected in this +scenario. The confinement condition invoked in this model changes +the spectral shape of the injected proton population by constraining +the minimum energy limit. +We estimate the minimum energy limit of the injected proton pop- +ulation by assuming that the escape energy is a decreasing function +of the shock radius (Makino et al. 2019). This approach is based +on the assumption that the maximum energy of CR protons, E𝑝 +𝑚𝑎𝑥 +is expected to increase up to knee energy (≈ 1015.5 eV) until the +beginning of the Sedov phase, and then decrease from that epoch +(Gabici et al. 2009; Ohira et al. 2012). The minimum energy re- +quired by protons to escape the confinement region can be given by +the phenomenological relation, +𝐸 𝑝 +𝑒𝑠𝑐 = 𝐸 𝑝 +𝑚𝑎𝑥 +� +𝑅𝑠ℎ +𝑅𝑆𝑒𝑑𝑜𝑣 +�−𝛼 +, +(4) +where 𝛼 signifies the evolution of the maximum energy during the +Sedov phase (Makino et al. 2019). We treat 𝛼 as a free parameter in +this work. After putting R𝑠ℎ ≈ R𝑒𝑠𝑐 = R𝑀𝐶 at the time of collision, +we find the minimum energy required to escape the confinement +zone, which also gives us the minimum energy threshold for the +proton population that gets injected inside the surrounding MCs, i.e., +E𝑝 +𝑒𝑠𝑐 = E𝑝 +𝑚𝑖𝑛. Since protons are accelerated by DSA mechanism, we +can expect the CR proton spectrum at the shock front ∝ E−𝑠. Then, +in an escape-limited particle acceleration scenario, the protons with +energies greater than 𝐸 𝑝 +𝑒𝑠𝑐 have a spectrum (Ohira et al. 2010), +𝑁 𝑝 +𝑒𝑠𝑐(𝐸) ∝ 𝐸−[𝑠+(𝛽/𝛼)], +(5) +where 𝛽 represents a thermal leakage model of CR injection and +is given by 𝛽 = 3(3–s)/2 (Makino et al. 2019). For a typical value of +s = 2, we get the value of 𝛽 = 1.5. Note that the spectral shape as +MNRAS 000, 1–6 (2022) + +Supernova connection of LHAASO J2108+5157 +3 +well as the minimum energy of the proton population are calculated +at the time when the escape boundary hits the surrounding MCs at t += t𝑐𝑜𝑙𝑙. +At t > t𝑐𝑜𝑙𝑙, the shock enters the momentum conserving “snow- +plow” phase. The time evolution of the radius of the shocked shell +R𝑠ℎ𝑒𝑙𝑙 (t) inside the MCs can be found using momentum conserva- +tion equation (Fujita et al. 2009; De Sarkar & Gupta 2022), +4𝜋 +3 +� +𝑛𝑀𝐶 (𝑅𝑠ℎ𝑒𝑙𝑙(𝑡)3 − 𝑅𝑠ℎ(𝑡𝑐𝑜𝑙𝑙)3) + 𝑛𝑐𝑎𝑣 𝑅𝑠ℎ(𝑡𝑐𝑜𝑙𝑙)3� +�𝑅𝑠ℎ𝑒𝑙𝑙(𝑡) += 4𝜋 +3 𝑛𝑐𝑎𝑣 𝑅𝑠ℎ(𝑡𝑐𝑜𝑙𝑙)3𝑣𝑠ℎ(𝑡𝑐𝑜𝑙𝑙), +(6) +with R𝑠ℎ𝑒𝑙𝑙 = R𝑀𝐶 at t = t𝑐𝑜𝑙𝑙, n𝑀𝐶 is the number density of +the MCs, and n𝑐𝑎𝑣 (≈ 1 cm−3) is the number density inside the +cavity of the shell. We solve equation 6 numerically for t > t𝑐𝑜𝑙𝑙, +to estimate the current age of the SNR. We estimate the current +age by considering the fact that the velocity of the shocked shell +at the current age must be similar or even smaller than the internal +gas velocity of the MCs. This approach takes into account the non- +detection of any SNR shell in unidentified UHE gamma-ray sources +discussed above, as the shell of the SNR becomes invisible owing to +the higher internal gas velocity of the MCs as compared to that of the +shocked shell. We consider the above discussed proton population +and total number density of the cold protons inside the surrounding +MCs (n𝑀𝐶) to calculate total gamma-ray flux produced through +hadronic interaction (Kafexhiu et al. 2014). +Similar to protons, electrons can also get accelerated in the SNR +shock front and subsequently escape the confinement region to get +injected in the associated MCs. Moreover, electrons also lose energy +through radiative cooling very efficiently. Hence, the injected electron +population was considered to be escape-limited, as well as loss- +limited (Yamazaki et al. 2006). We consider the spectral index of +the escaped electron population to be same as that of protons (Ohira +et al. 2012; De Sarkar & Gupta 2022). To take into accout loss-limited +nature of injected electron population, we consider a power law with +exponential cutoff as the spectral shape of the escaped electrons, +𝑁𝑒 +𝑒𝑠𝑐(𝐸) ∝ 𝐸−[𝑠+(𝛽/𝛼)]𝑒𝑥𝑝(−𝐸/𝐸𝑒 +𝑚𝑎𝑥), +(7) +where, maximum energy of the electron population has been de- +termined by synchrotron cooling (Yamazaki et al. 2006; Fujita et al. +2009), +𝐸𝑒 +𝑚𝑎𝑥 = 14ℎ−1/2 +� +𝑣𝑠ℎ +108 cm/s +� � +𝐵 +10 𝜇G +�−1/2 +TeV, +(8) +where, v𝑠ℎ is the velocity of the shock front and B is the down- +stream magnetic field. The parameter h (= 0.05𝑟 ( 𝑓 +𝑟𝑔) +𝑟−1 +, where r is +the density compression ratio, f and g are functions of shock angle +and gyro-factors) is used as a factor to calculate the acceleration time +scale of DSA. We take h ∼ 1, considering the SNR in Sedov phase +and neglecting non-linear effects, similar to Yamazaki et al. (2006). +We consider v𝑠ℎ = v𝑠ℎ (t𝑐𝑜𝑙𝑙) since we calculate the maximum en- +ergy of the lepton population at the collision time and B = B𝑀𝐶, +the magnetic field inside the MCs. The minimum energy of the elec- +tron population was considered to be E𝑒 +𝑚𝑖𝑛 ≈ 500 MeV (De Sarkar +& Gupta 2022). Furthermore, we consider bremsstrahlung, Inverse- +Compton (IC) and synchrotron cooling (Blumenthal & Gould 1970; +Ghisellini et al. 1988; Baring et al. 1999) of the injected lepton +population to calculate the gamma-ray flux produced. For IC inter- +action, we consider interstellar radiation field (ISRF) from Popescu +et al. (2017) at the source position, and the Cosmic Microwave Back- +ground (temperature T𝐶𝑀 𝐵 = 2.7 K, energy density U𝐶𝑀 𝐵 = 0.25 +eV cm−3) contribution as well. The number density was considered +to be same as that of the MCs. +Finally we note that in this particular model, we have neglected +the effect of diffusion of particles inside the MCs and assumed that +the CR particles, both protons and electrons, lose energy through +rapid cooling before escaping the cloud. This assumption can be +realized by considering the idea that inside MCs, the diffusion is +considerably suppressed (D ≈ 1025−26 cm2 s−1) as compared to that +observed in the ISM (D ≈ 1028 cm2 s−1) (Gabici et al. 2007, 2009; +Fujita et al. 2009; De Sarkar et al. 2021). Generation of plasma waves +by CR streaming can be the reason behind the slow diffusion inside +the MCs (Wentzel 1974). On the other hand, if the trapping of CR +particles occurs due to some particular orientation of the magnetic +field inside the MCs, then also the escape of the particles from the +MCs will not be effective and can be neglected (Makino et al. 2019). +Consequently, we have considered a steady-state proton and electron +population to explain the SED of LHAASO J2108+5157, details of +which are given in the next section. +3 APPLICATION OF THE MODEL: LHAASO J2108+5157 +LHAASO J2108+5157 is an UHE gamma-ray source detected by +LHAASO at R.A. = 317.22◦ ± 0.07◦ +𝑠𝑡𝑎𝑡 and decl. = 51.95◦ ± 0.05◦ +𝑠𝑡𝑎𝑡 +(Cao et al. 2021c) with a significance of 6.4𝜎 above 100 TeV. The +source is reported to have a 95% confidence level extension upper +limit of 0.26◦ with a 2D symmetrical Gaussian template, and its +spectrum above 25 TeV can be well described by a power law with a +photon index of 2.83 ± 0.18 (Cao et al. 2021c). Although no X-ray +counterpart within 0.26◦ radius of the source was found, a spatially +extended, HE counterpart 4FGL J2108.0+5155e (extension ∼ 0.48◦) +(Abdollahi et al. 2020) was observed to be situated at an angular +distance of 0.13◦ (Cao et al. 2021c). A new hard spectrum GeV +source was also found at l = 92.35◦ and b = 2.56◦ by Fermi-LAT +data analysis (Abe et al. 2022), but its large angular separation (∼ +0.27◦) from the LHAASO source indicates that this new source can +hardly be a counterpart. Although no VHE component within 0.5◦ +radius was confirmed previously, recent observations by LST-CTA +has hinted towards an existence of a source with 3.67𝜎 detection +significance in the energy range of 3 - 100 TeV (Abe et al. 2022). +Future observations may confirm an existence of a VHE counterpart +with hard spectral index. The UHE source is located near the center +of a GMC labeled [MML2017]4607 (Miville-Deschênes et al. 2017), +which has an average angular radius and mass of 0.236◦ and 8469 +M⊙, respectively, and is situated at a distance of 3.28 kpc from +Earth. The average number density of the GMC was estimated to +be n𝑀𝐶 ≈ 30 cm−3 (Cao et al. 2021c). The presence of the GMC, +spatially coincident with the UHE gamma-ray source points towards +the hadronic origin, but leptonic origin can not be neglected. The +absence of any energetic pulsar, its wind nebula or SNR warrants a +cautious approach in unveiling the true nature of emission regarding +this UHE source. +Two young open stellar clusters Kronberger 80 and Kronberger 82 +are in the vicinity of the LHAASO source (with angular distances of +0.62◦ and 0.45◦, respectively) (Cao et al. 2021c). But large angular +separation between these clusters and LHAASO source centroid, as +well as absence of proper distance estimation hint that the contri- +bution of these clusters are unlikely (Cao et al. 2021c; Abe et al. +MNRAS 000, 1–6 (2022) + +4 +A. De Sarkar +2022). Cao et al. (2021c) suggested that UHE gamma-ray emission +is due to an interaction of escaping CRs with MCs, whereas the GeV +counterpart maybe due to an old SNR. However, Abe et al. (2022) +pointed out that photon index of GeV counterpart spectrum is too +soft compared to the observations of old SNRs interacting with MCs +(Yuan et al. 2012), and to produce UHE gamma-ray spectrum, the +required spectral index of the proton population has to be very hard +as compared to the standard DSA theory. Instead, Abe et al. (2022) +proposed an alternate leptonic scenario, in which UHE gamma-ray +emission is due to TeV halo emission, and the GeV counterpart is due +to a tentative, previously undetected pulsar. But a very low associated +magnetic field (even lower than the average Galactic magnetic field), +and non-detection of a pulsar make the TeV halo interpretation ques- +tionable, and open the source up for further exploration. To that end, +we apply the model discussed in Section 2 to explain the gamma-ray +data from HE to UHE energy range, while being consistent with the +X-ray 2𝜎 upper limits. We note that these 2𝜎 X-ray upper limits +correspond to a uniform, circular source with a radius of 6′ centered +on the position of the LHAASO source (Abe et al. 2022). We explain +the VHE-UHE gamma-ray data with hadronic component produced +from the interaction between protons, accelerated and escaped at an +early time from a now old SNR shock front, with protons inside +the surrounding MCs, whereas the HE gamma-ray data is explained +using bremsstrahlung cooling of accelerated and escaped electrons +inside the medium of the MCs. Our model also shows that the main +contribution in X-ray range comes from the synchrotron cooling of +the same accelerated and escaped electrons. +In this work, we have considered the free parameter 𝛼 = 1.875, and +then let the total energy budgets of proton and electron populations +vary to explain the MWL SED. Considering the value of 𝛼, and the +values of s and 𝛽 discussed in Section 2, we get the spectral indices +of the escaped electron and proton populations as p = [s + (𝛽/𝛼)] += 2.8. The distance of the source was taken to be d ∼ 3 kpc. The +model spectrum components, as well as the considered MWL SED +are shown in Figure 1. Also, we calculate the time evolution of SNR +shocked shell inside the associated MCs using equation 6, and find +that the SNR, with a final radius of ∼ 30 pc, has to be ∼ 4.4 × 105 +years old, for the shock velocity to be lower than the internal gas +velocity of MC [MML2017]4607 (∼ 13 km s−1) (Cao et al. 2021c), +and the SNR shell to disappear. The time evolution of the shocked +shell is shown in Figure 2. Finally, the model parameters required to +explain the gamma-ray data are shown in Table 1. We have used open +source code GAMERA (Hahn 2016) to calculate the model spectrum +of different components. +4 DISCUSSION AND CONCLUSION +In this letter, we have discussed and applied a simple, analytical and +phenomenological model to explain the HE-VHE-UHE gamma-ray +data observed from the direction of LHAASO J2108+5157. By only +adjusting the index 𝛼, not only we show that the model components +are consistent with gamma-ray and X-ray observations, the results +also naturally explain the observed morphology of the source re- +gion, e.g., the disappearance of the SNR at current age. As expected, +the SNR was found be old (> 105 years). This also explains why +no pulsar has been seen in the source region, as the pulsar is ex- +pected to leave the source region due to its natal kick velocity (∼ +400-500 km s−1) (Gaensler & Slane 2006). Similar nature and emis- +sion were also found in another UHE gamma-ray source, LHAASO +J1908+0621, details of which were explained by this model in De +Sarkar & Gupta (2022). The fact that the emission of multiple UHE +10 +12 +10 +10 +10 +8 +10 +6 +10 +4 +10 +2 +100 +102 +Energy (TeV) +10 +16 +10 +15 +10 +14 +10 +13 +10 +12 +10 +11 +E2 J(E)[erg cm +2 s +1] +pp +synchrotron +bremsstrahlung +inverse-compton +Fermi-LAT (Abe et al. 2022) +Fermi-LAT (Cao et al. 2021) +LHAASO +LST-CTA +XMM-Newton +Figure 1. MWL SED of LHAASO J2108+5157. Gamma-ray data points and +upper limits obtained from different observatories such as Fermi-LAT (red +(Abe et al. 2022), purple (Cao et al. 2021c)), LHAASO (blue (Cao et al. +2021c)), and LST-CTA (green (Abe et al. 2022)) are shown in the figure. The +XMM-Newton X-ray 2𝜎 upper limits (Abe et al. 2022) are given in teal. The +model p-p interaction (solid line), bremsstrahlung (dashed), IC (dotted), and +synchrotron (dot-dashed) components are also shown in the figure. +103 +104 +105 +106 +Time (years) +16 +18 +20 +22 +24 +26 +28 +30 +32 +Shock radius (pc) +LHAASO J2108+5157 +Figure 2. Time evolution of the shocked shell associated with the old SNR, +inside the surrounding MCs. +gamma-ray sources were explained by the same model hints towards +its validity in a larger context. Interestingly, another unidentified +UHE gamma-ray source, LHAASO J0341+5258, also shows similar +characteristics shown by LHAASO J2108+5157 (Cao et al. 2021b). +It is very likely that this model is applicable in that case as well. +However, in that case, the VHE counterpart has not been properly +constrained, and the High Altitude Water Cherenkov (HAWC) upper +limit provided in Cao et al. (2021b) corresponds to only a 2𝜎 detec- +tion significance. Further observations by CTA and detailed analysis +by Fermi-LAT will be necessary to properly constrain the emission +of LHAASO J0341+5258. +From Figure 1, we can see that the hadronic component adequately +explain the VHE-UHE gamma-ray data, whereas the bremsstrahlung +component, originated from the cooling of the electron population, +explains the gamma-ray data in the HE range. The bremsstrahlung +MNRAS 000, 1–6 (2022) + +Supernova connection of LHAASO J2108+5157 +5 +Table 1. Parameters Used in The Model. +Definition +Parameter +Value +SNR/MC structure and evolution: +Initial shock velocity +v𝑖 (cm/s) +109 +Time at the start of Sedov phase +t𝑆𝑒𝑑𝑜𝑣 (years) +210 +Shock radius at the start of Sedov phase R𝑆𝑒𝑑𝑜𝑣 (pc) +2.1 +Time of collision +t𝑐𝑜𝑙𝑙 (years) +3.83 × 103 +Shock radius at time of collision +R𝑠ℎ (t𝑐𝑜𝑙𝑙) (pc) +16.77 (= R𝑀𝐶) +Shock velocity at time of collision +v𝑠ℎ (t𝑐𝑜𝑙𝑙) (cm/s) 1.75 × 108 +Current age of SNR +t𝑎𝑔𝑒 (years) +4.4 × 105 +Final radius of shock +R𝑠ℎ (t𝑎𝑔𝑒) (pc) +30 +Final velocity of shock +v𝑠ℎ (t𝑎𝑔𝑒) (cm/s) 1.2 × 106 +Distance +d (kpc) +3 +MC number density +n𝑀𝐶 (cm−3) +30 +MC magnetic field +B𝑀𝐶 (𝜇G) +25 +Cavity number density +n𝑐𝑎𝑣 (cm−3) +1 +Hadronic component: +Minimum energy +E𝑝 +𝑚𝑖𝑛 (TeV) +63 +Maximum energy +E𝑝 +𝑚𝑎𝑥 (TeV) +3.1 × 103 +Spectral index +p +2.8 +Energy budget +W𝑝 (erg) +3.6 × 1047 +Leptonic component: +Minimum energy +E𝑒 +𝑚𝑖𝑛 (TeV) +5 × 10−4 +Maximum energy +E𝑒𝑚𝑎𝑥 (TeV) +15.5 +Spectral index +p +2.8 +Energy budget +W𝑒 (erg) +3.6 × 1047 +component is expected to dominate the IC component, as the in- +teraction is taking place inside MCs with a high number density +of cold protons. Moreover, the synchrotron component does not +violate the X-ray 2𝜎 upper limits. We note that no proper radio +counterpart has been associated with the LHAASO J2108+5157 yet. +An extended radio source associated with nearby star-forming re- +gion (Cao et al. 2021c), as well as point-like radio source NVSS +210803+515255 or WENSS B2106.4+5140 (Abe et al. 2022) were +found within 95% extension upper limit of LHAASO J2108+5157 +and 4FGL J2108.0+5155e. Since no proper association was estab- +lished between these sources and the gamma-ray source, we refrain +from including their radio data in this study to further constrain the +model, and we follow the MWL SED discussed in (Abe et al. 2022) +to ascertain the feasibility of the model discussed in this letter. +As discussed earlier, we have neglected the effect of particle diffu- +sion in this model. We note that such an assumption may likely lead +to an overestimation, and the aspect of suppressed diffusion inside +the MCs is highly uncertain (Xu et al. 2016; Dogiel et al. 2015). In- +troducing an energy-independent diffusion coefficient, as discussed +in Dogiel et al. (2015), will lead to higher energy budgets required +by the electron and proton populations to explain the data. The sup- +pressed diffusion coefficient introduced by Gabici et al. (2009) has +similar energy dependence as to that observed in ISM, but the exact +energy dependence of diffusion coefficient inside clouds is not well +constrained. So, to avoid further complications, we have neglected +the effect of diffusion in this model, similar to Fujita et al. (2009); +Makino et al. (2019), and assumed that the injected particles quickly +cool down before escaping the MC medium. +We further note that we do not consider the contribution of accel- +erated and escaped particles, when the shock front is within the MC +medium, in calculating the total gamma-ray SED. Even if the SNR +is still in the Sedov phase when the shock is within the MCs, the +corresponding contribution was found to be negligible. Moreover, +the acceleration and subsequent escape of particles, in that case, will +depend on the evolution of the confinement region within the high- +density, turbulent medium of the MCs, details of which is beyond the +scope of the simple model discussed in this letter. Furthermore, as +the SNR enters its radiative phase at t𝑟𝑎𝑑 ∼ 4 × 104 years (Blondin +et al. 1998), the particle acceleration becomes ineffective as the small +shock velocity at that age, as obtained from equation 6 (< 1.1 × 107 +cm/s), prevents full ionization of the pre-shock gas (Shull & McKee +1979). So no significant contribution to the total gamma-ray SED is +expected in the radiative phase of the SNR as well. +Since hadronic component primarily dominates in the VHE-UHE +gamma-ray range, neutrinos can be produced from the hadronic in- +teraction as well. This neutrino flux can be a smoking gun evidence +for the dominant hadronic interaction. We have calculated the neu- +trino flux resulting from the hadronic interaction discussed above, +and found that the corresponding neutrino flux is too low to be de- +tected by current generation neutrino telescope such as ICECUBE. +Furthermore, we have found that the model neutrino flux does not +exceed the 5𝜎 discovery potential after 10 years of observation by +next generation neutrino observatory ICECUBE-Gen2 for two de- +clinations, 𝛿 = 0◦ and 30◦ (Aartsen et al. 2021), which indicates +that it is unlikely to confirm the hadronic nature of UHE gamma-ray +emission through neutrino observations, even in the near future, for +this source. +In conclusion, in this letter, we have shown that by essentially +tuning the 𝛼 index, the emission of the LHAASO source can be +explained. We note that we do not intend to “fit” the MWL SED, +as the SED, in various energy ranges (VHE, X-ray, radio), is poorly +constrained and in need of further observations. In this work, we +have only applied a simple phenomenological model, while also +minimizing the free parameters, which naturally explains the spec- +tral features and spatial morphology of LHAASO J2108+5157. Fu- +ture observations can confirm the viability of this model to ex- +plain LHAASO J2108+5157, or other unidentified UHE gamma-ray +source LHAASO 0341+5258, and sources detected in future as well, +which show similar nature and emission signatures. If confirmed, +then it can be posited that SNRs as a source class, similar to PWNe, +can likely be a strong candidate for being the Galactic PeVatrons. +ACKNOWLEDGEMENTS +I thank the anonymous reviewer for helpful comments and construc- +tive criticism. I thank Nayantara Gupta for encouragement. +DATA AVAILABILITY +The simulated data underlying this paper will be shared on reasonable +request to the corresponding author. +REFERENCES +Aartsen M. 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G., 1974, ARA&A, 12, 71 +Xu S., Yan H., Lazarian A., 2016, ApJ, 826, 166 +Yamazaki R., Kohri K., Bamba A., Yoshida T., Tsuribe T., Takahara F., 2006, +MNRAS, 371, 1975 +Yuan Q., Liu S., Bi X., 2012, ApJ, 761, 133 +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–6 (2022) + diff --git a/BNFQT4oBgHgl3EQf9jeh/content/tmp_files/load_file.txt b/BNFQT4oBgHgl3EQf9jeh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..791db2c93f13790739f0d53a8574c8e8623a1738 --- /dev/null +++ b/BNFQT4oBgHgl3EQf9jeh/content/tmp_files/load_file.txt @@ -0,0 +1,483 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf,len=482 +page_content='MNRAS 000, 1–6 (2022) Preprint 1 February 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='0 Supernova connection of unidentified ultra high energy gamma-ray source LHAASO J2108+5157 Agnibha De Sarkar,1★ 1Astronomy & Astrophysics group, Raman Research Institute C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Raman Avenue, 5th Cross Road, Sadashivanagar, Bengaluru 560080, Karnataka, India Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' in original form ZZZ ABSTRACT We present a simple phenomenological model of hadronic interaction between protons accelerated in an old supernova remnant (SNR) and cold protons situated within the associated molecular clouds (MCs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' The accelerated protons from the old SNR escaped the SNR shock front, and got injected into the MCs at an earlier time, producing ultra high energy gamma-rays and neutrinos through inelastic proton-proton interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' We also take into account the acceleration and subsequent escape of electrons from the SNR shock front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' The escaped electrons produce gamma-rays through various radiative cooling mechanisms, after getting injected into the MCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' We use the model discussed in this letter to explain the multiwavelength (MWL) spectral energy distribution (SED) of unidentified Galactic ultra high energy gamma-ray source LHAASO J2108+5157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' We also discuss the feasibility of applying this model in other cases as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Future observations can test the viability of the model discussed in this letter, which will in turn confirm that the SNRs can, in fact, accelerate particles up to PeV energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Key words: radiation mechanisms: non-thermal – ISM: individual objects: LHAASO J2108+5157 – gamma-rays: ISM – ISM: supernova remnants 1 INTRODUCTION Observations by the Large High Altitude Air Shower Observatory (LHAASO), located in China, have opened a new era of gamma- ray astrophysics (Cao 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Since it has become operational on 2020 April, LHAASO has detected more than a dozen ultra high energy (UHE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' E𝛾 ≥ 100 TeV) gamma-ray sources, most of which are unidentified (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' The detection of these UHE gamma- ray sources indicates the presence of cosmic ray (CR) accelerators in our Milky Way Galaxy, which can accelerate particles up to PeV (= 1015 eV) energies, more commonly known as “PeVatrons”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Several classes of Galactic sources such as supernova remnants (SNRs), pul- sar wind nebulae (PWNe), young stellar clusters have been posited to be potential PeVatron candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Although it is still an open question as to what class of source is responsible for accelerating particles up to PeV energies, most of the UHE gamma-ray sources detected by LHAASO, along with their high energy (HE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' E𝛾 < 100 GeV) and very high energy (VHE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 100 GeV ≤ E𝛾 < 100 TeV) gamma-ray counterparts, have been associated with PWNe in previous studies, due to their close proximity with an energetic pulsar, and their typ- ically extended spatial morphology (Abdalla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' This idea gained steam after it was confirmed that Crab pulsar wind nebula is indeed a PeVatron source (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' However, in spite of the notion that energetic pulsars with high spin-down luminosity ( �𝐸 > 1036 erg s−1) coinciding or remaining in a very close spatial proximity of UHE gamma-ray sources, may be a universal feature ★ E-mail: agnibha@rri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='in (Albert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2021), further theoretical analyses of these LHAASO detected UHE gamma-ray sources seem to tell a different story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Recent studies have modeled a few of the significantly detected LHAASO sources in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' For example, De Sarkar & Gupta (2022) found that the UHE gamma-ray emission observed from LHAASO J1908+0621 is most likely hadronic in origin, emanated from the interaction between SNR G40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='5-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='5 and the associated MCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' On the other hand, in De Sarkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' (2022), another significantly detected source, LHAASO J2226+6057, was extensively modeled assuming that the UHE gamma-ray emission is coming from the PWN associ- ated with PSR J2229+6114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' As caveats of the model, it was found that the PWN interpretation of LHAASO J2226+6057 leads to a very high radius of PWN, as well as a very small value of magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Naturally, these results are in contrast with the observational results (Ge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2022), thus indicating that the PWN may not be the contributing source to power the UHE gamma-ray source detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' This indicates interaction between SNRs and associated MCs may be the primary reason behind particle acceleration to PeV energies in Galactic sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' With this factors in mind, we focus on the emission of the recent LHAASO detected unidentified UHE gamma-ray sources: LHAASO J2108+5157 (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2021c) and LHAASO J0341+5258 (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Both of these sources were found to be associated with MCs, but no apparent association with energetic pulsars or SNRs were established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Scenarios including leptonic emission from TeV halo (Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2022), injection of particles from past explosions (Kar & Gupta 2022), hadronic interaction between SNR and MCs (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2021c) were discussed in previous literatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' But most of these models do not explain the HE-VHE-UHE gamma-ray spectral © 2022 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='13451v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='HE] 31 Jan 2023 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' De Sarkar features entirely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Moreover, recent reveal of VHE gamma-ray upper limits observed by the Large-Sized Telescope - Cherenkov Telescope Array (LST-CTA) (Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2022) has overruled some of these models for the case of LHAASO J2108+5157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' The absence of a powerful pulsar or supernova remnant adds to the mystery as well, leaving one asking what is the possible emission mechanism at play in case of these unidentified UHE gamma-ray sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' To that end, in this letter, we discuss and apply a phenomeno- logical model, in which accelerated particles, escaped from an old, shell-type SNR (now invisible), interact with the associated MCs to produce the observed HE-VHE-UHE gamma-ray data for the case of LHAASO J2108+5157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' We also provide the possible age of the old SNR, and account for the disappearance of the SNR in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Our simple model is also consistent with the X-ray 2𝜎 upper limits given by Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' We also discuss the applicability of the model in other unidentified Galactic UHE gamma-ray source such as LHAASO J0341+5258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Furthermore, we report that the neutrino flux produced from the hadronic interaction considered in this model, will be non-detectable, even by the next generation observatory such as ICECUBE-Gen2 (Aartsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2 THE MODEL In this section, we discuss the essentials of the model used to calculate the hadronic and leptonic components produced from the interaction between an old, now invisible SNR and the associated MCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' A more detailed discussion of the model can be found in De Sarkar & Gupta (2022), where we developed and applied our model to explain the peculiar HE-VHE-UHE gamma-ray SED of LHAASO J1908+0621.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Our simple model assumes that the supernova had exploded at the center of the cavity of a shell-like structure, which is surrounded by dense MCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' After this explosion, the SNR shock front expands inside the shell cavity, and finally hits the surrounding MCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' During the collision between the shock front and associated MCs, the accelerated particles get injected into the MCs to produce further emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' After the explosion, the supernova (SN) shock front expands freely during its free expansion phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' When the amount of swept-up inter- stellar medium (ISM) material becomes equal to that of the ejected material at t = t𝑆𝑒𝑑𝑜𝑣, the SN enters the adiabatic Sedov phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Fi- nally after t = t𝑟𝑎𝑑, the SN enters the radiative phase, in which the cooling timescales is less than the dynamic timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' During its evolution, the time dependence of the shock velocity and radius can be given by the following simple relations (Fujita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Ohira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' De Sarkar & Gupta 2022), 𝑣𝑠ℎ(𝑡) = � 𝑣𝑖 (𝑡 < 𝑡𝑆𝑒𝑑𝑜𝑣) 𝑣𝑖(𝑡/𝑡𝑆𝑒𝑑𝑜𝑣)−3/5 (𝑡𝑆𝑒𝑑𝑜𝑣 < 𝑡) (1) and, 𝑅𝑠ℎ(𝑡) ∝ � (𝑡/𝑡𝑆𝑒𝑑𝑜𝑣) (𝑡 < 𝑡𝑆𝑒𝑑𝑜𝑣) (𝑡/𝑡𝑆𝑒𝑑𝑜𝑣)2/5 (𝑡𝑆𝑒𝑑𝑜𝑣 < 𝑡) (2) We note that for the entirety of this work, we have assumed the following values: initial velocity of the shock v𝑖 = 109 cm s−1 (Fujita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2009), radius of the shock and time at the beginning of the Sedov phase, R𝑆𝑒𝑑𝑜𝑣 and t𝑆𝑒𝑑𝑜𝑣, to be 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='1 pc and 210 yr, respectively (Ohira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Makino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' The CR protons are accelerated through Diffusive Shock Acceler- ation (DSA) mechanism when the SN is in the Sedov phase, where the CR protons accelerate by scattering back and forth across the shock front, while the shock is expanding towards the surround- ing MCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' We assume an escape-limited acceleration scenario (Ohira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2010), in which the CR protons need to escape a geometrical confinement region around the SN shock front produced by strong magnetic turbulence, in order to get injected into the MCs and take part in further interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' The radius of the outermost boundary of this confinement region (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=', the escape boundary) is called the escape radius, and it can be denoted by, 𝑅𝑒𝑠𝑐(𝑡) = (1 + 𝜅)𝑅𝑠ℎ(𝑡), (3) where 𝜅 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='04 (Ohira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Makino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2019), and is defined by the geometrical confinement condition D𝑠ℎ/v𝑠ℎ ∼ l𝑒𝑠𝑐 = 𝜅R𝑠ℎ, where l𝑒𝑠𝑐 is the distance of the escape boundary from the shock front and D𝑠ℎ is the diffusion coefficient around the shock (Ohira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' After the explosion, the escape boundary in front of the shock front eventually hits the surrounding MCs after traversing a distance of R𝑀𝐶, the distance of MC surface from the cavity center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' This essentially means that at the time of collision t𝑐𝑜𝑙𝑙, the escape ra- dius is equal to the MC surface distance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' R𝑒𝑠𝑐 (t𝑐𝑜𝑙𝑙) = R𝑀𝐶 ≈ R𝑠ℎ (t𝑐𝑜𝑙𝑙), and at the time of collision, the velocity of the shock is denoted by v𝑠ℎ(t𝑐𝑜𝑙𝑙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' We assume that the particle acceleration stops at t = t𝑐𝑜𝑙𝑙 (Fujita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Consequently, protons accel- erated at t ≤ t𝑐𝑜𝑙𝑙 take part in further interactions inside the MCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Moreover, only the protons with sufficiently high energies will es- cape the confinement region around the SNR shock front, whereas the low energy protons will remain confined around the SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' So a suppression of fluxes in the lower energies, as well as a dominant con- tribution of fluxes in the highest energies should be expected in this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' The confinement condition invoked in this model changes the spectral shape of the injected proton population by constraining the minimum energy limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' We estimate the minimum energy limit of the injected proton pop- ulation by assuming that the escape energy is a decreasing function of the shock radius (Makino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' This approach is based on the assumption that the maximum energy of CR protons, E𝑝 𝑚𝑎𝑥 is expected to increase up to knee energy (≈ 1015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='5 eV) until the beginning of the Sedov phase, and then decrease from that epoch (Gabici et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Ohira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' The minimum energy re- quired by protons to escape the confinement region can be given by the phenomenological relation, 𝐸 𝑝 𝑒𝑠𝑐 = 𝐸 𝑝 𝑚𝑎𝑥 � 𝑅𝑠ℎ 𝑅𝑆𝑒𝑑𝑜𝑣 �−𝛼 , (4) where 𝛼 signifies the evolution of the maximum energy during the Sedov phase (Makino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' We treat 𝛼 as a free parameter in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' After putting R𝑠ℎ ≈ R𝑒𝑠𝑐 = R𝑀𝐶 at the time of collision, we find the minimum energy required to escape the confinement zone, which also gives us the minimum energy threshold for the proton population that gets injected inside the surrounding MCs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=', E𝑝 𝑒𝑠𝑐 = E𝑝 𝑚𝑖𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Since protons are accelerated by DSA mechanism, we can expect the CR proton spectrum at the shock front ∝ E−𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Then, in an escape-limited particle acceleration scenario, the protons with energies greater than 𝐸 𝑝 𝑒𝑠𝑐 have a spectrum (Ohira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2010), 𝑁 𝑝 𝑒𝑠𝑐(𝐸) ∝ 𝐸−[𝑠+(𝛽/𝛼)], (5) where 𝛽 represents a thermal leakage model of CR injection and is given by 𝛽 = 3(3–s)/2 (Makino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' For a typical value of s = 2, we get the value of 𝛽 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Note that the spectral shape as MNRAS 000, 1–6 (2022) Supernova connection of LHAASO J2108+5157 3 well as the minimum energy of the proton population are calculated at the time when the escape boundary hits the surrounding MCs at t = t𝑐𝑜𝑙𝑙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' At t > t𝑐𝑜𝑙𝑙, the shock enters the momentum conserving “snow- plow” phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' The time evolution of the radius of the shocked shell R𝑠ℎ𝑒𝑙𝑙 (t) inside the MCs can be found using momentum conserva- tion equation (Fujita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' De Sarkar & Gupta 2022), 4𝜋 3 � 𝑛𝑀𝐶 (𝑅𝑠ℎ𝑒𝑙𝑙(𝑡)3 − 𝑅𝑠ℎ(𝑡𝑐𝑜𝑙𝑙)3) + 𝑛𝑐𝑎𝑣 𝑅𝑠ℎ(𝑡𝑐𝑜𝑙𝑙)3� �𝑅𝑠ℎ𝑒𝑙𝑙(𝑡) = 4𝜋 3 𝑛𝑐𝑎𝑣 𝑅𝑠ℎ(𝑡𝑐𝑜𝑙𝑙)3𝑣𝑠ℎ(𝑡𝑐𝑜𝑙𝑙), (6) with R𝑠ℎ𝑒𝑙𝑙 = R𝑀𝐶 at t = t𝑐𝑜𝑙𝑙, n𝑀𝐶 is the number density of the MCs, and n𝑐𝑎𝑣 (≈ 1 cm−3) is the number density inside the cavity of the shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' We solve equation 6 numerically for t > t𝑐𝑜𝑙𝑙, to estimate the current age of the SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' We estimate the current age by considering the fact that the velocity of the shocked shell at the current age must be similar or even smaller than the internal gas velocity of the MCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' This approach takes into account the non- detection of any SNR shell in unidentified UHE gamma-ray sources discussed above, as the shell of the SNR becomes invisible owing to the higher internal gas velocity of the MCs as compared to that of the shocked shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' We consider the above discussed proton population and total number density of the cold protons inside the surrounding MCs (n𝑀𝐶) to calculate total gamma-ray flux produced through hadronic interaction (Kafexhiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Similar to protons, electrons can also get accelerated in the SNR shock front and subsequently escape the confinement region to get injected in the associated MCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Moreover, electrons also lose energy through radiative cooling very efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Hence, the injected electron population was considered to be escape-limited, as well as loss- limited (Yamazaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' We consider the spectral index of the escaped electron population to be same as that of protons (Ohira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' De Sarkar & Gupta 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' To take into accout loss-limited nature of injected electron population, we consider a power law with exponential cutoff as the spectral shape of the escaped electrons, 𝑁𝑒 𝑒𝑠𝑐(𝐸) ∝ 𝐸−[𝑠+(𝛽/𝛼)]𝑒𝑥𝑝(−𝐸/𝐸𝑒 𝑚𝑎𝑥), (7) where, maximum energy of the electron population has been de- termined by synchrotron cooling (Yamazaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Fujita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2009), 𝐸𝑒 𝑚𝑎𝑥 = 14ℎ−1/2 � 𝑣𝑠ℎ 108 cm/s � � 𝐵 10 𝜇G �−1/2 TeV, (8) where, v𝑠ℎ is the velocity of the shock front and B is the down- stream magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' The parameter h (= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='05𝑟 ( 𝑓 +𝑟𝑔) 𝑟−1 , where r is the density compression ratio, f and g are functions of shock angle and gyro-factors) is used as a factor to calculate the acceleration time scale of DSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' We take h ∼ 1, considering the SNR in Sedov phase and neglecting non-linear effects, similar to Yamazaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' We consider v𝑠ℎ = v𝑠ℎ (t𝑐𝑜𝑙𝑙) since we calculate the maximum en- ergy of the lepton population at the collision time and B = B𝑀𝐶, the magnetic field inside the MCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' The minimum energy of the elec- tron population was considered to be E𝑒 𝑚𝑖𝑛 ≈ 500 MeV (De Sarkar & Gupta 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Furthermore, we consider bremsstrahlung, Inverse- Compton (IC) and synchrotron cooling (Blumenthal & Gould 1970;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Ghisellini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Baring et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 1999) of the injected lepton population to calculate the gamma-ray flux produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' For IC inter- action, we consider interstellar radiation field (ISRF) from Popescu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' (2017) at the source position, and the Cosmic Microwave Back- ground (temperature T𝐶𝑀 𝐵 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='7 K, energy density U𝐶𝑀 𝐵 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='25 eV cm−3) contribution as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' The number density was considered to be same as that of the MCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Finally we note that in this particular model, we have neglected the effect of diffusion of particles inside the MCs and assumed that the CR particles, both protons and electrons, lose energy through rapid cooling before escaping the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' This assumption can be realized by considering the idea that inside MCs, the diffusion is considerably suppressed (D ≈ 1025−26 cm2 s−1) as compared to that observed in the ISM (D ≈ 1028 cm2 s−1) (Gabici et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2007, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Fujita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' De Sarkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Generation of plasma waves by CR streaming can be the reason behind the slow diffusion inside the MCs (Wentzel 1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' On the other hand, if the trapping of CR particles occurs due to some particular orientation of the magnetic field inside the MCs, then also the escape of the particles from the MCs will not be effective and can be neglected (Makino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Consequently, we have considered a steady-state proton and electron population to explain the SED of LHAASO J2108+5157, details of which are given in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 3 APPLICATION OF THE MODEL: LHAASO J2108+5157 LHAASO J2108+5157 is an UHE gamma-ray source detected by LHAASO at R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' = 317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='22◦ ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='07◦ 𝑠𝑡𝑎𝑡 and decl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' = 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='95◦ ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='05◦ 𝑠𝑡𝑎𝑡 (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2021c) with a significance of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='4𝜎 above 100 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' The source is reported to have a 95% confidence level extension upper limit of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='26◦ with a 2D symmetrical Gaussian template, and its spectrum above 25 TeV can be well described by a power law with a photon index of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='18 (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2021c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Although no X-ray counterpart within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='26◦ radius of the source was found, a spatially extended, HE counterpart 4FGL J2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='0+5155e (extension ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='48◦) (Abdollahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2020) was observed to be situated at an angular distance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='13◦ (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2021c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' A new hard spectrum GeV source was also found at l = 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='35◦ and b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='56◦ by Fermi-LAT data analysis (Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2022), but its large angular separation (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='27◦) from the LHAASO source indicates that this new source can hardly be a counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Although no VHE component within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='5◦ radius was confirmed previously, recent observations by LST-CTA has hinted towards an existence of a source with 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='67𝜎 detection significance in the energy range of 3 - 100 TeV (Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Future observations may confirm an existence of a VHE counterpart with hard spectral index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' The UHE source is located near the center of a GMC labeled [MML2017]4607 (Miville-Deschênes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2017), which has an average angular radius and mass of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='236◦ and 8469 M⊙, respectively, and is situated at a distance of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='28 kpc from Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' The average number density of the GMC was estimated to be n𝑀𝐶 ≈ 30 cm−3 (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2021c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' The presence of the GMC, spatially coincident with the UHE gamma-ray source points towards the hadronic origin, but leptonic origin can not be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' The absence of any energetic pulsar, its wind nebula or SNR warrants a cautious approach in unveiling the true nature of emission regarding this UHE source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Two young open stellar clusters Kronberger 80 and Kronberger 82 are in the vicinity of the LHAASO source (with angular distances of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='62◦ and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='45◦, respectively) (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2021c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' But large angular separation between these clusters and LHAASO source centroid, as well as absence of proper distance estimation hint that the contri- bution of these clusters are unlikely (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2021c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' MNRAS 000, 1–6 (2022) 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' De Sarkar 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' (2021c) suggested that UHE gamma-ray emission is due to an interaction of escaping CRs with MCs, whereas the GeV counterpart maybe due to an old SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' However, Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' (2022) pointed out that photon index of GeV counterpart spectrum is too soft compared to the observations of old SNRs interacting with MCs (Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2012), and to produce UHE gamma-ray spectrum, the required spectral index of the proton population has to be very hard as compared to the standard DSA theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Instead, Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' (2022) proposed an alternate leptonic scenario, in which UHE gamma-ray emission is due to TeV halo emission, and the GeV counterpart is due to a tentative, previously undetected pulsar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' But a very low associated magnetic field (even lower than the average Galactic magnetic field), and non-detection of a pulsar make the TeV halo interpretation ques- tionable, and open the source up for further exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' To that end, we apply the model discussed in Section 2 to explain the gamma-ray data from HE to UHE energy range, while being consistent with the X-ray 2𝜎 upper limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' We note that these 2𝜎 X-ray upper limits correspond to a uniform, circular source with a radius of 6′ centered on the position of the LHAASO source (Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' We explain the VHE-UHE gamma-ray data with hadronic component produced from the interaction between protons, accelerated and escaped at an early time from a now old SNR shock front, with protons inside the surrounding MCs, whereas the HE gamma-ray data is explained using bremsstrahlung cooling of accelerated and escaped electrons inside the medium of the MCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Our model also shows that the main contribution in X-ray range comes from the synchrotron cooling of the same accelerated and escaped electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' In this work, we have considered the free parameter 𝛼 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='875, and then let the total energy budgets of proton and electron populations vary to explain the MWL SED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Considering the value of 𝛼, and the values of s and 𝛽 discussed in Section 2, we get the spectral indices of the escaped electron and proton populations as p = [s + (𝛽/𝛼)] = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' The distance of the source was taken to be d ∼ 3 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' The model spectrum components, as well as the considered MWL SED are shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Also, we calculate the time evolution of SNR shocked shell inside the associated MCs using equation 6, and find that the SNR, with a final radius of ∼ 30 pc, has to be ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='4 × 105 years old, for the shock velocity to be lower than the internal gas velocity of MC [MML2017]4607 (∼ 13 km s−1) (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2021c), and the SNR shell to disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' The time evolution of the shocked shell is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Finally, the model parameters required to explain the gamma-ray data are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' We have used open source code GAMERA (Hahn 2016) to calculate the model spectrum of different components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 4 DISCUSSION AND CONCLUSION In this letter, we have discussed and applied a simple, analytical and phenomenological model to explain the HE-VHE-UHE gamma-ray data observed from the direction of LHAASO J2108+5157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' By only adjusting the index 𝛼, not only we show that the model components are consistent with gamma-ray and X-ray observations, the results also naturally explain the observed morphology of the source re- gion, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=', the disappearance of the SNR at current age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' As expected, the SNR was found be old (> 105 years).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' This also explains why no pulsar has been seen in the source region, as the pulsar is ex- pected to leave the source region due to its natal kick velocity (∼ 400-500 km s−1) (Gaensler & Slane 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Similar nature and emis- sion were also found in another UHE gamma-ray source, LHAASO J1908+0621, details of which were explained by this model in De Sarkar & Gupta (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' The fact that the emission of multiple UHE 10 12 10 10 10 8 10 6 10 4 10 2 100 102 Energy (TeV) 10 16 10 15 10 14 10 13 10 12 10 11 E2 J(E)[erg cm 2 s 1] pp synchrotron bremsstrahlung inverse-compton Fermi-LAT (Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2022) Fermi-LAT (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2021) LHAASO LST-CTA XMM-Newton Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' MWL SED of LHAASO J2108+5157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Gamma-ray data points and upper limits obtained from different observatories such as Fermi-LAT (red (Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2022), purple (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2021c)), LHAASO (blue (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2021c)), and LST-CTA (green (Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2022)) are shown in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' The XMM-Newton X-ray 2𝜎 upper limits (Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2022) are given in teal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' The model p-p interaction (solid line), bremsstrahlung (dashed), IC (dotted), and synchrotron (dot-dashed) components are also shown in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 103 104 105 106 Time (years) 16 18 20 22 24 26 28 30 32 Shock radius (pc) LHAASO J2108+5157 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Time evolution of the shocked shell associated with the old SNR, inside the surrounding MCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' gamma-ray sources were explained by the same model hints towards its validity in a larger context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Interestingly, another unidentified UHE gamma-ray source, LHAASO J0341+5258, also shows similar characteristics shown by LHAASO J2108+5157 (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' It is very likely that this model is applicable in that case as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' However, in that case, the VHE counterpart has not been properly constrained, and the High Altitude Water Cherenkov (HAWC) upper limit provided in Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' (2021b) corresponds to only a 2𝜎 detec- tion significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Further observations by CTA and detailed analysis by Fermi-LAT will be necessary to properly constrain the emission of LHAASO J0341+5258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' From Figure 1, we can see that the hadronic component adequately explain the VHE-UHE gamma-ray data, whereas the bremsstrahlung component, originated from the cooling of the electron population, explains the gamma-ray data in the HE range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' The bremsstrahlung MNRAS 000, 1–6 (2022) Supernova connection of LHAASO J2108+5157 5 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Parameters Used in The Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Definition Parameter Value SNR/MC structure and evolution: Initial shock velocity v𝑖 (cm/s) 109 Time at the start of Sedov phase t𝑆𝑒𝑑𝑜𝑣 (years) 210 Shock radius at the start of Sedov phase R𝑆𝑒𝑑𝑜𝑣 (pc) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='1 Time of collision t𝑐𝑜𝑙𝑙 (years) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='83 × 103 Shock radius at time of collision R𝑠ℎ (t𝑐𝑜𝑙𝑙) (pc) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='77 (= R𝑀𝐶) Shock velocity at time of collision v𝑠ℎ (t𝑐𝑜𝑙𝑙) (cm/s) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='75 × 108 Current age of SNR t𝑎𝑔𝑒 (years) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='4 × 105 Final radius of shock R𝑠ℎ (t𝑎𝑔𝑒) (pc) 30 Final velocity of shock v𝑠ℎ (t𝑎𝑔𝑒) (cm/s) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='2 × 106 Distance d (kpc) 3 MC number density n𝑀𝐶 (cm−3) 30 MC magnetic field B𝑀𝐶 (𝜇G) 25 Cavity number density n𝑐𝑎𝑣 (cm−3) 1 Hadronic component: Minimum energy E𝑝 𝑚𝑖𝑛 (TeV) 63 Maximum energy E𝑝 𝑚𝑎𝑥 (TeV) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='1 × 103 Spectral index p 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='8 Energy budget W𝑝 (erg) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='6 × 1047 Leptonic component: Minimum energy E𝑒 𝑚𝑖𝑛 (TeV) 5 × 10−4 Maximum energy E𝑒𝑚𝑎𝑥 (TeV) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='5 Spectral index p 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='8 Energy budget W𝑒 (erg) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='6 × 1047 component is expected to dominate the IC component, as the in- teraction is taking place inside MCs with a high number density of cold protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Moreover, the synchrotron component does not violate the X-ray 2𝜎 upper limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' We note that no proper radio counterpart has been associated with the LHAASO J2108+5157 yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' An extended radio source associated with nearby star-forming re- gion (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2021c), as well as point-like radio source NVSS 210803+515255 or WENSS B2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='4+5140 (Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2022) were found within 95% extension upper limit of LHAASO J2108+5157 and 4FGL J2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='0+5155e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Since no proper association was estab- lished between these sources and the gamma-ray source, we refrain from including their radio data in this study to further constrain the model, and we follow the MWL SED discussed in (Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2022) to ascertain the feasibility of the model discussed in this letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' As discussed earlier, we have neglected the effect of particle diffu- sion in this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' We note that such an assumption may likely lead to an overestimation, and the aspect of suppressed diffusion inside the MCs is highly uncertain (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Dogiel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' In- troducing an energy-independent diffusion coefficient, as discussed in Dogiel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' (2015), will lead to higher energy budgets required by the electron and proton populations to explain the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' The sup- pressed diffusion coefficient introduced by Gabici et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' (2009) has similar energy dependence as to that observed in ISM, but the exact energy dependence of diffusion coefficient inside clouds is not well constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' So, to avoid further complications, we have neglected the effect of diffusion in this model, similar to Fujita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' (2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Makino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' (2019), and assumed that the injected particles quickly cool down before escaping the MC medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' We further note that we do not consider the contribution of accel- erated and escaped particles, when the shock front is within the MC medium, in calculating the total gamma-ray SED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Even if the SNR is still in the Sedov phase when the shock is within the MCs, the corresponding contribution was found to be negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Moreover, the acceleration and subsequent escape of particles, in that case, will depend on the evolution of the confinement region within the high- density, turbulent medium of the MCs, details of which is beyond the scope of the simple model discussed in this letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Furthermore, as the SNR enters its radiative phase at t𝑟𝑎𝑑 ∼ 4 × 104 years (Blondin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 1998), the particle acceleration becomes ineffective as the small shock velocity at that age, as obtained from equation 6 (< 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content='1 × 107 cm/s), prevents full ionization of the pre-shock gas (Shull & McKee 1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' So no significant contribution to the total gamma-ray SED is expected in the radiative phase of the SNR as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Since hadronic component primarily dominates in the VHE-UHE gamma-ray range, neutrinos can be produced from the hadronic in- teraction as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' This neutrino flux can be a smoking gun evidence for the dominant hadronic interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' We have calculated the neu- trino flux resulting from the hadronic interaction discussed above, and found that the corresponding neutrino flux is too low to be de- tected by current generation neutrino telescope such as ICECUBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Furthermore, we have found that the model neutrino flux does not exceed the 5𝜎 discovery potential after 10 years of observation by next generation neutrino observatory ICECUBE-Gen2 for two de- clinations, 𝛿 = 0◦ and 30◦ (Aartsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' 2021), which indicates that it is unlikely to confirm the hadronic nature of UHE gamma-ray emission through neutrino observations, even in the near future, for this source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' In conclusion, in this letter, we have shown that by essentially tuning the 𝛼 index, the emission of the LHAASO source can be explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' We note that we do not intend to “fit” the MWL SED, as the SED, in various energy ranges (VHE, X-ray, radio), is poorly constrained and in need of further observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' In this work, we have only applied a simple phenomenological model, while also minimizing the free parameters, which naturally explains the spec- tral features and spatial morphology of LHAASO J2108+5157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' Fu- ture observations can confirm the viability of this model to ex- plain LHAASO J2108+5157, or other unidentified UHE gamma-ray source LHAASO 0341+5258, and sources detected in future as well, which show similar nature and emission signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' If confirmed, then it can be posited that SNRs as a source class, similar to PWNe, can likely be a strong candidate for being the Galactic PeVatrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' ACKNOWLEDGEMENTS I thank the anonymous reviewer for helpful comments and construc- tive criticism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' I thank Nayantara Gupta for encouragement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' DATA AVAILABILITY The simulated data underlying this paper will be shared on reasonable request to the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' REFERENCES Aartsen M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQf9jeh/content/2301.13451v1.pdf'} +page_content=' G.' metadata={'source': 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Bhatt +Theoretical Physics Division +Physical Research Laboratory, Ahmedabad. +to the +Discipline of Physics +Indian Institute of Technology Gandhinagar, +Gujarat 382355, India. +arXiv:2301.02655v1 [astro-ph.CO] 6 Jan 2023 + +★★★★★ +List of Acronyms +ΛCDM +Λ Cold Dark Matter +BBN +Big-Bang Nucleosynthesis +WMAP +Wilkinson Microwave Anisotropy Probe +COBE +COsmic Background Explorer +QCD +Quantum ChromoDynamic +EoR +Epoch of Reionization +IGM +InterGalactic Medium +JWST +James Webb Space Telescope +EDGES +Experiment to Detect the Global Epoch of Reionization Sig- +nature +ISM +InterStellar Medium +CMB +Cosmic Microwave Background +SIDM +Self-Interacting Dark Matter +WDM +Warm Dark Matter +THESEUS +Transient High Energy Sky and Early Universe Surveyor +NuSTAR +Nuclear Spectroscopic Telescope Array +CMBR +Cosmic Microwave Background Radiation +LIGO +Laser Interferometer Gravitational-Wave Observatory +PBH +Primordial Black Hole +AGN +Active Galactic Nuclei +LUX +Large Underground Xenon + +PandaX +Particle and astrophysical Xenon +CRESST +Cryogenic Rare Event Search with Superconducting Ther- +mometers +PICO +PICASSO and COUPP +PICASSO +Project in CAnada to Search for Super-symmetric Objects +FIRAS +Far Infrared Absolute Spectrophotometer +COUPP +Chicagoland Observatory for Underground Particle Physics +AMEGO +All-sky Medium Energy Gamma-ray Observatory +PMFs +Primordial Magnetic Fields +MHD +MagnetoHydroDynamics +ARCADE +Absolute Radiometer for Cosmology, Astrophysics and Dif- +fuse Emission +LWA +Long Wavelength Array +HESS +High Energy Stereoscopic System + +Contents +List of Acronyms +iii +1 +Introduction +1 +1.1 +Evolution of our Universe +. . . . . . . . . . . . . . . +3 +1.1.1 +Big-Bang nucleosynthesis . . . . . . . . . . . . +5 +1.1.2 +Recombination and photon decoupling +. . . . +7 +1.1.3 +Through the dark ages to the present day +. . +8 +1.2 +21 cm line as a probe during end of darkness . . . . . +9 +1.2.1 +21 cm differential brightness temperature . . . +12 +1.2.2 +Evolution of the global 21 cm signal . . . . . . +15 +1.3 +21 cm line as a probe of new physics +. . . . . . . . . +17 +1.3.1 +Sterile neutrino dark matter — Chapter 2 +. . +20 +1.3.2 +Primordial black hole dark matter — Chapter 3 20 +1.3.3 +Primordial magnetic fields — Chapter 4 & 5 . +21 +2 +Sterile Neutrino Dark Matter +23 +v + +vi +2.1 +Sterile neutrinos as dark matter . . . . . . . . . . . . +25 +2.2 +Existing bounds on sterile neutrinos . . . . . . . . . . +28 +2.3 +Radiative decay of sterile neutrinos . . . . . . . . . . +29 +2.4 +Impact on the thermal and ionization history . . . . . +30 +2.5 +Bounds on the sterile neutrinos +. . . . . . . . . . . . +33 +2.6 +Summary +. . . . . . . . . . . . . . . . . . . . . . . . +40 +2.7 +Additional study +. . . . . . . . . . . . . . . . . . . . +41 +2.7.1 +Bounds in light of varying T21 and redshift . . +41 +3 +Primordial Black Hole Dark Matter +45 +3.1 +Primordial black holes as dark matter . . . . . . . . . +47 +3.1.1 +Signature of Primordial Black Holes . . . . . . +48 +3.2 +Existing bounds on Primordial Black Holes . . . . . . +49 +3.3 +Impact on the thermal and ionization history . . . . . +49 +3.4 +Results and Discussion . . . . . . . . . . . . . . . . . +52 +3.5 +Conclusions . . . . . . . . . . . . . . . . . . . . . . . +59 +3.6 +Additional study +. . . . . . . . . . . . . . . . . . . . +60 +3.6.1 +Bounds in light of varying T21 and redshift . . +60 +4 +PMFs & Excess Radio Background +63 +4.1 +Generation of primordial magnetic fields +. . . . . . . +64 +4.2 +Existing bounds on primordial magnetic fields . . . . +66 +4.3 +Evolution of PMFs after recombination . . . . . . . . +66 +4.4 +Background excess radio radiation . . . . . . . . . . . +67 + +vii +4.4.1 +Excess radiation during the cosmic dawn . . . +69 +4.4.2 +Phenomenological model for excess radiation . +70 +4.5 +Impact on the thermal and ionization history due to +primordial magnetic fields +. . . . . . . . . . . . . . . +71 +4.6 +Impact on the thermal and ionization history due to +background radiation . . . . . . . . . . . . . . . . . . +73 +4.7 +Result and discussion . . . . . . . . . . . . . . . . . . +74 +4.8 +Conclusions . . . . . . . . . . . . . . . . . . . . . . . +87 +5 +PMFs & Baryon-Dark matter Interaction +89 +5.1 +Baryon-dark matter interaction in presence of mag- +netic fields . . . . . . . . . . . . . . . . . . . . . . . . +91 +5.2 +Results and Discussion . . . . . . . . . . . . . . . . . +93 +5.2.1 +Correlation between dark matter mass and baryon- +dark matter cross section . . . . . . . . . . . . +96 +5.2.2 +Effect of primordial magnetic fields on the global +21 cm signal . . . . . . . . . . . . . . . . . . . +99 +5.3 +Conclusions . . . . . . . . . . . . . . . . . . . . . . . +101 +6 +Summary and Future outlook +103 +6.1 +Summary +. . . . . . . . . . . . . . . . . . . . . . . . +103 +6.1.1 +Bounds on dark matter candidates +. . . . . . +105 +6.1.2 +Primordial Magnetic Fields +. . . . . . . . . . +107 +A Appendix +109 + +viii +A.1 Spin temperature of hydrogen . . . . . . . . . . . . . +109 +A.2 Emergent brightness temperature . . . . . . . . . . . +111 +A.3 Optical depth of hydrogen medium . . . . . . . . . . +111 +References +115 + +“It might seem limited to impose our human per- +ception to try to deduce the grandest cosmic code. +But we are the product of the universe and I think +it can be argued that the entire cosmic code is im- +printed in us. Just as our genes carry the mem- +ory of our biological ancestors, our logic carries +the memory of our cosmological ancestry. +We +are not just imposing human-centric notions on +a cosmos independent of us. We are progeny of +the cosmos and our ability to understand it is an +inheritance.” +Janna Levin, How the Universe Got its Spots +(2002) +1 +Introduction +In the 21st century, our knowledge of the Universe has proliferated— thanks to the +tremendous progress of observational instruments in the last three decades. Espe- +cially the precision cosmology has grown remarkably in the past three decades as a +result of an ample amount of high-quality Cosmic Microwave Background (CMB) +data, in addition to the data and comprehensive studies of supernovae, stars and +nearby galaxies. It is the greatest triumph of precision cosmology that we now +know the age of our Universe, and it is only the tip of the iceberg. As another +example, we know that observable/baryonic matter in the Universe is only about 5 +percent and the leftover energy component consists the dark matter (∼ 26 percent) +and dark energy (∼ 69 percent) based on the ΛCDM model of cosmology— the +standard model of cosmology [48]. Here, Λ represents the dark energy, and CDM +represents the cold dark matter. +The ΛCDM model, together with the cosmo- +ahttp://lambda.gsfc.nasa.gov/product/cobe/ + +21 cm Line Astronomy and Constraining New Physics +Figure 1.1: +The graphic represents the evolution of precision cosmology over +three decades— comparison between CMB temperature maps reported by each +satellite. +Left to right: Cosmic Background Explorer (COBE)a— launched in +1989, WMAP— launched in 2001 and Planck— launched in 2009. Image credits: +NASA/JPL-Caltech/ESA, https://www.nasa.gov/mission pages/planck. +logical inflation, can provide a complete picture of the evolution of our Universe +from the beginning. The CMB data from Wilkinson Microwave Anisotropy Probe +(WMAP)b played a crucial role in establishing the ΛCDM model. It is also sup- +ported by the Planckc observations [48, 53]. The ΛCDM model is widely accepted +now, and there are various good reasons to believe this model: N-body simulations +of structure formation based on the ΛCDM framework can explain the observed +large scale structure of the Universe [54], it can also explain the CMB anisotropies +& polarization [48, 53, 55–57] and accelerating expansion of the Universe caused +by cosmological constant Λ [48, 58, 59]d. In addition to this, the predictions for the +helium and deuterium fractions by the standard Big-Bang Nucleosynthesis (BBN) +for ΛCDM cosmology agree very well with observations. +bhttps://map.gsfc.nasa.gov/ +chttps://www.esa.int/Science Exploration/Space Science/Planck +dSaul Perlmutter with Brian P. Schmidt and Adam G. Riess received the Nobel Prize in +Physics for 2011 “for the discovery of the accelerating expansion of the Universe through obser- +vations of distant supernovae.” +Chapter 1 +Introduction +2 + +COBE +WMAP +Planck21 cm Line Astronomy and Constraining New Physics +Figure 1.2: The evolution of the Universe from it’s beginning. +Image credits: +European Space Agency (ESA)e. +1.1 +Evolution of our Universe +Before going into 21 cm cosmology, we briefly review cosmic history from the +beginning of our Universe to the present day. The figure (1.2), shows a schematic +picture of the evolution of our Universe from the beginning. The best current +widely agreed model of the origin and evolution of our Universe is the Big-Bang +model. According to this model, our Universe came into existence with a Big-Bang +about 13.8 billion years ago [48]. Observations of CMB also support this theory +[48, 60]f. According to our best present-day understanding, the early Universe +had an exponential expansion after the Big-Bang— it is known as the inflationary +epochg [61, 62]. There are several reasons to believe the inflation model: It can +solve the three technical problems of the Big-Bang model— the horizon problem, +flatness problem and the magnetic monopole problem [61, 62]: +ehttps://www.esa.int/ +fResults from the COBE were honoured with the Nobel Prize in Physics 2006. +gAlan H. Guth, Andrei D. Linde and Alexei A. Starobinsky received 2014 KAVLI prize in +Astrophysics “for pioneering the theory of cosmic inflation.” +3 +Introduction +Chapter 1 + +10-32 seconds +1 second +100 seconds +380 000 years +300-500 million years +Billions of years +13.8 billion years +Beginning +of the +Universe +Light and matter +Dark ages +Inflation +Formation of +Light and matter +First stars +Galaxy evolution +The present Universe +Accelerated expansion +light and matter +are coupled +separate +Atoms start feeling +The first stars and +of the Universe +Dark matter evolves +- Protons and electrons +the gravity of the +galaxies form in the +independently: it starts +form atoms +cosmic web of dark +densest knots of the +clumping and forming +matter +- Light starts travelling +cosmic web +a web of structures +freely: it will become the +Cosmic Microwave +Background (CMB) +- Tiny fluctuations: +Frequent collisions +As the Universe expands, +Last scattering of +The Universe is dark as +Light from first stars and +Light can interact +the seeds of future +particles collide less +light off electrons +stars and galaxies are +again with electrons +structures +and light +frequently +yet to form +→ Polarisation +→ Polarisation +- Gravitational waves? +the Universe21 cm Line Astronomy and Constraining New Physics +• The Big-Bang model fails to explain why causally disconnected regions ap- +pear homogeneous. The observation shows that the CMB temperature is +uniform up to a scale of ∆T/T ≈ 10−5 even when observed in opposite di- +rections. Here, ∆T is the temperature difference between the two regions +of the sky, and T is the average temperature over the whole sky. Assuming +the standard Big-Bang model, opposite directions were so far separated that +they always have been acausal. Then, why does CMB appear so uniform? +It is known as the horizon problem. +• The second one is the flatness problem: The present-day total energy den- +sity of the Universe is equal to the critical energy density of the Universe. +Any departure from the critical density will result in the curvature of the +Universe. The observation shows dimensionless curvature energy density of +the Universe Ωk = 0.001 ± 0.002 [48]. It implies a flat Universe. A slight +deviation of total energy density from critical energy density would have re- +sulted in extreme effects on the flatness of the Universe over the cosmic time. +Therefore, a flat universe like ours requires extreme fine-tuning conditions in +the beginning. It is known as the flatness problem. +• The Grand Unified Theories (GUT) predict the existence of magnetic monopoles +as at a very high temperature as the electromagnetic, weak and strong forces +are not fundamental forces. Therefore, there can exist many stable magnetic +monopoles in the Universe. No monopoles have been observed yet. It is +known as the monopole problem. +These problems of the Big-Bang model can be circumvented by introducing the +cosmic inflation model [61, 62]. Additionally, the inflation can give an idea of the +origin of the observed structures in the Universe. The quantum fluctuations, prior +to inflation, embedded in the initial energy density might have grown to astronom- +ical scales over the cosmic time. Later, the dense regions might have condensed +into structures like stars, galaxies and clusters of galaxies. The inflation epoch ends +Chapter 1 +Introduction +4 + +21 cm Line Astronomy and Constraining New Physics +when inflation potential steepens, and the inflation field acquires kinetic energy. +Then inflation sector energy creates the standard model particles. This process is +known as reheating. As the Universe expands continuously, it cools down. Then, +Baryogenesis (excess of baryons over antibaryons)h, electroweak phase transition +(100 GeV) and QCD phase transition (150 MeV) takes place. The table (1.1), rep- +resents the time scale, redshift and temperature for various events in the Universe. +The decoupling and freeze-out of various species can be understood by comparing +the rate of interaction (Γ) and Hubble expansion (H). If tΓ ≪ tH, then particle +interactions dominates over expansion. Here, t∗ ≡ 1/∗ is the time scale for corre- +sponding rate (∗ ≡ Γ or H). Therefore, local thermal equilibrium can be reached. +As Universe cools down, the value of tΓ increases faster than tH. At tΓ ∼ tH +particles starts to decouple from thermal equilibrium. Different species decouple +at different times as tΓ varies from species to species. If the mass (m) of particles +becomes larger than their temperature (T), the distribution function is exponen- +tially suppressed, ∝ e−m/T and particles freeze out. For example, the cross-section +for weak interaction is σ ∼ G2 +F T 2; GF = 1.17 × 10−5 GeV−2 is Fermi constant. +It implies Γ/H ∼ (T/MeV)3. For example, the neutrinos interact through weak +interaction only and they decouple around T ∼ 1 MeV from primordial plasma. +1.1.1 +Big-Bang nucleosynthesis +When plasma cools down below ∼ 100 KeV, around three minutes after the be- +ginning of the Universe, Big-Bang nucleosynthesis takes place. In this phase, light +elements were formed. The neutrons and protons start to form deuterium via the +process, +n + p ↔ D + γ . +(1.1) +hThe exact time and mechanism for Baryogenesis are not exactly known yet. +5 +Introduction +Chapter 1 + +21 cm Line Astronomy and Constraining New Physics +Event +time +redshift +Temperature +Inflation +10−36 sec +- +- +Baryogenesis +? +? +? +Electroweak phase transition +20 ps +1015 +100 GeV +QCD phase transition +20 µs +1012 +150 MeV +Dark matter freeze-out +? +? +? +Neutrino Decoupling +1 sec +6 × 109 +1 MeV +Electron-positron annihilation +6 sec +2 × 109 +500 KeV +Big-Bang nucleosynthesis +3 minute +4 × 108 +100 KeV +Matter-radiation equality +60 Kyr +3400 +0.75 eV +Recombination +260−380 Kyr +1400 − 1100 +0.33 − 0.26 eV +Photon decoupling +∼ 380 Kyr +∼ 1100 +∼ 0.27 eV +First stars formation +∼ 100 Myr +∼ 30 +∼ 7 meV +Reionization +∼ 400 Myr +∼ 11 +∼ 2.6 meV +Dark energy-matter equality +9 Gyr +0.4 +0.33 meV +Present +13.8 Gyr +0 +0.24 meV +Table 1.1: Approximate time scale, redshift and temperature for various events in +the Universe. Table credit: Daniel Baumann, “Lecture notes on cosmology: Part +III Mathematical Tripos.” +Chapter 1 +Introduction +6 + +21 cm Line Astronomy and Constraining New Physics +Now, these formed nuclei can form the heavier nuclei via the process, +D + p ↔ He3 + γ +and +D + He3 ↔ He4 + p . +(1.2) +The number density ratio of these elements can be found easily. For example: in +equation (1.1), µn + µp = µD as µγ = 0. Here, µ is the chemical potential for the +corresponding species. It implies the number densities ratio to be, +� nD +nn np +� +eq += 3 +4 +� mD +mn mp +2 π +T +�3/2 +e−(mD−mn−mp)/T , +(1.3) +here, T is the plasma temperature. mD, mn and mp are masses of deuterium, +neutron and proton, respectively. +1.1.2 +Recombination and photon decoupling +Within the ΛCDM cosmology, free electrons and protons cool sufficiently after +∼ 3 × 105 years of Big-Bang to form neutral hydrogen atoms. Recombination +occurs around redshift 1100. During this epoch, electrons and protons combine to +form hydrogen atoms via the process, +e− + p ↔ H + γ . +(1.4) +When the plasma temperature was above 1 eV, there were still free electrons and +protons in the plasma. Photons remain tightly coupled to electrons due to Comp- +ton scattering, and electrons were coupled to protons due to Coulomb scattering. +In turn, there was only a small density of neutral hydrogen atoms. When the +plasma temperature decreased sufficiently, electrons and protons combined and +formed hydrogen atoms. Subsequently, the free electron density fell rapidly. As +the number density of free electrons decreased adequately, the mean free path of +photons increased sharply; and photons decoupled from plasma. As discussed in +7 +Introduction +Chapter 1 + +21 cm Line Astronomy and Constraining New Physics +the end of section (1.1), one can estimate the photon decoupling redshift by rela- +tion, tΓγ(zdec) ∼ H(zdec) . Here, Γγ(zdec) = ne(zdec) σT is the photon interaction +rate or photon mean free path at the time of decoupling, zdec is the redshift of +photon decoupling from plasma and σT is Thomson cross-section. The electron +number density (ne) can be found by using the Saha equation for the process +in equation (1.4). +After solving the relation, we can find zdec ∼ 1100 or cor- +responding time to 380,000 yr after Big-Bang. +We can also estimate that the +free electron fraction in the plasma remains only about one percent— the plasma +becomes mostly transparent for photons. This time is known as the surface of +last-scattering. After decoupling, these photons stream freely and are known as +CMB radiation (CMBR). +1.1.3 +Through the dark ages to the present day +After photon decoupling from baryonic matter, there were no luminous objects— +this epoch is known as the dark ages. The Universe was predominantly neutral +during this era. This period of darkness ensued until the first luminous object was +not formed in the Universe for about a hundred million years after the Big-Bang. +During this era, overdensity was growing in the dark matter perturbations already. +Later, these overdensities reached a critical value and collapsed to form dark matter +halos— a gravitationally bound structure [63]. The first generation of luminous +objects sprung up around redshift 30 inside dark matter halos— this period is +known as the Cosmic Dawn. As of now, it is not clear that these objects were +either quasars or stars. As the first stars formed in very different circumstances, +they probably were very different from our nearby stars. After the formation of +the first luminous objects, their radiation start to ionize the gas in the Universe. +This era is known as the epoch of reionization (EoR). Three-year WMAP obser- +vations of CMB suggest that reionization starts around redshift 11 and ends by +∼ 7 [56]. Planck observations suggest instantaneous reionization with mid-point +Chapter 1 +Introduction +8 + +21 cm Line Astronomy and Constraining New Physics +redshift of reionization 7.68 ± 0.79 [48]. Supernovae observations suggest that the +Universe enters into an accelerated expansion phase around redshift ∼ 0.5 [59]. +This accelerating expansion can not be explained only by matter in the Universe. +To explain, one requires the existence of dark energy [59, 64]. Then, we reach the +present-day after 13.8 billion years from the Big-Bang. +The first complexity in the physics, after the dark ages, emerged with the event +of the formation of the first luminous objects. As of now, this era is not observed +due to the lack of our instrumental capability. The recently launched James Webb +Space Telescope (JWST)i will be able to probe the Universe back to redshift ∼ 20. +One of the best pre-eminent and promising methods to probe the cosmic dawn era +is the observation of the redshifted radiation from the hyperfine transition in the +ground state of the neutral hydrogen atom. The low-frequency radio telescopes, +sensitive to a frequency of as low as 40 MHz, can help to explore this era. +1.2 +21 cm line as a probe during end of darkness +The 21 cm signal appears to be a treasure trove to provide an insight into the period +when the first luminous objects were formed; hereafter we will refer these objects +as first stars. The 21 cm line has been actively used to trace the neutral hydrogen +in Milky Way for more than seven decades since its first observation in 1951 [1]. +It was first suggested by H. C. van de Hulst in 1945 that a 21 cm line might be +observable in the galactic radiation spectrum [65]. However, probing the neutral +hydrogen during and pre cosmic dawn via the 21 cm signal is different. These +periods are observed in the form of absorption/emission by the neutral hydrogen +medium relative to the CMBR or background radiation at a reference wavelength +of 21 cm. It is referred as the 21 cm differential brightness temperature— we will +discuss it later. +The 21 cm line corresponds to the wavelength for hyperfine transition between +ihttps://jwst.nasa.gov/ +9 +Introduction +Chapter 1 + +21 cm Line Astronomy and Constraining New Physics +1S singlet and triplet states of the neutral hydrogen atom. The corresponding +frequency for the 21 cm line is 1420.4 MHz. For a transition at redshift z, the +frequency can be mapped for a present-day observed frequency as 1420.4/(1 + z). +Hydrogen is the dominating fraction in the Inter-Galactic-Medium (IGM) during +cosmic dawn. Therefore, it is convenient and advantageous to study IGM using +the 21 cm signal. The transition probability for the hyperfine state is once in ∼ 107 +years in the absence of any external sources. The presence of any exotic source +of energy can significantly affect the hyperfine transition, thus spin temperature +of the hydrogen gas. The spin temperature (TS) is characterized by the number +density ratio in 1S singlet and triplet states of the neutral hydrogen atom, +nT +nS += gT +gS +× exp +� +−2πνTS +TS +� +, νTS = 1420.4 MHz ≃ 1/(21 cm) , +(1.5) +here, nT and nS are the population of triplet and singlet states, respectively. Hy- +perfine splitting suppresses the singlet and lifts the triplet state. gT = 3 and gS = 1 +are the statistical or spin degeneracies of triplet and singlet states, respectively. +In the cosmological scenarios, there are three processes that can affect the spin +temperature: background radio radiation, Lyα radiation from the first stars and ++ +, +, +- +ν = 1420 MHz +λ = 21 cm +1s 2S1/2 + +Singlet +Triplet +Figure 1.3: A schematic diagram for hyperfine transition in ground state of neutral +hydrogen atom. +Chapter 1 +Introduction +10 + +21 cm Line Astronomy and Constraining New Physics +collisions of a hydrogen atom with another hydrogen atoms, residual electrons or +protons. In the presence of all these three effect, we can write the rate of change +in the population density of singlet state, +dnS +dt = −nS(P R +ST + P α +ST + P C +ST) + nT(P R +TS + P α +TS + P C +TS) , +(1.6) +here, PST and PTS are excitation and de-excitation coefficients, respectively. R, +α and C superscripts represent the excitation/de-excitation due to background +radio radiationj, Lyα radiation from first stars and collisions, respectively. In the +detailed balance between the population of 1S singlet and triplet states, by solving +the equation (1.6)— see appendix A.1, one can find the spin temperature as [2, 3], +T −1 +S += T −1 +R + xα T −1 +α ++ xc T −1 +gas +1 + xα + xc +, +(1.7) +here, Tα and TR is the colour temperature of Lyα radiation from first stars and +background radio radiation temperature, respectively. Tgas is the gas temperature. +It refers to the temperature of either neutral species, ions, electrons or protons— all +remain in thermal equilibrium. Before the first luminous objects formation, there +was no Lyα radiation implying xα & Tα = 0. After the first luminous objects +formation, their Lyα photons started repeatedly scatter with the gas, and brought +the Lyα radiation into a local thermal equilibrium with the gas. Therefore, during +the cosmic dawn era the colour temperature can be taken as gas temperature, Tα ≃ +Tgas [2, 3, 66]. xα = P α +TS/P R +TS is the Lyα coupling coefficient due to Wouthuysen- +Field effect [2, 4]. Here, P R +TS = (1 + TR/TTS) A10, TTS = 2 π νTS = 0.068 K and +A10 = 2.85 × 10−15 sec−1 is the Einstein coefficient for spontaneous emission from +triplet to singlet state. For the all presented scenarios in the thesis: TR ≳ 49 K ≫ +TTS at required redshift z ∼ 17. Thus, one can approximate P R +TS ≃ A10 ×(TR/TTS). +P α +TS = 4 Pα/27 and Pα is the rate of scattering of Lyα photons [3]. xc = P C +TS/P R +TS +jP R +TS includes both the induced emission due to background radio radiation and spontaneous +emission— equation (A.4). +11 +Introduction +Chapter 1 + +21 cm Line Astronomy and Constraining New Physics +is the collisional coupling coefficient due to scattering between hydrogen atoms +or scattering of hydrogen atoms with other species such as electrons and protons. +Hence, the Lyα and collisional coupling coefficients [3], +xα = P α +TS +P R +TS += +4 Pα +27 A10 +× TTS +TR +, +(1.8) +xc = P C +TS +P R +TS += P C +TS +A10 +× TTS +TR +. +(1.9) +Here, the de-excitation coefficient due to collisions in gas: P C +TS = nHI kHH +10 +ne kHe +10 + +np kHp +10 . nHI, ne and np are the number density of neutral hydrogen, electrons and +protons in the medium, respectively. kHH +10 is the rate of scattering between hydrogen +atoms. kHe +10 is the rate of scattering between hydrogen atoms and electrons. kHp +10 is +the rate of scattering between hydrogen atoms and protons. For a more detailed +review, see the review article by Pritchard and Loeb [3]. +1.2.1 +21 cm differential brightness temperature +Figure 1.4: A schematic diagram for the change in brightness temperature of a +light when it passes through a medium. +Chapter 1 +Introduction +12 + +21 cm Line Astronomy and Constraining New Physics +As discussed above, the 21 cm signal is observed in the form of differential +brightness temperature during the cosmic dawn era. If a light with initial intensity +(Iν0) & brightness temperature (TR) passes through a medium having optical depth +(τν) & excitation temperature (Texc), there can be an absorption or emission by +the medium resulting in a different final/emergent intensity (I′ +ν) and brightness +temperature (T ′ +R). The divergence of the emergent brightness temperature (T ′ +R) +from the initial brightness temperature (TR) is known as the differential brightness +temperature (observed temperature by antennas), +δTB = T ′ +R − TR . +(1.10) +In observation, we measure the specific intensity of radiation at some frequency. +As discussed above, the initial frequency ν of light at redshift z changes with time +due to the expansion of the Universe. For present-day, it will modify to ν/(1 + z). +Accordingly, the frequency of 1420.4 MHz of a light originated in the redshift range +z = 15 − 10 will suppress to O(105 Hz). While the CMB peak occurs around a +frequency of O(108 Hz)— this is much higher than the 21 cm line. Therefore, we +can approximate the blackbody spectrum as the Rayleigh-Jeans limit. In this limit +the observed specific intensity of radiation at a frequency ν, +Iν = +4 π ν3 +exp(2 π ν/T) − 1 +2πν/T ≪ 1 +−−−−−−→ +Iν ≡ 2 ν2 T , +(1.11) +T is the brightness temperature of the blackbody. The emergent brightness tem- +perature, T ′ +R in equation (1.10), is a combination of TR and Texc. We can find T ′ +R +by solving the equation of radiative transfer. If a light passes through a medium— +figure (1.4), the change in its intensity (dIν) due to the absorption or emission +with travelled distance (dl), +dIν +dl = jν − ανIν , +(1.12) +13 +Introduction +Chapter 1 + +21 cm Line Astronomy and Constraining New Physics +where, jν is emission of light by spontaneous, stimulated emission, etc. αν is the +absorption coefficient of medium at frequency ν. Here, we follow the review articles +by Pritchard et al. [3] and Furlanetto et al. [67]. Writing equation (1.12) as, +dIν +dτν += Sν − Iν , +(1.13) +here, dτν = αν dl and Sν = jν/αν. Therefore, +τν = +� +αν dl , +(1.14) +is the optical depth. Optical depth is a function of the absorption of light by the +medium with travelled distance in the medium. By solving the equation (1.12) +and using equation (1.11), we can find the T ′ +R— see the appendix A.2, +T ′ +R = Texc (1 − e−τν) + TR e−τν . +(1.15) +The differential brightness temperature, by equation (1.10), δTB = (Texc − TR) × +(1 − e−τν). For the expending Universe, the temperature of radiation is ∝ (1 + z). +Thus, the redshifted differential brightness temperature for present-day, +δTB = Texc − TR +1 + z +× (1 − e−τν) . +(1.16) +In our case, the medium is hydrogen gas and the Texc for the 21 cm line is TS— +defined in equation (1.7). The τν is ≪ 1 for neutral hydrogen gas— optically +thin. Hereafter, we will write δTB as T21 for the 21 cm line. Therefore, the 21 cm +differential brightness temperature [3], +T21 ≃ TS − TR +1 + z +× τν . +(1.17) +The optical depth can be found by solving the equation (1.14) for a hydrogen +Chapter 1 +Introduction +14 + +21 cm Line Astronomy and Constraining New Physics +medium and a line profile— see appendix A.3, +τν ≃ 27 xHI (1 + z) +�mK +TS +� � 0.15 +Ωm h2 +1 + z +10 +�1/2 �Ωb h2 +0.023 +� +, +(1.18) +here, xHI = nHI/nH is the fraction of neutral hydrogen in the Universe, and nH +is the total number density of hydrogen. +Ωm = ρM/ρcr and Ωb = ρb/ρcr are +the dimensionless energy density parameters for total matter and baryons in the +Universe, respectively. +ρM and ρb are the energy density for total matter and +baryons, respectively. ρcr = 3 H2/(8 π GN) is the critical energy density and GN is +the gravitational constant. h = H0/(100 Km sec−1 Mpc−1) and H0 is the present- +day value of Hubble parameter. Substituting the value of τν from equation (1.18) +into equation (1.17), we get the final expression for the global 21 cm differential +brightness temperature [3, 68–71], +T21 ≃ 27 xHI +� +1 − TR +TS +� � 0.15 +Ωm h2 +1 + z +10 +�1/2 �Ωb h2 +0.023 +� +mK . +(1.19) +Depending on the ratio TR/TS, there can be three scenarios for 21 cm signal: If +TS = TR then T21 = 0 and there will not be any signal; for the case when TS > TR, +emission spectra can be observed, and when TS < TR, it leaves an imprint of +absorption spectra. +1.2.2 +Evolution of the global 21 cm signal +Usually, in the ΛCDM model of cosmology, the contribution in the background +radiation is assumed to be solely by the CMB radiation, TR ≡ TCMB ; TCMB is the +CMBR temperature. Therefore, in this subsection, we discuss the evolution of the +global 21 cm signal when only CMBR is present as background radiation. +At the end of recombination, the baryon number density of the Universe is dom- +kThe position and amplitude of the second dip from the left (between redshift 30 − 15) may +modify depending on models of first-stars formation or x-ray heating of the gas. +15 +Introduction +Chapter 1 + +21 cm Line Astronomy and Constraining New Physics +Figure 1.5: The figures represents the evolution of fluctuation in the 21 cm signal +(above) and global 21 cm signal (below) when the background radiation is CMBRk. +Image credits: Pritchard & Loeb, Rep. Prog. Phys., 75, 086901, (2012) [3, 31]. +inated mainly by the neutral hydrogen, a small fraction of helium, residual free +electrons and protons. After recombination (z ∼ 1100) down to z ∼ 200, the resid- +ual free electrons undergo Compton scattering and maintain thermal equilibrium +between electrons and CMBR. The free electrons remain in thermal equilibrium +with other gas components implying Tgas ∼ TCMB [72]. Using equations (1.7) and +(1.19), we can find that T21 = 0 , and the 21 cm signal is not present during this era. +From z ∼ 200 until 40, the number density of free electrons decreases significantly +and this makes the Compton scattering insufficient. As a result, the gas decouples +from CMBR, and its temperature falls adiabatically: Tgas ∝ (1+z)2. The gas tem- +perature falls below CMBR implying an early 21 cm absorption signal— known +as the collisional absorption signal [3]. During this period, collisions among the +gas components dominate, i.e. xc ≫ 1, which implies TS ∼ Tgas — equation (1.7). +Nevertheless, this signal is not observed yet due to the poor sensitivity of present- +day available radio antennas as the sensitivity of antennas falls dramatically below +Chapter 1 +Introduction +16 + +10 million +100 million +250 million +500 million +1 billion +Time after +Big Bang +[Years] +.30 +Redshift= 160 +80 +40 +20 +15 +14 +13 +12 +11 +10 +9 +8 +7 +50 +First galaxies form +[mK] +0 +Reionization begins +Reionization ends +Brightness [ +-50 +Dark Ages +-100 +Heating begins +Cosmictime +-150 +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +200 +Frequency [MHz]21 cm Line Astronomy and Constraining New Physics +∼ 50 MHz. After z ∼ 40 to the formation of the first starl, number density and +temperature of the gas are very small, hence, xc → 0. Therefore, T21 ∼ 0 and no +signal is present there [3, 73]. After the first star formation, gas temperature cou- +ples again to the spin temperature due to Lyα radiation emitted from the first stars +by Wouthuysen-Field (WF) effect [2, 4, 66]. Therefore, xα ≫ 1, xc and absorption +spectra can be seen— equations (1.7 and 1.19). After z ∼ 15, the gas temperature +starts to rise due to x-ray radiation emitted from the first starsm. Consequently, +the temperature of gas rises above CMB temperature and the emission spectra can +be seen. As the reionization ends, neutral hydrogen fraction becomes very small +and no signal is observed. The small fraction of neutral hydrogen were left only +in dense regions of collapsed structures. These regions can be analysed by 21 cm +forest— an analogy to Lyα forest. +1.3 +21 cm line as a probe of new physics +As shown in the figure (1.6), the 21 cm signal can probe a large volume of the +history of our Universe— pink region. Currently, we are not able to probe the +high redshift Universe (z ≳ 30) as the sensitivity of presently available radio +antennas becomes very low below ∼ 50 MHz. We expect that the future advanced +technology for the 21 cm signal observation will be able to probe the Universe +above the redshift 25. In the thesis, we focus on the 21 cm signal between the +redshift range of 30 to 15. +After z ∼ 200 gas temperature falls adiabatically and reaches to ≃ 7 K at +z = 17.2, while the CMB temperature reaches to ≃ 49.6 K. From the equation +(1.19), this implies a value of absorptional amplitude of T21 to ∼ −220 mK in +absence of any heating effects on the IGM gas due to first stars. Here, to calculate +T21, we have taken xHI to unity. The xHI can be written as 1 − xe . In our case, at +lThe redshift of first stars formation is not well known and it could be around 35 to 25. +mIt is also not very clear when x-ray heating begins to dominate the temperature of the gas. +We use the fiducial models for x-ray heating considered in references [51, 74–76]. +17 +Introduction +Chapter 1 + +21 cm Line Astronomy and Constraining New Physics +z ∼ 17 the ionization fraction, xe ≲ O(10−3) implying xHI ≃ 1. Here, xe = ne/nH +is the ionization fraction and ne is the number density of residual free electrons. +The presence of any exotic source of energy can inject energy into IGM and heat +the gas. This in turn can modify the absorption amplitude in the global 21 cm +signal. This feature can provide a robust bound on the properties of such sources +of energy injection into IGM. In the thesis, the following four works has been +Figure 1.6: The CMB observations can only probe the thin outer shell (z ∼ 1100), +and the observation of large scale structures can probe a small fraction of volume +near the centre. We expect that the future advanced technology for the 21 cm +signal observation will be able to probe the entire pink region. In the thesis, we +focus on the 21 cm signal from the redshift 30 to 15. Image credits: With the +permission of Josh Dillon [32]; originally reproduced from Tegmark & Zaldarriaga +(2009) [33]. +Chapter 1 +Introduction +18 + +Modern +z= 12 +Z +50 +Z= +110021 cm Line Astronomy and Constraining New Physics +considered: sterile neutrinos and primordial black holes as dark matter candidates +and constrain their properties in the light of the global 21 cm signal. Another two +works discussed in the thesis are related to the constraining strength of primordial +magnetic fields that might have been generated in the early Universe. +In 2018, the Experiment to Detect the Global Epoch of Reionization Signature +(EDGES)n collaboration reported an absorption profile for the 21 cm signal in +the redshift range 15 − 20 [5]. +The EDGES collaboration reported T21 to be +−500+200 +−500 mK in the redshift range 15−20 centred at 78±1 MHz and in symmetric +“U” shaped form. This absorption amplitude is nearly two times smaller than +predicted by theoretical models based on ΛCDM framework (∼ −220 mK). It is +argued that to explain the EDGES observation, for the best fitting amplitude at +the centre of the “U” profile, either the cosmic background radiation temperature +TCMB ≳ 104 K for the standard Tgas evolution or Tgas ≲ 3.2 K in the absence of any +non-standard evolution of the TCMB [5]. Recently, many articles have questioned +the EDGES measurement [6–8, 77, 78]. For example, in Ref. [77], the authors have +questioned the fitting parameters for the foreground emission and data. There +is a possibility that the absorption feature in the EDGES observation can be a +ground screen artifact [7]. The absorption amplitude may modify depending on +the modelling of the foreground [8, 78]. In a recent article [6], authors claimed +that the EDGES observation might not be of an astrophysical origin. We revisit +the EDGES observation and controversies over it in the chapter (6) also. In the +light of these controversies, in the recent two articles (1.3.1 & 1.3.2), we do not +consider the absorption amplitude reported by the EDGES collaboration. In these +articles, we take 21 cm differential brightness temperature such that it does not +change, from its standard theoretical value (∼ −220 mK), by a factor of more than +1/4 (i.e. −150 mK) or 1/2 (i.e. −100 mK) at redshift 17.2 . While in the older +two articles (1.3.3), we have considered the absorption amplitude reported by the +nhttps://www.haystack.mit.edu/astronomy/astronomy-projects/edges-experiment-to-detect- +the-global-eor-signature/ +19 +Introduction +Chapter 1 + +21 cm Line Astronomy and Constraining New Physics +EDGES collaboration. +1.3.1 +Sterile neutrino dark matter — Chapter 2 +In the warm dark matter models, one of the theoretically well-motivated candidates +is KeV mass sterile neutrinos. Sterile neutrinos are radiatively unstable and can +inject photon energy into the IGM. The injection of energy into the IGM can +modify the temperature and ionization history of the IGM gas thus absorption +amplitude of 21 cm signal during cosmic dawn era. Therefore one can constraint +the lifetime of sterile neutrinos and the mixing angle of sterile neutrinos with active +neutrinos. +The article has been published as: Pravin Kumar Natwariya and Alekha C. +Nayak, “Bounds on sterile neutrino lifetime and mixing angle with active neutrinos +by global 21 cm signal”, Physics Letters B 827 (2022) 136955. +1.3.2 +Primordial black hole dark matter — Chapter 3 +Primordial black holes (PBHs) have attracted much interest in recent years and +have been a part of intense studies for more than five decades. +As PBHs are +massive, interact only gravitationally and are formed in the very early Universe, +they can be considered as a potential candidate for non-particle dark matter. +Hawking evaporation of PBHs can inject energy into the IGM and therefore be +constrained by the absorption feature in the global 21 cm signal. The mass and +spin are fundamental properties of a black hole, and they can substantially affect +the evaporation rate of the black hole. In this work, we derive an upper bound on +the dark matter fraction in the form of the primordial black holes with a non-zero +spin. +The article has been published as: Pravin Kumar Natwariya, Alekha C. Nayak +and Tripurari Srivastava, “Constraining spinning primordial black holes with global +21-cm signal”, Mon Not R Astron Soc 510, 4236–4241 (2022). +Chapter 1 +Introduction +20 + +21 cm Line Astronomy and Constraining New Physics +1.3.3 +Primordial magnetic fields — Chapter 4 & 5 +Observations suggest that the magnetic fields (MFs) are ubiquitous in the Universe– +from the length scale of planets and stars to the cluster of galaxies. The origin and +evolution of PMFs are one of the outstanding problems of cosmology. Decaying +PMFs can inject magnetic energy into thermal energy of the IGM and heat the +gas. As briefly mentioned earlier, one requires to cool the IGM gas during cosmic +dawn below the standard evolution or increase the radio background at required +redshift to explain the EDGES observation. Here, we explore the upper bounds on +the present-day strength of the PMFs in both the scenarios by considering different +models. The articles have been published as: +• Pravin Kumar Natwariya, “Constraint on Primordial Magnetic Fields In the +Light of ARCADE 2 and EDGES Observations”, Eur. Phys. J. C 81 (2021) +5, 394. +• Jitesh R. Bhatt, Pravin Kumar Natwariya, Alekha C. Nayak and Arun Ku- +mar Pandey, “Baryon-Dark matter interaction in presence of magnetic fields +in light of EDGES signal”, Eur. Phys. J. C 80 (2020) 4, 334. +Chapter 6 summarises the main results of the thesis. We also discuss pos- +sibilities of further extensions and future scopes of the results obtained in the +thesis. +21 +Introduction +Chapter 1 + + +“Would you tell me, please, which way I ought +to go from here?’ ‘That depends a good deal on +where you want to get to,’ said the Cat” +Lewis Carroll, Alice in Wonderland +“It is the nature of all greatness not to be exact” +Edmund Burke, speech “On American Taxation” +2 +Sterile Neutrino Dark Matter +Despite the searching for decades, the nature of dark matter is still unknown. It is +one of the biggest mysteries in particle physics and cosmology. Although ΛCDM +model of cosmology is highly successful in explaining Big-Bang nucleosynthesis, +CMB anisotropies and large scale structures of the Universe, it faces challenges +at a smaller length scale, ≲ 1 Mpc (for a detailed review see [79] and references +therein). These problems include the missing satellite or dwarf galaxy problem +[80, 81], the too-big-to-fail problem [82, 83] and the core-cusp problem [84]. In +the simulations, the cold dark matter scenario clusters hierarchically and predicts +a large number of satellite galaxies. However, the observations show less number +of satellite galaxies [80, 81]. For example, the Milky Way size halo simulations +show around 500 satellites, while observations show a far less number of satellite +galaxies [81, 85]. Subsequently, the missing satellite creates a new problem also: +The simulation of Galactic size haloes predicts a larger number of big satellites + +21 cm Line Astronomy and Constraining New Physics +that are so massive that there is no way not to host visible stars. Therefore, these +massive satellites should be visible. In contrast, the observations show no such +satellites consistent with the simulations [82, 83, 86]. N-body simulations of cold +dark matter also show the cuspy profile for dark matter density at the halo centre, +while the observation of rotation curves suggest the flat profile [84]. In the light of +these problems, alternatives to the cold dark matter model have been proposed, +for e.g. self-interacting dark matter [87–90], fuzzy cold dark matter [91, 92], warm +dark matter (WDM) [93–97], etc. The difference between cold, warm and hot dark +matter can be characterized in the form of their thermal velocities, v = +� +(3 T/m) . +Here, v, T and m are the speed, temperature and mass of the particle, respectively. +One can see that a larger speed implies a higher temperature for a fixed mass +of particles. Roughly, if their speed is less than ten percent of the light speed +(v ≲ 0.1), they can be considered cold dark matter candidates. If v is ≳ 0.1, they +can be considered hot dark matter candidates [98]. The WDM lies in between the +hot and warm dark matter. The WDM behaves similar to CDM on large length +scales. This scale can be characterized in the form of “free-streaming length”— the +other important concept to differentiate between hot, cold or warm dark matter. +Typically, the free-streaming length can be estimated by how far a particle has +travelled from beginning to matter-radiation equality [99], +λfs = +� teq +0 +v +a dt , +(2.1) +here, teq is the matter-radiation equality time. For a length scale larger than λfs, +WDM behaves as CDM— i.e. it makes structures hierarchically above λfs. While +below the length scale λfs, there is a possibility that WDM may create structures +“top-down”— i.e. small structures may emerge via the fragmentations of large +structures [95, 98, 99]. The free-streaming length can be found as [99], +λfs ∼ 0.4 +�mWDM +KeV +�−4/3 �ΩWDM h2 +0.135 +�1/3 +Mpc/h , +(2.2) +Chapter 2 +Sterile Neutrino Dark Matter +24 + +21 cm Line Astronomy and Constraining New Physics +here, mWDM is the mass of WDM particle and ΩWDM is the dimensionless energy +density parameter for WDM. The free-streaming scale is inversely proportional to +mass of particle. It implies that the size of formed-first-structures will increase +for a smaller particle mass— the numbers of small-length-scale structures will +suppress. +For example, if one considers the mass of the WDM particle to be +10 KeV, then the free-streaming scale will be ∼ 2 × 101 Kpc. +Therefore, one +can overcome the missing satellite problem by considering an adequate mass of +WDM. In the hot dark matter scenario, the free-streaming length typically is +so large that density fluctuations below cluster scale would get washed up, and +formed-first-structures would have been the size of superclusters. +Later, their +fragments might have formed the clusters, then galaxies. While the observation +shows that galaxies formed first, then emerged as clusters and then superclusters +due to their mutual gravitational attraction [98]. As discussed above, the nature +of dark matter has significant effects on structure formation. The WDM can also +solve the angular momentum problem— galaxies have smaller specific angular +momenta in CDM simulation compared to observations [100, 101]. Additionally, +by including the baryonic feedback with WDM can address the too-big-to-fail +and core-cusp problems also [102–104]. The two popular candidates for WDM +are sterile neutrinos and gravitinos. The presence of sterile neutrino warm dark +matter having KeV mass can also explain the recently observed unexpected and +unidentified emission line around 3.5 KeV in x-ray spectra of nearby galaxies and +clusters [97, 105–108]. In this chapter, we consider sterile neutrino and study its +lifetime and mixing angle with active neutrinos [9]. +2.1 +Sterile neutrinos as dark matter +Sterile neutrino with KeV mass is one of the exciting and well-motivated candidates +for WDM (Ref. [35, 109, 110] and Refs. therein). The standard model of particle +physics considers the neutrinos as massless. However, experiments and theoretical +25 +Sterile Neutrino Dark Matter +Chapter 2 + +21 cm Line Astronomy and Constraining New Physics +models questioned the standard model of particle physics over the past years. One +well-studied example is neutrino oscillation [111–114]. To explain the observations +of neutrino oscillations, one has to extend the standard model to introduce the +massive neutrinos (for more details, see the reviews [115, 116])a. There are three +flavours of active neutrinos— electron, muon and tau neutrino, but absolute value +of their masses are not very well known. Nevertheless, the square mass difference +between different flavours has been constrained by various oscillation experiments, +such as solar, atmospheric, reactor and accelerator (see the Ref. [121] and reviews +[122, 123]). In the standard model of particle physics, all particles get their mass +via Higgs Mechanism, but neutrinos remain massless. One of the mechanisms via +which neutrinos can get their mass is the Seesaw mechanism. As of now, active +neutrinos have been observed with only left-handed chirality [124]. To give mass +to neutrinos, we also require the right-handed counterpart of active neutrinos. +The right-handed neutrinos can have mass from a few eV to GUT scale [124]. +In the Seesaw mechanism, sterile neutrino naturally appears as an eigenstate of +the neutrino mass matrix. Introducing a new Yukawa interaction with new Weyl +fermions N β [123], +LY ⊃ −yαβ(i σ2 H∗) LαN β + h.c. , +(2.3) +here, α and β are summed over e, µ, τ and 1, 2, ..., n , respectively; n is the number +of fields of N β. i σ2 H∗ and Lα = (να, eα)T are SU(2)L doublet and carry opposite +hypercharges: +1/2 and -1/2, respectively. Therefore, their combination is total +singlet, implying N β to be total singlet also [123]. When Higgs field (H) acquires +vacuum expectation value (v), the neutrino mass term can be written as, +Lmass ⊃ −M αβ +D να N β + h.c. , +(2.4) +aTakaaki Kajita with Arthur B. McDonald received the Nobel Prize in Physics for 2015 “for +the discovery of neutrino oscillations, which shows that neutrinos have mass” [117–120]. +Chapter 2 +Sterile Neutrino Dark Matter +26 + +21 cm Line Astronomy and Constraining New Physics +the Dirac mass term M αβ +D +≡ yαβ v/ +√ +2 . +Since N β does not have any strong, +electromagnetic or weak coupling, it is called the sterile; and it can be considered +a dark matter candidate. να has weak coupling with standard model particles, +and it is called active neutrino. As sterile neutrinos are singlet, in principle we can +write a Majorana mass term for N β: Lmass = −(1/2) M αβ +M N α N β + h.c. . From +equation (2.4), +Lmass ⊃ −1 +2 nT M n ≡ −1 +2 nT +� +� 0 +MD +MT +D +MM +� +� n + h.c. , +(2.5) +here, n = (νe...ντ, N 1...N n)T, MD = M αβ +D and MM = M αβ +M . Assuming ||MM|| ≫ +||MD||, as MM is not protected by any symmetry and MD can not be larger than +electroweak scale because it will require Yukawa coupling ≫ 1, the eigenvalues of +mass matrix: mν +1 = O (M 2 +D/MM) and mN = O(MM) . We get the light neutrino +mass to mν ∼ 0.1 eV by taking ||MD|| ∼ 100 GeV and ||MM|| ∼ 1014 GeV. The +sterile neutrinos are stable— have a larger lifetime compared to the age of the +Universe. Therefore, they can make an excellent candidate for the warm dark +matter if they also have mass in the KeV range [122]. +One of the minimal extensions of the standard model of particle physics, where +neutrino mass and KeV sterile neutrinos in the context of dark matter are widely +explored via the Seesaw mechanism, is the Neutrino Minimal Standard Model +(νMSM) [122–125]. In this model, we can lower one of the eigenvalues of MM +to get KeV scale sterile neutrino while keeping others super-heavy. In the basis +(νa, νs, N); N represents the heavier sterile states, the mass matrix [123], +M = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +0 +0 +0 +M1 +s +M11 +D +M12 +D +0 +0 +0 +M2 +s +M21 +D +M22 +D +0 +0 +0 +M3 +s +M31 +D +M32 +D +M1 +s +M2 +s +M3 +s +µs +0 +0 +M11 +D +M21 +D +M31 +D +0 +M1 +M +0 +M12 +D +M22 +D +M32 +D +0 +0 +M2 +M +� +� +� +� +� +� +� +� +� +� +� +� +� +� +(2.6) +27 +Sterile Neutrino Dark Matter +Chapter 2 + +21 cm Line Astronomy and Constraining New Physics +here, νs and νa are sterile and active neutrinos, respectively. Applying the Seesaw +mechanism, we can find the mass of neutrinos: Mν ≃ −MD M −1 +M M T +D−Ms µ−1 +s M T +s ; +and the mass of light sterile neutrino: ms ≃ µs. +The νMSM model is a minimal extension of the standard model of particle +physics— with only three additional sterile neutrinos up to the Planck scale. One +having KeV scale mass— can account for dark matter. The other two heavier +sterile neutrinos can account for the observed light neutrino masses by the Seesaw +mechanism. They can also explain the baryon asymmetry in the Universe through +oscillation-induced leptogenesis if they are nearly degenerate in the mass range +150 MeV−100 GeV [109, 125]. More details about KeV sterile neutrino models +can be found in the review article by A. Merle [126]. +2.2 +Existing bounds on sterile neutrinos +The possibility of KeV mass range sterile neutrinos as a WDM candidate can +be explored and constrained by the observation of large scale structures in the +Universe [94]. There are several model-dependent mechanisms that can produce +sterile neutrinos in the early Universe [35, 127, 128]. In recent years, various tech- +niques have been proposed to probe the unexplored sterile neutrino dark matter +parameter space, for example, by mapping of x-ray intensity at different redshift +[129], by observing KeV energy photons using instruments onboard Transient High +Energy Sky and Early Universe Surveyor (THESEUS) mission (for the details of +instruments sensitivity of THESEUS, see the Ref. [130]), by exploring the im- +prints of sterile neutrino on solar neutrino fluxes [131], by testing the hypothesis +of decaying-sterile-neutrino [132, 133], etc. The lower bound on the mass of ster- +ile neutrinos can be obtained by the Pauli exclusion principle [35, 123]. These +bounds depends on momentum distribution and the dwarf galaxy used for astro- +nomical data [35, 123]. The authors of the ref. [134], finds the lower bound on the +mass of non-resonantly produced sterile neutrino to be > 1.7 KeV when all the +Chapter 2 +Sterile Neutrino Dark Matter +28 + +21 cm Line Astronomy and Constraining New Physics +dark matter is composed of sterile neutrinos. Additionally, the parameter space +of sterile neutrino dark matter has been constrained by various observations and +theoretical studies. The observations from Nuclear Spectroscopic Telescope Array +(NuSTAR)b did not find any sign of anomalous x-ray lines for sterile neutrino +mass range 10 − 40 KeV. The future updated version of NuSTAR will be able to +probe for sterile neutrino mass range 6 − 10 KeV [36]. In the context of EDGES +signal, authors of the reference [135], put a constraint on the Dodelson-Widrow +sterile neutrinos mass to 63+19 +−35 KeV. The WMAP, Lyα forest and x-ray observa- +tions constrain the sterile neutrino mass in the range from ∼ 2 KeV to ∼ 50 KeV +[136–140]. +The authors of the Refs. +[141, 142] compare the observed satellite +galaxy with conferred from WDM simulations of Galaxy-sized halo and constrain +the mass of sterile neutrino ≳ 2 KeV. Further, individual bounds on the sterile +neutrino parameter space can be found in the Refs. [39, 143–151]. +2.3 +Radiative decay of sterile neutrinos +Sterile neutrinos with KeV mass can decay to active neutrinos via two channels: +νs → νa νa ¯νa and νs → νa γ. In this work, we study the effect of radiative decay +of sterile neutrinos on the thermal and ionization history of the Universe, and +constrain the sterile neutrino decay time and mixing angle with active neutrinos. +The decay of sterile neutrino to active neutrino via the radiative process can inject +the photon energy into IGM and modify the absorption amplitude of the 21 cm +signal during cosmic dawn. Hence, we can constrain the sterile neutrino decay time +and mixing angle with the active neutrino using the 21 cm absorption signal. In +this process, half of the total energy of a sterile neutrino (mνs/2) is carried away by +a photon and remaining by an active neutrino. The decay width of sterile neutrino +bhttps://heasarc.gsfc.nasa.gov/docs/nustar/index.html +29 +Sterile Neutrino Dark Matter +Chapter 2 + +21 cm Line Astronomy and Constraining New Physics +for radiative process can be written as ([35, 152] and reference cited therein), +Γνs = Γνs→νaγ = 9 α G2 +F +1024 π4 sin2(2 θ) m5 +νs , +(2.7) +here, θ ≡ � +i=e, µ, τ |θi|2 is the total mixing angle between sterile and active neu- +trinos. In equation (2.7), GF and α are the Fermi and fine structure constant, +respectively. mνs stands for the mass of the sterile neutrino. The mixing angle +θ ≪ 1, therefore sin2(2 θ) ≃ 4 sin2(θ). We can write the decay width as [35, 152], +Γνs = τ −1 +νs ≃ 5.52 × 10−22 sin2(θ) +� mνs +KeV +�5 � 1 +sec +� +, +(2.8) +here, τνs is the lifetime or decay time of sterile neutrinos. For sterile neutrinos to +be dark matter candidate, their lifetime must be larger than age of the Universe, +4.4 × 1017 sec. Using this fact and equation (2.8), one can estimate the upper +bound on the total mixing angle. +2.4 +Impact on the thermal and ionization history +Evolution of the ionization fraction with redshift in the presence of energy injection +by decaying sterile neutrinos [153–159], +dxe +dz = +P +H (1 + z) × +� +nHx2 +e αB(Tgas) − (1 − xe) βB(Tgas) e−Eα/Tgas� +− +1 +H (1 + z) +� +1 +E0 +− 1 − P +Eα +� +(1 − xe) E +3 nH +, +(2.9) +where xe = ne/nH is the ionization fraction, ne is the free electron number density +and nH is the total hydrogen number density in the Universe. αB and βB are the +case-B recombination coefficient and photo-ionization rate, respectively [153, 154, +156]. E0 = 13.6 eV and Eα = (3/4) E0 are ground state binding energy and Lyα +transition energy for the hydrogen atom, respectively. P is the Peebles coefficient +Chapter 2 +Sterile Neutrino Dark Matter +30 + +21 cm Line Astronomy and Constraining New Physics +[156, 157, 160], +P = +1 + KH ΛH nH (1 − xe) +1 + KH (ΛH + βH) nH (1 − xe) , +(2.10) +here, KH = π2/(E3 +α H) and ΛH = 8.22/sec account for the redshifting of Lyα +photon due to expansion of the Universe and the 2S-1S level two photon decay +rate of the hydrogen atom, respectively [161]. The last term in equation (2.9), +describes the additional effect of sterile neutrinos decay on the ionization fraction. +E ≡ E(z, mνs) is the energy deposition rate per unit volume into IGM gas due to +decaying sterile neutrinos. It can be written as [156, 157, 162], +E(z, mνs) = FS fabs(z, mνs) × ρνs,o +τνs +(1 + z)3 +(2.11) +here, τνs is the lifetime of sterile neutrino to decay in a active neutrino and a photon. +FS is the fraction of the sterile neutrinos that are decaying. We consider that all +sterile neutrinos are decaying, i.e. FS = 1 . ρνs,0 = mνs nνs,0 is the present day +energy density of sterile neutrino. nνs,0 is the present day number density of sterile +neutrinos. For the present work, we consider that all the dark-matter is composed +of sterile neutrinos, ρνs,0 ≡ ρDM,0 , and ρDM,0 is the present day dark-matter +energy density [35, 128, 162, 163]. fabs(z, mνs) is the energy deposition efficiency +into IGM by decaying sterile neutrinos. The energy deposition happens due to +only radiative decay of sterile neutrino as active neutrinos interact very weakly +with matter. +Therefore, we consider only radiative decay of sterile neutrinos. +fabs(z, mνs) depends on the redshift and mass of sterile neutrino [162]. The mass +of decaying particles enters only through fabs(z, mνs). In the presence of energy +deposition into IGM, the gas temperature evolution with redshift [153–157, 159], +dTgas +dz += 2 Tgas +(1 + z)+ +ΓC +(1 + z) H (Tgas − TCMB) +− +2 +3 H (1 + z) × (1 + 2 xe) E +3 ntot +, +(2.12) +31 +Sterile Neutrino Dark Matter +Chapter 2 + +21 cm Line Astronomy and Constraining New Physics +here, ntot = nH (1 + fHe + xe) is the total number density of gas, fHe = nHe/nH +is the helium fraction, nHe is the helium number density. The first term in this +equation comes due to the expansion of the Universe. The matter temperature +falls with redshift adiabatically: ∝ (1 + z)2 when Compton scattering (second +term) becomes insufficient (z ≲ 200) and τνs → ∞. The Compton scattering rate +is defined as, +ΓC = +8 σT arT 4 +CMB xe +3 (1 + fHe + xe) me +, +(2.13) +where, σT, ar and me are the Thomson scattering cross-section, Stefan-Boltzmann +radiation constant and mass of electron, respectively. Above the redshift z ∼ 200, +the gas remains in thermal equilibrium with photons due to Compton scattering as +ΓC ≫ H. At z = 200, one can find that ΓC ≈ 1.4×10−14 sec−1 when E = 0, while, +H = 3.6×10−15 sec−1. As ΓC ∝ (1+z)4 and H ∝ (1+z)3/2 for matter dominated +era, the Compton scattering rate will dominate over H as one increase z above 200. +Therefore, the gas and CMB share same temperature above z ∼ 200 — as second +term dominates over the first term. Below z ∼ 200, the Compton scattering rate +becomes smaller compared to H resulting in an adiabatic evolution of the gas when +there is no last term present in equation (2.12). The last term corresponds to the +energy deposition into IGM due to radiative decay of sterile neutrinos. Following +the Refs. [156, 157, 164, 165], we consider the ‘SSCK’ approximation— in which +(1 − xe)/3 fraction of deposited energy goes into ionization, nearly same amount +goes into excitation, and remaining (1 + 2xe)/3 fraction goes into IGM heating. +We also discuss the projected bounds on sterile neutrinos after the inclusion of +the process of gas heating in the cosmic dawn era by CMBR using Ref. [34], in +subsequent discussion we call this process VDKZ18. Here, the energy transfer +between gas and CMBR is mediated by Lyα photons from the first stars. The +authors claim that it can increase the gas temperature by the order of (∼ 10%) +at z ∼ 17. Here, it is to be noted that we do not include the x-ray heating of the +gas due to the uncertainty of known physics of the first stars. For a fix value of +Chapter 2 +Sterile Neutrino Dark Matter +32 + +21 cm Line Astronomy and Constraining New Physics +T21 at a redshift, if we include the x-ray heating of the gas, the projected bounds +becomes stronger. Including the heating due to VDKZ18 effect, equation (2.12) +will modify as, +dTgas +dz += dTgas +dz +����� +[eq.(2.12)] +− +ΓR +(1 + z) (1 + fHe + Xe) , +(2.14) +where, dTgas/dz +�� +[eq.(2.12)] represents the temperature evolution in equation (2.12), +and heating rate due to energy transfer from CMB photons to the thermal energy +of gas by Lyα photons, +ΓR = xHI +A10 +2 H xR +�TR +TS +− 1 +� +T10 , +(2.15) +here, A10 = 2.86 × 10−15 sec−1 is the Einstein coefficient for spontaneous-emission +from triplet state to singlet state. xR = 1/τ21 × [1 − exp(−τ21)] and τ21 = 8.1 × +10−2 xHI [(1 + z)/20]1.5 (10 K/TS) is the 21 cm optical depth. +T10 = 2πν10 = +0.0682 K and xHI ≃ 1 − xe is the neutral hydrogen fraction in the Universe. +2.5 +Bounds on the sterile neutrinos +As described in the (1.3), we get an absorption profile in the 21 cm signal around +redshift z ∼ 17 with an amplitude of T21 ∼ −220 mK in the theoretical models +based on ΛCDM framework of cosmology. We take 21 cm differential brightness +temperature such that it does not change, from its standard value (∼ −220 mK), +by more than about a factor of 1/4 (i.e. −150 mK) or 1/2 (i.e. −100 mK) at +redshift 17.2 . We solve the coupled equations (2.9) and (2.12) for different mass +and lifetime of sterile neutrino to get xHI and Tgas at redshift z = 17.2 . To get any +absorption signal in redshift range 15−20, the gas temperature should be less than +CMB temperature in shaded region. By requiring T21 ≃ −150 mK or −100 mK at +z=17.2, we can put the projected constraints on the lifetime of sterile neutrinos. +33 +Sterile Neutrino Dark Matter +Chapter 2 + +21 cm Line Astronomy and Constraining New Physics +10-1 +100 +101 +102 +103 +104 +101 +102 +103 +TCMB +Tgas (K) +z +τνs = 2x1026 sec +τνs = 6x1026 sec +τνs = 1x1027 sec +(a) +-200 +-150 +-100 +-50 + 0 + 50 + 100 + 5 + 10 + 15 + 20 + 25 + 30 +T21 (mK) +z +τνs = 2x1026 sec +τνs = 6x1026 sec +τνs = 1x1027 sec +(b) +Figure 2.1: The gas temperature evolution with redshift in the presence of decaying +sterile neutrinos. The red dashed line represents the CMB temperature evolution. +The black solid line depicts the Tgas when there is no sterile neutrino decay. The +shaded region corresponds to EDGES absorption signal, i.e. 15 ≤ z ≤ 20. In these +figures, we keep mass of sterile neutrino fix to 10 KeV and vary lifetime. In figure +(2.1b), we plot evolution of 21 cm differential brightness temperature as a function +of redshift for the cases represented in figure (2.1a). +Subsequently, using equation (2.8), we can also put projected constraints on the +mixing angle of sterile neutrinos with active neutrinos. +In the figures (2.1a), (2.2a) and (2.3a), we plot the gas temperature evolution +as a function of redshift for different mass and lifetime of sterile neutrino. The red +dashed line in all plots represents the CMB temperature evolution with redshift. +The black solid line represents the gas temperature evolution when there is no effect +of decaying sterile neutrino on the IGM gas. The shaded pink region corresponds +to redshift range 15 ≤ z ≤ 20 . We obtain these results by considering fabs(z, mνs) +from Ref. [162]. In figures (2.1b), (2.2b) and (2.3b), we plot the evolution of the +21 cm differential brightness temperature as a function of redshift for the scenarios +discussed in figures (2.1a), (2.2a) and (2.3a), respectively. We consider the tanh +parametrization model for the Wouthuysen-Field coupling coefficient (xα) to get +Chapter 2 +Sterile Neutrino Dark Matter +34 + +21 cm Line Astronomy and Constraining New Physics +10-1 +100 +101 +102 +103 +104 +101 +102 +103 +TCMB +Tgas (K) +z +mνs = 2 KeV +mνs = 6 KeV +mνs = 10 KeV +mνs = 25 KeV +(a) +-200 +-150 +-100 +-50 + 0 + 50 + 100 + 5 + 10 + 15 + 20 + 25 + 30 +T21 (mK) +z +mνs = 2 KeV +mνs = 6 KeV +mνs = 10 KeV +mνs = 25 KeV +(b) +Figure 2.2: The figure caption is same as in figure (2.1), except here, we consider +τνs constant to 6 × 1026 sec and vary mass of sterile neutrino. +10-1 +100 +101 +102 +103 +104 +101 +102 +103 +TCMB +Tgas (K) +z +τνs = 2x1026 sec +τνs = 6x1026 sec +τνs = 1x1027 sec +(a) +-200 +-150 +-100 +-50 + 0 + 50 + 100 + 5 + 10 + 15 + 20 + 25 + 30 +T21 (mK) +z +τνs = 2x1026 sec +τνs = 6x1026 sec +τνs = 1x1027 sec +(b) +Figure 2.3: The figure caption is same as in figure (2.1), except here, we keep +fabs(z, mνs) = 1/2 and vary lifetime of sterile neutrino. +35 +Sterile Neutrino Dark Matter +Chapter 2 + +21 cm Line Astronomy and Constraining New Physics +T21 profiles [51, 74, 75]. In the shaded region of the figures, the spin temperature +can be approximated as gas temperature. Therefore, when the gas temperature is +lower than CMB temperature, we get the absorption profile, i.e. T21 < 0. When +the gas temperature rises above the CMB temperature, T21 becomes positive, and +we see an emission profile. +In all figures (2.1b), (2.2b) and (2.3b), above the +redshift z ≳ 25, xα, xc < 1, therefore, the spin temperature is dominated by CMB +temperature, i.e. T21 ≈ 0. To get the absorption profile at z ∼ 17, one has to keep +Tgas < TCMB. +In figure (2.1a), we keep the mass of sterile neutrino (mνs) fix to 10 KeV. +The violet solid line depicts the gas temperature evolution when lifetime of sterile +neutrino is 2 × 1026 sec. As we increase the lifetime of sterile neutrino from 2 × +1026 sec to 1 × 1027 sec, the gas temperature decreases— shown by green and cyan +curves. It happens because by increasing the τνs, the radiative decay of sterile +neutrinos decreases and the number of photons injected into IGM also decreases. +Therefore, we get less heating of IGM by increasing the τνs, and it results in a +smaller amplitude (larger dip) of T21 shown by figure (2.1b). +In plot (2.2a), lifetime of sterile neutrino is fixed to 6 × 1026 sec and the values +of mνs varies from 2 KeV (violet solid line) to 25 KeV (yellow solid line). +If +one increases the sterile neutrino mass from 2 KeV (violet line) to 6 KeV (green +line), the heating of IGM decreases significantly in the shaded region. It happens +because ρνs = mνsnνs, nνs is the number density of sterile neutrinos. Therefore +at a particular redshift, when one increases mνs the number density of sterile +neutrino decreases, and we get less photon injunction, produced from decaying +sterile neutrinos, into the IGM. Hence, one gets less heating of IGM when the +mass of sterile neutrino increases, and it results in a smaller amplitude (larger dip) +of T21 shown by figure (2.2b). +If one considers the immediate and complete absorption of the photon energy +into IGM, then energy deposition efficiency, fabs = 1/2 — half of the total energy +of sterile neutrino will be carried away by active neutrino [162, 166]. The mass +Chapter 2 +Sterile Neutrino Dark Matter +36 + +21 cm Line Astronomy and Constraining New Physics +of the sterile neutrino in the equations (2.9) and (2.12), enters through only fabs. +Therefore, the energy deposition rate, equation (2.11), will depend only on the +lifetime of sterile neutrinos. This case has been depicted in figure (2.3a) for the +different values of τνs. In this case, as expected, the heating of IGM increases +more compared to the cases in figure (2.1a). The corresponding profiles for 21 cm +signal are shown in figure (2.3b). In figure (2.3a), the gas temperature for τνs = +2×1026 sec is higher than the CMB temperature in the shaded region– (violet line). +Therefore, we get a emission profile for T21 in figure (2.3b) for τνs = 2 × 1026 sec. +For τνs = 6 × 1026 sec, at redshift ∼ 17, the gas temperature is comparable to the +CMB temperature, therefore we do not see any absorption/emission in the 21 cm +signal. Above the redshift ∼ 17, temperature of gas is lower than CMB, therefore, +we see a small absorption in the profile. Below the redshift ∼ 17, the temperature +of the gas is higher than CMB, therefore, we see a emission profile. For the case +with τνs = 1 × 1027 sec, in the shaded region, temperature of the gas is smaller +than the CMB (cyan line), therefore, we get an absorption profile for the 21 cm +signal— figure (2.3b). +In figure (2.4), we plot the lower projected constraints on lifetime as a function +of mνs by requiring T21 such that it does not suppress the standard theoretical +value of T21(z = 17.2) ≈ −220 mK more than about a factor of 1/4 or 1/2. +Considering T21 < −150 mK, will further strengthen our projected bounds. The +red coloured curves depict the lower projected constraints on τνs when T21 ≃ +−150 mK, while the black coloured curves represent the lower projected constraint +on τνs when T21 ≃ −100 mK. To get the dashed line, we do not take into account +the VDKZ18 heating of the gas. For the dotted line we consider VDKZ18 heating +of the gas. Inclusion of VDKZ18, gives more stringent projected constraint on τνs +as gas temperature rises due to the energy transfer from CMB photons mediated +by Lyα photons. In figure (2.5), we obtained the upper projected constraint on +mixing angle of sterile neutrinos with active neutrinos as a function of mass. For +reference, we have also plotted the x-ray constraint on mixing angle as function +37 +Sterile Neutrino Dark Matter +Chapter 2 + +21 cm Line Astronomy and Constraining New Physics +of mνs. The constraint is obtained by assuming solely radiative decay of sterile +neutrinos. x-ray constraint comes from the fact that no such x-rays have been seen +in observations [35]. The red and black coloured curves depict the upper projected +constraint on mixing angle when T21 ≃ −150 mK and -100 mK, respectively. To +get the dashed curves, we do not take into account the VDKZ18 heating of the gas. +For the dotted line we have included the VDKZ18 heating of the gas. Here, it is to +noted that these bounds do not depend on dark-matter clustering. Therefore, the +bounds are free of astrophysical parameters such as density profile or mass function +of dark-matter halos, etc. To obtain these bounds, we do not consider any non- +standard cooling mechanism to cool the IGM or any source of radio photons. The +results in figure (2.5) are comparable with the x-ray constraint for the higher mass +of sterile neutrinos, while we get more stronger bounds for lower mass. +1025 +1026 +1027 +1028 +1029 +101 +τνs (sec) +mνs (KeV) +T21 ≃ -150 mK +T21 ≃ -100 mK +Figure 2.4: The figure represents lower projected bounds on the lifetime of sterile +neutrinos as a function of mass of sterile neutrinos by keeping 21 cm differential +brightness temperature, T21 ≃ −150 and −100 mK. The dotted (dashed) line +represents the case when energy transfer from CMB photons to gas is included +(excluded) [34]. +Chapter 2 +Sterile Neutrino Dark Matter +38 + +21 cm Line Astronomy and Constraining New Physics +10-14 +10-13 +10-12 +10-11 +10-10 +10-9 +10-8 +10-7 +10-6 + 2 + 4 + 6 + 8 + 10 + 20 + 30 + 40 + 50 +NuSTAR 20 +NuSTAR 22 +x-ray +Swift-XRT 22 +XMM 21 +sin2(θ) +mνs (KeV) +Z = 17.2 +T21 ≃ -150 mK +T21 ≃ -100 mK +Figure 2.5: The figure represents upper projected bounds on the mixing angle of +sterile neutrinos with active neutrinos as a function of mass of sterile neutrinos +by keeping 21 cm differential brightness temperature, T21 ≃ −150 and −100 mK. +The dotted (dashed) line represents the case when energy transfer from CMB +photons to gas is included (excluded) [34]. The shaded regions are excluded for +corresponding observations. The x-ray constraint on mixing angle (cyan shaded +region) has been taken from the Ref. [35]. The red shaded region depicts the +upper bounds on sin2(θ) from NuSTAR observations [36, 37]. Here, we have also +plotted the recently reported bounds (after publication of our article) on sin2(θ) +by NuSTAR— represented by NuSTAR 22 [37] and by Swift-XRT— represented +by Swift-XRT 22 [38]. The grey shaded region is excluded by XMM-Newton [39]. +39 +Sterile Neutrino Dark Matter +Chapter 2 + +21 cm Line Astronomy and Constraining New Physics +2.6 +Summary +We have constrained the sterile neutrino dark matter lifetime and mixing angle +with active neutrino as a function of sterile neutrino mass, such that energy in- +jection from radiative decay of sterile neutrino does not change the standard 21 +cm absorption signal (∼ −220 mK) more than about a factor of 1/4 (−150 mK) +or 1/2 (−100 mK) at the redshift, z = 17.2 . We have considered the two scenar- +ios to get the bounds: First, IGM evolution without the heat transfer from the +background radiation to gas mediated by Lyα photons (VDKZ18 effect). Next, we +have considered the VDKZ18 effect on the IGM gas. The following summarises +our results for T21 = −150 mK : +In the first scenario, the lower bound on the sterile neutrino lifetime varies +from 8.3 × 1027 sec to 9.4 × 1025 sec by varying sterile neutrino mass from 2 KeV +to 50 KeV. The lifetime of sterile neutrino decrease when one increases the mass +of the sterile neutrino. It happens because ρνs = mνsnνs. At a particular redshift, +when one increases mνs, the nνs decreases. Consecutively, one gets less radiative +decay of sterile neutrinos. Therefore, we get more window to increase the gas +temperature, i.e. we can decrease the lifetime of sterile neutrinos. The upper +bound on the mixing angle (sin2 θ) varies from 6.8×10−9 to 6.1×10−14 by varying +sterile neutrino mass from 2 KeV to 50 KeV. +In the second scenario, the lower bound on the sterile neutrino lifetime varies +from 1.5 × 1028 sec to 1.7 × 1026 sec by varying sterile neutrino mass from 2 KeV +to 50 KeV. While the upper bound on the mixing angle varies from 3.8 × 10−9 to +3.42 × 10−14 by varying sterile neutrino mass from 2 KeV to 50 KeV. +We have also plotted the x-ray constraint to rule out some parameter space for +mixing angle of the sterile neutrinos with active neutrinos [35]. Although we have +considered that sterile neutrinos account for all the dark matter in the Universe, +sterile-neutrino may account for only a fraction of the dark matter abundance. In +this scenario, the bounds on the sterile neutrino lifetime and mixing angle with +Chapter 2 +Sterile Neutrino Dark Matter +40 + +21 cm Line Astronomy and Constraining New Physics +active neutrino may modify. +2.7 +Additional study +2.7.1 +Bounds in light of varying T21 and redshift +We have also studied the projected constraints on τνs and sin2(θ) by varying the +absorption amplitude of T21 between 0 mK (no signal) to −200 mK at z = 17. +If we increase the value of T21 above 0, it gives a emission signal instead of an +absorption signal. Therefore, we restrict the maximum value of T21 to 0. Also, we +do not take the values of T21 below ∼ −200 mK, as the sterile neutrino term in +the temperature evolution equation becomes insignificant compared to adiabatic +and Compton scattering term. In the ΛCDM model without invoking any new +physics, we get T21(z = 17) = −220.215 for the cosmological parameters Ωm = +0.31, Ωb = 0.048 and h = 0.68 . For the demonstration purpose of this, we take +mνs = 2 KeV. After inclusion of physics of decaying sterile neutrinos, we get +T21(z = 17) = −220.213 mK for τνs = 4 × 1032 sec. If we increase the value of T21 +only by 9.7 × 10−2 percent (from T21 = −220.213 mK to −220 mK), the value of +τνs changes significantly from 4 × 1032 sec to 3.78 × 1030 sec— decreases by more +than a factor of hundred. Therefore, we do not consider bounds on the τνs and +sin2(θ) near the maximal absorptional value of T21; and vary the value of T21 from +−200 to 0 mK. +In figure (2.6a), we have plotted the lower projected bounds on the lifetime +of sterile neutrinos as a function of mass for various values of T21 at z = 17. In +figure (2.6b), we have plotted the upper projected bounds on the mixing angle +of sterile neutrinos with active neutrinos as a function of mass for various values +of T21 at z = 17. NuSTAR 22 bound is reported in July 2022 [37] and Swift- +XRT 22 is reported in August 22 [38]. The observational bounds indicate that the +decay of sterile neutrinos will not significantly impact the thermal history of the +41 +Sterile Neutrino Dark Matter +Chapter 2 + +21 cm Line Astronomy and Constraining New Physics +1022 +1023 +1024 +1025 +1026 +1027 +1028 +1029 + 2 + 4 + 6 + 8 + 10 + 20 + 30 + 40 + 50 +τνs (sec) +mνs (KeV) +Z = 17 +T21 ≃ 0 mK +T21 ≃ -50 mK +T21 ≃ -100 mK +T21 ≃ -150 mK +T21 ≃ -200 mK +(a) +10-14 +10-13 +10-12 +10-11 +10-10 +10-9 +10-8 +10-7 +10-6 + 2 + 4 + 6 + 8 + 10 + 20 + 30 + 40 + 50 +NuSTAR 20 +NuSTAR 22 +x-ray +Swift-XRT 22 +XMM 21 +sin2(θ) +mνs (KeV) +Z = 17 +T21 ≃ 0 mK +T21 ≃ -50 mK +T21 ≃ -100 mK +T21 ≃ -150 mK +T21 ≃ -200 mK +(b) +Figure 2.6: Plot (2.6a), shows lower projected bounds on the lifetime of sterile +neutrinos as a function of mass, while plot (2.6b), shows upper projected bounds +on the mixing angle of sterile neutrinos with active neutrinos as a function of mass +of sterile neutrinos for varying 21 cm differential brightness temperature (T21) +at z = 17. In figure (2.6b), the shaded regions are excluded for corresponding +observations. The cyan shaded region represents the x-ray constraint [35]. The red +shaded region depicts the upper bounds on sin2(θ) from NuSTAR observations [36, +37]. Here, we have also included the recently reported bounds (after publication of +our article) on sin2(θ) by NuSTAR— represented by NuSTAR 22 [37] and by Swift- +XRT— represented by Swift-XRT 22 [38]. The grey shaded region is excluded by +XMM-Newton [39]. +Chapter 2 +Sterile Neutrino Dark Matter +42 + +21 cm Line Astronomy and Constraining New Physics +1024 +1025 +1026 +1027 +1028 +1029 + 2 + 4 + 6 + 8 + 10 + 20 + 30 + 40 + 50 +τνs (sec) +mνs (KeV) +T21 ≃ -150 mK + Z = 15 +Z = 17 +Z = 19 +(a) +10-14 +10-13 +10-12 +10-11 +10-10 +10-9 +10-8 +10-7 + 2 + 4 + 6 + 8 + 10 + 20 + 30 + 40 + 50 +sin2(θ) +mνs (KeV) +T21 ≃ -150 mK +Z = 15 +Z = 17 +Z = 19 +(b) +Figure 2.7: Plot (2.7a), shows lower projected bounds on the lifetime of sterile +neutrinos, while plot (2.7b), shows upper projected bounds on the mixing angle +of sterile neutrinos with active neutrinos by keeping T21 to −150 mK for different +values of redshift. +Universe as the parameter space is excluded more stringently by observations for +a higher mass of sterile neutrinos. For example, XMM 21 excludes the values of +sin2(θ) ≳ 2 × 10−12 for mνs ≃ 6 KeV. If one wants to exclude this parameter space +using 21-cm signal, it requires to consider T21 < −200 mK— i.e. no significant +modification in the thermal and ionization history of the Universe. +For further analysis with variation of redshift values, we have also added +the plots for the case with different values of redshift keeping the value of T21 +constant— presented in figure (2.7). Here, we vary redshift between 15 to 19. As +it is shown in figure (4.4), for fiducial models for Lyα coupling and x-ray heating, +we can not take the spin temperature to be gas temperature above z ∼ 17 and +also x-ray starts to dominate below z ∼ 17. Therefore, we restrict ourselves about +redshift 17 and take a range from 15 to 19. Here, we do not see significant variation +in the projected bounds of lifetime and mixing angle with variation of the values +of redshift. +In figure (2.8), we vary both the value of redshift and T21, and plot the lower +43 +Sterile Neutrino Dark Matter +Chapter 2 + +21 cm Line Astronomy and Constraining New Physics +1024 +1025 +1026 +1027 +1028 +1029 + 2 + 4 + 6 + 8 + 10 + 20 + 30 + 40 + 50 +τνs (sec) +mνs (KeV) + Z = 15, T21 ≃ -178 mK +Z = 17, T21 ≃ -165 mK +Z = 19, T21 ≃ -155 mK +(a) +10-14 +10-13 +10-12 +10-11 +10-10 +10-9 +10-8 +10-7 + 2 + 4 + 6 + 8 + 10 + 20 + 30 + 40 + 50 +sin2(θ) +mνs (KeV) +Z = 15, T21 ≃ -178 mK +Z = 17, T21 ≃ -165 mK +Z = 19, T21 ≃ -155 mK +(b) +Figure 2.8: Plot (2.7a), shows lower projected bounds on the lifetime of sterile +neutrinos, while plot (2.7b), shows upper projected bounds on the mixing angle of +sterile neutrinos with active neutrinos by keeping T21 such that it does not change +more than a factor of 1/4 from the minimum possible amplitude based on ΛCDM +model for corresponding values of redshift. +projected bounds on τνs in figure (2.8a) and upper projected bounds on sin2(θ) +in figure (2.8b). Here, we choose the value of T21, such that, it does not change +more than a factor of 1/4 from the minimum possible amplitude based on ΛCDM +model. For the cosmological parameters, given above, we get the minimum pos- +sible amplitude of T Min +21 +to be −236.7 mK at z = 15, −220.2 mK at z = 17 and +−206.1 mK at z = 19. +Chapter 2 +Sterile Neutrino Dark Matter +44 + +“The universe doesn‘t allow perfection.” +Stephen Hawking, A Brief History of Time +3 +Primordial Black Hole Dark Matter +Primordial black holes have attracted much interest in recent years and have been +a part of intense studies for more than five decades. The idea of the black hole +goes back to the 18th century. In 1784, John Michell proposed that there could be +such supermassive bodies that light could not pass them, or all light emitted would +return towards them [10–12]. Later, in 1915 Albert Einstein developed the general +theory of relativity. In 1916, Karl Schwarzschild found the solution of black holes +by solving the Einstein field equations for a point mass [167]. Subsequently, in +1963, Roy Kerr found the solution of rotating black holes [168]. In 1965, the more +general solution of a rotating and charged black hole was found [169]. There is a +possibility that a colossal number of black holes might have been formed in the very +early Universe— known as primordial black holes (PBHs). PBHs can be created +by various mechanisms. It was first suggested by Zel’dovich and Novikov that the +presence of initial inhomogeneities in the Universe can form PBHs [170]. There is + +21 cm Line Astronomy and Constraining New Physics +a possibility that for many regions in the space, gravitational energy of the initial +density fluctuations can exceed the kinetic energy. These regions would have a +gravitational collapse instead of the expanding with Universe creating collapsed +objects with a minimum mass of ∼ 10−5 g [171–173]. There are various mechanisms +that can produce inhomogeneities in the early Universe. For example, such density +fluctuations can be generated due to the vacuum strings produced during the grand +unification phase transition [174]. Indeed, these fluctuations were present in the +very early Universe, as evident from observations of structures in the Universe. +The other explanations of PBHs formation include the collapse of cosmic string +loops, collisions of bubbles, etc. The cosmic string loops can disappear in two ways: +First, they can shrink into scalar and gauge particles. Second, some loops with +specific initial shapes may disappear by collapsing in size below their Schwarzschild +radius and form black holes [175–178]. In the article [179], the authors consider +the formation of PBHs due to collapsing cosmic strings and argue that PBHs can +significantly contribute to the dark matter density if their relic mass is larger than +103 mpl, here mpl is the Planck mass. In another scenario, the collapse of the cusps +neighbourhood of cosmic strings loops can also form a large number of spinning +PBHs [180]. The collisions between the bubbles during various phase transitions +in the Universe can also give rise to the formation of PBHs [181–183]. PBHs can +also be produced in various inflation models [184–186]. +Depending on the formation time (t), PBHs can have a wide range of masses +(in most of the cases roughly order of the Hubble horizon mass at the formation +time) [187, 188], +MPBH ∼ 1015 +� +t +10−23 sec +� +g. +(3.1) +For example, PBHs with mass 1015 g might have formed at t ∼ 10−23 after big- +bang. In another example, PBHs formed during the QCD phase transition (t ∼ +10−5 sec) might have a mass comparable to a solar mass [189]. +PBHs formed +Chapter 3 +Primordial Black Hole Dark Matter +46 + +21 cm Line Astronomy and Constraining New Physics +around neutrino decoupling (t ∼ 1 sec) can have a mass about 105 M⊙ . +3.1 +Primordial black holes as dark matter +In the last decades, many particle-dark matter models have been proposed to +explain the various astrophysical observations, as discussed in chapter (2). The +laboratory experiments for direct detection of dark matter have not observed any +signature yet, for example, DarkSide-50, LUX, XENON1T, PandaX-II, CRESST, +PICO, etc. +[190–195]. +In this situation, it is desirable to look for alternative +scenarios where dark matter may not be an elementary particle. As PBHs are +massive, interact only gravitationally and are formed in the very early Universe, +they can be considered as a potential candidate for non-particle dark matter. +Recently, PBHs have gathered much attention in the scientific community after +the black hole binary merger detection by Virgo and LIGO collaborations, and +these events suggest that PBHs may constitute a fraction of dark matter [13– +15, 196–198]. PBHs having a mass below ∼ 1022 g can explain all the dark matter +in the Universe as they are not ruled out by microlensing constraints [199]. We +will discuss other constraints on dark matter fractions in the form of PBHs with +mass below ∼ 1022 g later in the sections (3.2) and (3.4). One can explain the +existence of dark matter in the form of PBHs without considering physics beyond +the standard model (BSM) of particle physics by considering standard model Higgs +fluctuations during inflation as instability can occur in Higgs potential at a scale +O(1011 GeV) [200]. In the article [184], authors consider the double inflation model +to explain the formation of PBHs between two inflations and argue that PBHs can +be accounted for dark matter in the Universe. PBHs as missing matter or dark +matter in the context of galaxy formation has been explored in old literature also +[201, 202]. Authors of the Ref. [185], consider the formation of PBH dark matter +due to the mild-waterfall phase of hybrid inflation and discuss how the tail of the +mass distribution of PBHs can explain the origin for the supermassive black holes +47 +Primordial Black Hole Dark Matter +Chapter 3 + +21 cm Line Astronomy and Constraining New Physics +observed at galactic centres. These massive back holes can also provide the seed +for present-day observed structures [185, 203]. A fraction/all of dark matter in +the form of PBHs can produce the r-process nucleosynthesis— a process that is +responsible for producing about half of the heavier nuclei than iron, in the mass +range 10−14 M⊙ < MPBH < 10−8 M⊙ [204]. Black holes can lose their mass by the +emission of energetic particles due to Hawking evaporation [205]. For non-rotating +and non-charged black holes formed in the very early Universe, their evaporation +time scale can be given by [187], +τ(MPBH) ∼ +�MPBH +1015 g +�3 +Gyr. +(3.2) +Therefore, PBHs having mass larger than 1015 g can survive the Hawking evapo- +ration and account for present-day dark matter density [206]. +3.1.1 +Signature of Primordial Black Holes +It is possible that a fraction of PBHs can grow to intermediate-mass black holes and +explain the ultraluminous x-ray sources reported in various observations [185, 207– +209]. There are several hints that indicate the presence of PBHs, such as dynam- +ics and star clusters of ultra-faint-dwarf-galaxies, correlations between x-ray and +infrared cosmic backgrounds, etc. (for a detailed review, see Ref. [210]). The pres- +ence of evaporating PBHs can explain the galactic/extra-galactic γ-ray background +radiation [211–214], short-duration γ-ray bursts [215, 216], and reionization by in- +jection of energetic photons and e± radiations into IGM [217, 218]. The emission +of nucleons by evaporating PBHs can explain the observed baryon number density +if more baryons are produced compared to antibaryons— in a baryon-symmetric +Universe [213]. Clustering between PBHs can provide the seeds for galaxy forma- +tion. PBHs evaporation can explain the observed point-like γ-ray sources [217]. +The presence of massive PBHs can also serve as seeds for active galactic nuclei +(AGN) [217]. +Chapter 3 +Primordial Black Hole Dark Matter +48 + +21 cm Line Astronomy and Constraining New Physics +3.2 +Existing bounds on Primordial Black Holes +The fraction of dark matter in the form of PBHs (fPBH ≡ ΩPBH/ΩDM) is con- +strained from various astrophysical observations and theoretical predictions. Here, +ΩPBH and ΩDM are the dimensionless density parameters for PBHs and dark mat- +ter, respectively. PBHs with mass smaller than ∼ O(1015 g) may have evaporated +as of now and can be constrained from the impact on big bang nucleosynthesis by +evaporated particles, background radiation etc. Higher mass PBHs can be con- +strained by the effect on large-scale structures, gravitational wave and lensing, and +impact on thermal and ionization history of the IGM (for details, see the recent re- +views [187, 219, 220] and the references cited therein). In the context of the 21 cm +signal, the upper bound on the fPBH can be found in Refs. [221–228]. Angular mo- +mentum is a fundamental property of a black hole, and it can modify the Hawking +evaporation drastically [40, 229–231]. In the case of rotating PBHs, authors of the +Refs. [41, 232] have reported the various types of bound on fPBH as a function of +PBHs mass and spin. Future collaboration, All-sky Medium Energy Gamma-ray +Observatory (AMEGO)a will be able to constrain some parameter space for the +rotating PBHs [42]. We discuss more bounds on the fraction of PBH dark matter +in the result and discussion section (3.4). In this chapter, we consider the rotating +PBHs and constrain dark matter fraction in the form of PBHs as a function of +their mass for various values of angular momentum in the light of global 21 cm +signal [16]. +3.3 +Impact on the thermal and ionization history +During the cosmic dawn era, the evolution of the gas temperature and ionization +fraction of the Universe are well-known [153, 154]. The addition of any exotic +source of energy during the cosmic dawn era can significantly impact the ionization +ahttps://asd.gsfc.nasa.gov/amego/index.html +49 +Primordial Black Hole Dark Matter +Chapter 3 + +21 cm Line Astronomy and Constraining New Physics +and thermal history of the Universe. Therefore, we can constrain the properties of +such exotic sources from the observations during the cosmic dawn era. Evaporating +PBHs can heat the gas and modify the free electron fraction in the IGM [46, +232]. Rotating PBHs can emit more particles into IGM and substantially affect +the IGM evolution compared to non-rotating PBHs [214, 229, 233]. Therefore, +it is important to study the properties of spinning PBHs. Black holes can get +their spin depending on generation mechanisms, merger or accretion [234–244]. +PBHs with higher mass can have a lifetime larger/comparable than the age of the +Universe. Therefore, they have enough time to accrete mass and spin up [245]. +In the present work, we consider the Hawking emission of PBHs into background +radiations (photons and electron/positron) and provide the projected constraints +on the fraction of dark matter in the form of PBHs (fPBH) as a function of mass and +spin. We analyse projected bounds on spinning PBHs such that 21 cm differential +brightness temperature does not change by more than a factor of 1/4 from the +ΛCDM model prediction (|T21| ∼ 220 mK). +A rotating black hole with angular momentum JPBH and having mass MPBH +can be defined with a rotation parameter, a∗ = JPBH/(GN M 2 +PBH) [233], where GN +is the gravitational constant. Rotating black hole with higher spin (a∗ → 1) injects +more energy into IGM and evaporates faster than non-rotating ones [40, 229–231]. +Therefore, we expect that the bounds on fPBH to be more stringent compared to +non-rotating PBHs. The energy injection per unit volume per unit time due to +e± and photons into IGM, for monochromatic mass distribution of PBHs, can be +written as [232, 246], +Γe± +PBH(z, a∗) = 2 +� � +f e +c (E − me, z) (E − me) +� d2Ne +dt dE +� � +nPBH dE , +(3.3) +Γγ +PBH(z, a∗) = +� � +f γ +c (E, z) E +� d2Nγ +dt dE +� � +nPBH dE . +(3.4) +Energy injection into IGM happens by three processes: heating, ionization, and +excitation of the gas [247–249]. +f i +c represents the energy deposition efficiency +Chapter 3 +Primordial Black Hole Dark Matter +50 + +21 cm Line Astronomy and Constraining New Physics +into IGM. Here, c stands for above-mentioned three channels and i ≡ (electron/ +positron, photon) stands for different types of injected particles. The factor of 2 +in equation (3.3) accounts for the total contribution of electrons and positrons. +nPBH = fPBH (ρDM/MPBH) is the number density of the PBHs, and ρDM is the dark +matter energy density. +d2N i/(dt dE) ≡ d2N i/(dt dE) +� +E, MPBH, a∗ +� +represents +the number of i particles emitted by black hole per unit time per unit energy +[232, 233, 250, 251], +d2N i +dt dE = 1 +2 π +� +dof +Γi(E, MPBH, a∗) +eE′/TPBH ± 1 +, +(3.5) +here, Γi is the greybody factor— defines the probability of emitted particle i from +black hole to overcome its gravitational potential well. dof represents the degree +of freedom [251]. Moreover, E is the total energy of emitted particle i and E′ = +E − nΩ. While, n is the axial quantum number and Ω is the angular velocity at +black hole horizon. We use the BlackHawk codeb to calculate the spectra due to +photons, electrons and positrons; we take both the primary and secondary Hawking +evaporation spectra into account— i.e. emitted final particle j per unit time and +per unit energy [251, 252] +d2N j +dt dE = +� +i +d2N i +dt dE′′ +dN i +j +dE dE′′ , +(3.6) +here, dN i +j is the hadronization table accounts for the transformation of the primary +spectra into secondary spectra [251–253]. +In the presence of Hawking radiation, the thermal evolution of the gas can be +written as[157, 254], +dTgas +dz += 2 Tgas +1 + z + +Γc +(1 + z) H (Tgas − TCMB) − +2 ΓPBH +3 ntot(1 + z) H , +(3.7) +bhttps://blackhawk.hepforge.org/ +51 +Primordial Black Hole Dark Matter +Chapter 3 + +21 cm Line Astronomy and Constraining New Physics +here, ΓPBH = Γe± +PBH + Γγ +PBH is the total energy injection per unit time and per unit +volume into IGM. We consider the following numerical values of the cosmological +parameters: h = 0.674, ΩM = 0.315, Ωb = 0.049 and TCMB|z=0 = 2.725 K [48, 255]. +To compute the energy deposition efficiency, thermal and ionization history of the +Universe, we use DarkHistoryc package with necessary modifications [249]. +3.4 +Results and Discussion +We take 21 cm differential brightness temperature such that it does not change, +from its ΛCDM value (∼ 220 mK), by more than a factor of 1/4 at redshift +17.2 . We solve the coupled equations (3.7) and (2.9— replacing E with ΓPBH) +for different mass, spin and fraction of PBH dark matter to get xHI and Tgas at +redshift z = 17.2 . To get any absorption signal in redshift range 15 − 20, the gas +temperature should be less than CMB temperature in shaded region— redshift +range from 15 to 20. By requiring T21 ≃ −150 mK at z=17.2, we constrain the +parameter space of PBH dark matter. In the present chapter, we do not consider +x-ray heating of the gas due to the uncertainty in the physics of the first stars— +as we discussed earlier. For a fix value of T21 at a redshift, if one includes the +x-ray heating of gas, our projected upper constraints on PBH dark matter fraction +becomes stronger. Here, it is to be noted that the gas temperature may increase +due to the energy transfer from the background radiation to the thermal motions +of the gas mediated by Lyα radiation from the first stars [34]. However, again due +to the uncertainty in physics of the first star formation, we do not include this +effect also. The inclusion of this effect will also further strengthen our projected +upper bounds on fPBH— similar to discussed in chapter (2). +In order to understand how spin, fraction and mass of PBH dark matter can +affect the thermal evolution of the gas, we plot the figures (3.1), (3.2) and (3.3), +respectively. The shaded region corresponds redshift range, 15 − 20 . The red +chttps://darkhistory.readthedocs.io/en/master/ +Chapter 3 +Primordial Black Hole Dark Matter +52 + +21 cm Line Astronomy and Constraining New Physics +10-1 +100 +101 +102 +103 +104 +101 +102 +TCMB +Tgas (No PBH) +Tgas (K) +1+Z +MPBH = 1x1015 g, fPBH = 10-7 +a* = 0 +a* = 0.5 +a* = 0.99 +Figure 3.1: The gas temperature evolution with redshift for evaporating primordial +black hole. The red dashed lines represent the CMB temperature evolution. The +black dashed lines depicts the Tgas when there is no PBHs. The shaded region +corresponds to the redshift 15 ≤ z ≤ 20 (EDGES observed signal). In this figure, +we consider PBHs mass and fPBH to 1 × 1015 g and 10−7, respectively, and vary +the spin of PBHs. +10-1 +100 +101 +102 +103 +104 +101 +102 +TCMB +Tgas (No PBH) +Tgas (K) +1+Z +MPBH = 1x1015 g, a* = 0.5 +fPBH = 1 x 10-6 +fPBH = 1 x 10-7 +fPBH = 1 x 10-8 +Figure 3.2: The caption is the same as in Figure (3.1), except here, we keep +MPBH = 1 × 1015 g and a∗ = 0.5 constant and vary fPBH. +53 +Primordial Black Hole Dark Matter +Chapter 3 + +21 cm Line Astronomy and Constraining New Physics +10-1 +100 +101 +102 +103 +104 +101 +102 +TCMB +Tgas (No PBH) +Tgas (K) +1+Z + a* = 0.5, fPBH = 10-7 +MPBH = 1 x 1015 g +MPBH = 3 x 1015 g +MPBH = 5 x 1015 g +Figure 3.3: The caption is the same as in Figure (3.1), except here, we vary the +mass of PBHs and keep spin and fPBH to 0.5 and 10−7, respectively. +dashed curves in all plots depict the CMB temperature evolution, while the black +dashed line represents the gas temperature when there are no evaporating PBHs. +In Figure (3.1), we keep mass to 1 × 1015 g and fPBH = 10−7, and vary the spin +of PBHs. As expected, when we increase the spin of PBHs, the gas temperature +rises significantly in the shaded region. The solid violet curve represents the case +when the spin of PBHs is 0. +Increasing the spin to 0.5 (solid green line), the +gas temperature increases. Further increasing a∗ to 0.99 (solid cyan line), the gas +temperature rises further. In Figure (3.2), we keep MPBH = 1 × 1015 g, spin to 0.5 +and vary fPBH. In this plot, as we increase the fPBH from 10−8 (solid cyan line) +to 10−6 (solid violet line), the IGM heating rises rapidly. If the gas temperature +becomes larger than the CMB temperature in the shaded region, it can erase the +21 cm absorption signal; instead, it may give an emission signal. Therefore, at +desired redshift (in our scenario z = 17.2), one has to keep Tgas < TCMB to get an +absorption signal. Increasing fPBH, for a given mass, the number density of PBHs +increases resulting in more energy injection into IGM by Hawking evaporation of +PBHs. Therefore, fPBH plays a significant role in deciding whether one gets an +Chapter 3 +Primordial Black Hole Dark Matter +54 + +21 cm Line Astronomy and Constraining New Physics +absorption profile or emission. In Figure (3.3), we vary the mass of PBHs and keep +spin and fPBH constants to 0.5 and 10−7, respectively. In this plot, as we increase +the mass of PBHs from 1 × 1015 g (solid violet line) to 5 × 1015 g (solid cyan line), +the gas temperature decreases. It happens for two reasons: (i) Increasing the mass +of PBHs leads to a decrease in the total power contributions from Hawking evapo- +ration of PBHs [250]. (ii) Ignoring the integral dependency in equations (3.3) and +(3.4), Γe± +PBH and Γγ +PBH are proportional to nPBH = fPBH (ρDM/MPBH). For a fixed +dark-matter energy density and fPBH, the number density of PBHs increases by +decreasing the black hole mass. Thus, energy injection into IGM per unit volume +and time (ΓPBH) increases, and one gets more heating of the gas. +55 +Primordial Black Hole Dark Matter +Chapter 3 + +21 cm Line Astronomy and Constraining New Physics +10-10 +10-9 +10-8 +10-7 +10-6 +10-5 +1015 +1016 +IGRB, a* = 0.9 +Planck, a*=0 +COMPTEL, a*=0 +fPBH +MPBH (g) +a* = 0 +a* = 0.5 +a* = 0.9 +a* = 0.9999 +(a) +10-5 +10-4 +10-3 +10-2 +10-1 +100 +1016 +1017 +1018 +INTEGRAL, +a* = 0.9 +Leo T, a*=0 +AMEGO, a*=0 +(forecast) +AMEGO, a*=0.9999 +(forecast) +fPBH +MPBH (g) +a* = 0 +a* = 0.5 +a* = 0.9 +a* = 0.9999 +(b) +Chapter 3 +Primordial Black Hole Dark Matter +56 + +21 cm Line Astronomy and Constraining New Physics +Figure 3.4 (previous page): The projected upper bounds on the dark fraction of +matter in the form PBHs (fPBH = ΩPBH/ΩDM) as a function of PBHs mass for +different values of a∗. The shaded regions are excluded from our analysis for fPBH +when a∗ = 0 (dotted black line), 0.5 (dot-dashed black line), 0.9 (dashed black line) +and 0.9999 (solid black line). The dashed blue curve depicts the upper constraint +on fPBH by observations of the diffuse Isotropic Gamma-Ray Background (IGRB) +for a∗ = 0.9 [40]. The double-dot-dashed blue curve represents the upper constraint +on fPBH from Diffuse Supernova Neutrino Background (DSNB) searches at Super- +Kamiokande, while the solid blue line represents the INTErnational Gamma-Ray +Astrophysical Laboratory (INTEGRAL) observation of 511 KeV γ-ray lines at +Galactic centre constraint on fPBH for a∗ = 0.9 [41]. The double-dot-dashed ma- +genta (red) line represents the AMEGO forecast for a∗ = 0 (a∗ = 0.9999) [42]. +Near future, AMEGO collaboration will be able to probe the parameter-space +above the magenta (red) double-dot-dashed curve for a∗ = 0 (a∗ = 0.9999). The +solid green line stands for 95% confidence level bound from INTEGRAL obser- +vation of Galactic gamma-ray flux for non-spinning PBHs [43]. Solid cyan curve +depicts the upper bound from observing the 511 KeV γ-ray lines at the Galactic +centre by assuming all the PBHs within a 3 Kpc radius of the Galactic centre for +non-spinning PBHs [44]. The magenta solid line represents the Planck constraint +[45]. The red solid line depicts the dwarf galaxy Leo T constraint [46] and the +green dashed line shows the COMPTEL bound [47] for non-spinning PBHs. +In Figure (3.4), we plot the upper projected bounds on the fraction of dark +matter in the form of PBHs as a function of PBHs mass for different spins. Here, +we have considered that 21 cm differential brightness temperature, T21, remains +−150 mK at redshift z = 17.2. We vary the mass of PBHs from 1015 g to 1018 g. +The shaded regions in both the plots are excluded for the corresponding PBH spins. +The dashed blue curve represents the upper constraint on fPBH by observations +57 +Primordial Black Hole Dark Matter +Chapter 3 + +21 cm Line Astronomy and Constraining New Physics +of the diffuse Isotropic Gamma-Ray Background (IGRB) [40]. The double-dot- +dashed blue curve represents the upper constraint on fPBH from Diffuse Supernova +Neutrino Background (DSNB) searches at Super-Kamiokande, while the solid blue +line represents the INTErnational Gamma-Ray Astrophysical Laboratory (INTE- +GRAL) observation of 511 KeV γ-ray line at Galactic centre constraint on fPBH +for a∗ = 0.9 [41]. For a∗ = 0, the observation at the Jiangmen Underground Neu- +trino Observatory (JUNO) will be able to place a 20 times stronger bound on the +upper allowed value of fPBH for MPBH = 1015 g compared to Super-Kamiokande +[41, 256]. The double-dot-dashed magenta (red) line represents the AMEGO fore- +cast for a∗ = 0 (a∗ = 0.9999) [42]. In the near future, AMEGO collaboration will +be able to probe the parameter-space above the magenta (red) double-dot-dashed +curve for a∗ = 0 (a∗ = 0.9999). Solid green line stands for 95% confidence level +bound from INTEGRAL observation of Galactic γ-ray flux for non-spinning PBHs +[43]. The solid cyan curve depicts the upper bound from the observation of 511 +KeV γ-ray lines at the Galactic centre by assuming all the PBHs within a 3 Kpc +radius of the Galactic centre for non-spinning PBHs [44]. For the comparison, we +have also plotted the bounds from Planck [45], Leo T [46] and COMPTEL [47] +observations for non-spinning PBHs. In Figure (3.4a), fPBH varies from 1 × 10−10 +to 1 × 10−5, while, in Figure (3.4b), it varies from 1 × 10−5 to its maximum al- +lowed value 1 (ΩPBH = ΩDM). In Figure (3.4), as we increase the value of spin +from 0 to its extremal value, 0.9999, the upper bounds become more stringent. +This is due to an increment in evaporation of PBHs, and it results in more energy +injection into the IGM [233, 257, 258]. As discussed earlier, increasing the mass of +PBHs, energy injection into IGM decreases. Subsequently, one gets more window +to increase the gas temperature or fPBH, and the upper bound becomes weaker. +Therefore, in Figure (3.4), the upper bound on fPBH weakens as we increase the +mass. +Our upper projected constraint on fPBH for a∗ = 0.9 is comparable to +the INTEGRAL observation of 511 KeV γ-ray lines for PBHs mass larger than +∼ 8×1016 and becomes stronger for smaller PBH masses. Also, compared to IGRB +Chapter 3 +Primordial Black Hole Dark Matter +58 + +21 cm Line Astronomy and Constraining New Physics +[40] and DSNB [41], our projected bounds are stringent for the considered mass +range of PBHs. We find the most robust lower projected constraint on the mass +of PBHs, which is allowed to constitute the entire dark matter, to 1.5 × 1017 g, +1.9 × 1017 g, 3.9 × 1017 g and 6.7 × 1017 g for PBH spins 0, 0.5, 0.9 and 0.9999, +respectively. The lower bound on MPBH for ΩPBH = ΩDM, for extremal spinning +PBHs is nearly four times larger than non-spinning PBHs. +3.5 +Conclusions +Spinning primordial black holes can substantially affect the ionization and thermal +history of the Universe. Subsequently, it can modify the 21 cm absorption signal +in the cosmic dawn era by injecting energy due to Hawking evaporation. +We +study the upper projected bounds on the fraction of dark matter in the form +of PBHs as a function of mass and spin, considering that the 21 cm differential +brightness temperature does not change by more than a factor of 1/4 from the +theoretical prediction based on the ΛCDM framework. Our projected constraints +are stringent compared to DSNB, INTEGRAL observation of the 511 KeV line, +IGRB, Planck, Leo T and COMPTEL. In the near future, AMEGO collaboration +will be able to probe some parameter space in our considered mass range of PBHs. +In the present work, we have considered the monochromatic mass distribution of +PBHs. The allowed parameter space can also be explored for different PBHs mass +distributions such as log-normal, power-law, critical collapse, etc. [251]. Here, it +is to be noted that we have not considered heating of IGM gas due to x-ray from +the first stars in the vague of known physics of the first stars. For a fix value of +T21 at a redshift, if one includes the x-ray heating of the gas, the projected bounds +becomes stronger. +59 +Primordial Black Hole Dark Matter +Chapter 3 + +21 cm Line Astronomy and Constraining New Physics +3.6 +Additional study +3.6.1 +Bounds in light of varying T21 and redshift +We also study upper projected bounds on the fraction of the dark matter in the +from of PBHs by varying the amplitude of 21 cm differential brightness tempera- +ture and redshift. In figure (3.5), we have plotted upper bounds on the fraction of +dark matter in the form of primordial black holes as a function of mass for various +values of T21 at z = 17. To understand that how bounds change on fPBH with T21, +we consider two scenarios for the spin of PBHs: a∗ = 0 (figure 3.5c) and a∗ = 0.9 +(figure 3.5d). In both of the plots, we notice that when we change the value of T21 +from −200 mK to −150 mK the bound relaxes with a factor of ∼ 4.3 . By changing +the T21 from −150 mK to −100 mK bounds relax by a factor of ∼ 2.3, and, going +from −100 mK to −50 mK the bounds relax by a factor of ∼ 2.25. By changing +the T21 from −50 mK to 0, bounds relax by a factor of ∼ 3.1 . This similar pattern +10-10 +10-8 +10-6 +10-4 +10-2 +100 +1015 +1016 +1017 +fPBH +MPBH (g) +a* = 0, Z = 17 +T21 ≃ 0 mK +T21 ≃ -50 mK +T21 ≃ -100 mK +T21 ≃ -150 mK +T21 ≃ -200 mK +(c) +10-10 +10-8 +10-6 +10-4 +10-2 +100 +1015 +1016 +1017 +fPBH +MPBH (g) +a* = 0.9, Z = 17 +T21 ≃ 0 mK +T21 ≃ -50 mK +T21 ≃ -100 mK +T21 ≃ -150 mK +T21 ≃ -200 mK +(d) +Figure 3.5: The plots represent the upper projected bounds on the fraction of dark +matter in the form of primordial black holes (fPBH) as a function of mass of PBHs +(MPBH) for varying 21 cm differential brightness temperature (T21) at z = 17. +Figure (3.5c) represents the case when spin of PBHs: a∗ = 0, while, figure (3.5d) +represents the case with a∗ = 0.9 . +Chapter 3 +Primordial Black Hole Dark Matter +60 + +21 cm Line Astronomy and Constraining New Physics +10-10 +10-8 +10-6 +10-4 +10-2 +100 +1015 +1016 +1017 +fPBH +MPBH (g) +T21 ≃ -150 mK, a* = 0 +Z = 15 +Z = 17 +Z = 19 +(a) +10-10 +10-8 +10-6 +10-4 +10-2 +100 +1015 +1016 +1017 +fPBH +MPBH (g) +a* = 0 +Z = 15, T21 ≃ -179 mK +Z = 17, T21 ≃ -166 mK +Z = 19, T21 ≃ -155 mK +(b) +Figure 3.6: Figure (3.6a) represents upper projected bounds on fPBH when T21 ≃ +−150 mK for different values of redshift (z). Figure (3.6b) represents upper bounds +on fPBH when T21 does not change more than a factor of 1/4 from the minimum +possible amplitude based on ΛCDM model for corresponding values of redshift +(T Min +21 (z = 15) ≃ −238 mK, T Min +21 (z = 17) ≃ −221.2 mK and T Min +21 (z = 19) ≃ +−207 mK). Here, the cosmological parameters are: h = 0.674, ΩM = 0.315, Ωb = +0.049 [48]. Both figures obtained for a∗ = 0. +61 +Primordial Black Hole Dark Matter +Chapter 3 + +21 cm Line Astronomy and Constraining New Physics +also occurs for the case of sterile neutrinos and the factors remain same. Therefore, +we can also find the constraints on fPBH for other values of spin when the bound +on fPBH is given for any value of T21 ∈ {0, −50, −100, −150, −200} mK. +In figure (3.6a), similarly to sterile neutrino case, we see that bounds do not +change significantly for a fix value of T21 at different values of redshift. In figure +(3.6b), upper bounds on fPBH are obtained such that T21 does not change more +than a factor of 1/4 from the minimum possible amplitude based on ΛCDM model +for corresponding values of redshift. In this chapter, we consider the following +numerical values of the cosmological parameters: h = 0.674, ΩM = 0.315, Ωb = +0.049 and TCMB|z=0 = 2.725 K [48, 255]. Therefore, we get minimum possible +value of T21 based on ΛCDM model to −238 mK at z = 15, −221.2 mK at z = 17 +and −207 mK at z = 19. +Chapter 3 +Primordial Black Hole Dark Matter +62 + +“Astronomy and Pure Mathematics are the mag- +netic poles toward which the compass of my mind +ever turns.” +Carl Friedrich Gauss, In Letter to Bolyai (30 +Jun 1803) +4 +Primordial Magnetic Fields and Excess +Radio Background +Observations suggest that the magnetic fields are ubiquitous in the Universe— +from the length scale of planets and stars to the cluster of galaxies [17–20]. Fermia +and High Energy Stereoscopic System (HESS)b gamma-ray observation suggests +that even voids could host magnetic fields with strength O(10−16 G) with a typical +coherent scale of Mpc [21, 22]. Magnetic fields can also play a significant role in +reionization, relic electron density and structure formation [259]. The presence of +magnetic fields can substantially affect the evolution and dynamics of structures +in the Universe as they can contribute to the total pressure against gravitational +collapse. +This could modify the total matter power spectrum on small scales, +ahttps://fermi.gsfc.nasa.gov/ +bhttps://www.mpi-hd.mpg.de/hfm/HESS/ + +21 cm Line Astronomy and Constraining New Physics +≲ 1 Mpc [259–264]. The presence of magnetic fields during recombination can +also have important consequences, such as it could have lead to the collapse of +gas clouds after recombination, formation of first pre-galactic stars, quasars [265]. +The Earth has a magnetic field of the order of O(G), and it is sustained for years +by some dynamo mechanism. Similarly, other astronomical objects near to Earth, +such as Sun and other solar system planets, also show the presence of magnetic +fields [266]. Our home galaxy Milky Way, other spiral galaxies and their interstellar +medium (ISM) contain magnetic fields with the strength O(µG) [261, 264, 266– +268]. Moreover, galaxy clusters, intercluster medium, filaments, IGM, etc., also +show the magnetic fields [19, 21, 269–271]. These magnetic fields are likely to be +seeded by primordial magnetic fields (PMFs). These PMFs might have originated +in the very early Universe, and subsequently amplified in the small scale structures +by some mechanisms [21, 24]. +4.1 +Generation of primordial magnetic fields +The origin and evolution of PMFs is one of the outstanding problems of modern +cosmology (Ref. [23, 24] and references cited therein). It would be very difficult to +explain the magnetic fields in the voids and high redshift galaxies with only late- +time astrophysical processes without magnetic fields from the very early Universe. +Therefore, these magnetic fields indeed may have a primordial origin [21, 272– +274]. There are several theoretical models that can generate the magnetic field +in the early Universe with a large coherent scale. The two scenarios to generate +PMFs that are vastly discussed in the literature are phase transitions in the early +Universe and various models of inflation (for details, see the recent review [24]). +In the Ref. [275], the authors discuss how the inflation model can generate large +scale, ∼ O(Mpc), magnetic fields. The generated magnetic fields have a small +strength. To amplify the field, one has to break the conformal invariance of the +electromagnetic field. The authors consider three mechanisms to break the con- +Chapter 4 +PMFs & Excess Radio Background +64 + +21 cm Line Astronomy and Constraining New Physics +formal invariance: Coupling of the photon to the axions, gravitational field and +massless-charged-nonconformally invariant scalar field [275]. Authors of the Ref. +[276], extend the inflation model by introducing the coupling between the Maxwell +field and the scalar field (Φ) responsible for inflation (∝ eαΦFµνF µν), here, Fµν is +the electromagnetic field tensor. This scenario can generate magnetic fields with +a present-day strength up to nG with the coherence scale of a few Mpc depending +on the parameter α. A similar mechanism to generate the magnetic fields during +inflation is based on the superstring cosmology [277, 278]. The Lagrangian, simi- +lar to considered by [276] with α = −1, naturally arises from the effective action +in low-energy string theory. Here, inflation is driven by the kinetic part of the +dilaton scalar field— Φ′. Whereas in the article [276], it is driven by the false vac- +uum scalar field potential— which is too steep for producing the slow-roll inflation +[272, 277]. In the article [279], authors argue that the back reaction of generated +magnetic fields via inflation can spoil the inflation. Considering the backreaction, +the authors put an upper bound on the present-day strength of magnetic fields to +10−32 G on the Mpc scale. This strength seems too small for galactic dynamos to +amplify to explain the observed magnetic fields. In the recent article [280], it is +shown that this issue can be circumvented for some parameter space. The authors +find that magnetic fields with a present-day strength of ∼ 10−13 G with a scale of +Mpc can be generated while keeping the backreactions under control. Magnetic +fields can also arise during electroweak [281] and quantum-chromo-dynamics [282] +phase transitions. Other mechanisms include cosmic strings [283, 284], primor- +dial plasma vorticity [285], etc. In this chapter, we obtain the upper bounds on +present-day strength of PMFs for various values of spectral index in the light of +EDGESc observation and excess radio background observed by the ARCADE 2 +& LWA 1 observation [25]. Here, we obtain the bounds on PMFs in both the +presence and absence of heating effects due to first stars. +cRecently, the EDGES signal has been questioned in many articles. We discuss this point in +chapter (6). +65 +PMFs & Excess Radio Background +Chapter 4 + +21 cm Line Astronomy and Constraining New Physics +4.2 +Existing bounds on primordial magnetic fields +The present-day strength, spectral index and coherence scale of PMFs depends on +their generation mechanisms. Therefore, the constraints on PMFs can give a hint of +the early Universe physics. Recently in the Ref. [286], authors show that PMFs can +be used as a remedy to resolve the Hubble tension between different observations. +The present-day amplitude of PMFs is constrained from the BBN, formation of +structures and temperature anisotropies & polarization of CMB [259, 287, 288]. +Authors of the Ref. [50], put an upper constraint to ∼ 10−10 G on 1 Mpc scale +by considering Tgas ≲ TCMB (i.e. +T21 ≲ 0) so that, PMFs can not erase the +T21 absorptional signal in the redshift range 15 ≲ z ≲ 20. Planck 2015 results +put individual upper constraints of the O(nG) for different cosmological scenarios +on 1 Mpc scale [49]. The authors of the Ref. [289], in the context of EDGES +observation, put an upper and lower constraint on the PMFs to be 6 × 10−3 nG +and 5 × 10−4 nG respectively. Also, the lower bound on the present-day strength +of PMFs found in Refs. [290–292]. Further, in the Ref. [21], authors put a lower +bound on the strength of intergalactic magnetic fields of the order of 3 × 10−16 G +using Fermi observations of TeV blazars. Authors of the reference [293], report +upper bound of 2 × 109 G at the end of BBN. Presence of PMFs can modify the +present-day relic abundance of He4 and other light elements. Therefore, magnetic +fields can be constrained by observations of light element abundances [259, 294– +297]. The authors of the Ref. [298], put an upper bound of 47 pG for scale-invariant +PMFs by comparing CMB anisotropies, reported by the Wilkinson Microwave +Anisotropy Probe (WMAP) and Planck, with calculated CMB anisotropies. +4.3 +Evolution of PMFs after recombination +The generation of the magnetic fields in the early Universe for the various cosmo- +logical scenarios has been studied in the earlier literature (for example see Refs. +Chapter 4 +PMFs & Excess Radio Background +66 + +21 cm Line Astronomy and Constraining New Physics +[272, 282, 290, 299, 300]). It is to be noted that decaying magnetic fields has been +studied in several literatures. In these works, the authors consider the decay of the +PMFs by ambipolar diffusion and turbulent decay [26, 50, 254, 259, 301]. Ambipo- +lar diffusion of magnetic fields is important in neutral medium as it is inversely +proportional to free-electron fraction (xe) and xe ∼ 10−4 after redshift z ≲ 100 +[160, 254, 259]. The presence of PMFs can induce the Lorentz force in the gas. +The force exerts only on free electrons and ions leaving the neutral components +unaffected. This can result in creating a velocity difference between charged and +neutral components. The velocity difference can enhance the collision frequency in +the gas, resulting in a dissipation of magnetic energy into the gas— known as the +ambipolar diffusion of magnetic fields [302]. After the recombination (z ∼ 1100), +the radiative viscosity of fluid dramatically decreases, and velocity perturbations +are no longer damped. Therefore, the tangled magnetic fields having length scale +smaller than the magnetic Jeans length can dissipate via another mode— tur- +bulent decay [254, 259, 303]. Magnetic heating of the gas due to the turbulent +decay decreases with redshift but later when ionization fraction decreases, heat- +ing increases due to ambipolar diffusion [254, 259]. We further discuss about the +ambipolar and turbulent decay in section (4.5). Decaying PMFs can inject mag- +netic energy into the thermal energy of the IGM and heat the gas above 6.7 K at +z = 17, and even it can erase the EDGES absorption signal [50, 254, 259]. Still, +the EDGES absorption signal can be explained by considering the possible early +excess of radio radiation [304]. +4.4 +Background excess radio radiation +The Absolute Radiometer for Cosmology, Astrophysics and Diffuse Emission (AR- +CADE 2) collaborationd, a double-nulled balloon-borne instrument with seven ra- +diometers, measured the absolute sky temperature in a frequency range of 3 − +dhttps://asd.gsfc.nasa.gov/archive/arcade/ +67 +PMFs & Excess Radio Background +Chapter 4 + +21 cm Line Astronomy and Constraining New Physics +90 GHz. The observation reported excess radio radiation in a frequency range of +3 − 10 GHz [27], +T(ν) = T0 + Tr (ν/ν0)β , +(4.1) +here, ARCADE 2 observation fitted the parameters as: T0 = 2.731 ± 0.004 K, +β = −2.6, Tr = 21.1 ± 3.0 K and ν0 = 310 MHz. By combining ARCADE 2 with +the Low-frequency data [305–308] and Far Infrared Absolute Spectrophotometer +(FIRAS) data [309], the parameters can be fitted as: T0 = 2.725 ± 0.001 K, +β = −2.599 ± 0.036, Tr = 24.1 ± 2.1 K and ν0 = 310 MHz in a frequency range +of 22 MHz−10 GHz [27]. This is measured at present-day (z = 0). The radiation +temperature maps with redshift as: ∝ (1 + z), we can multiply T(ν) with (1 + z) +for past [304, 310–314], +T(z) = T0 (1 + z) +� +1 + Tr +T0 +� 78 +310 +�β +× +� +ν +78 MHz +�β +� +. +(4.2) +This radiation is several times larger than the observed radio counts due to the +known Galactic and extragalactic radio processes and sources, such as star-forming +galaxies, AGN-driven sources— quasars and radio galaxies, etc. [28, 29]. The +presence of early excess radiation can not be completely ruled out at the time +of cosmic dawn. +For example, in the redshift range z ≈ 30 to 16, accretion +onto the first intermediate-mass black holes can produce a radio radiation [315]. +Accreting supermassive black holes [316] or supernovae [317] can also produce radio +background due to synchrotron emission at the time of cosmic down by accelerated +electrons in the presence of the magnetic field. The enhancement in the background +radiation is also supported by the first station of the Long Wavelength Array (LWA +1)e in frequency range 40 − 80 MHz. The excess observed by LWA 1 can also be +fitted by the same model given by equation (4.1). After the inclusion of LWA +ehttps://leo.phys.unm.edu/ lwa/ +Chapter 4 +PMFs & Excess Radio Background +68 + +21 cm Line Astronomy and Constraining New Physics +1 data with ARCADE 2 [27] and Low-frequency data [305–308], the parameters +change as: T0 = 2.722 ± 0.022 K, β = −2.58 ± 0.05 and Tr = 30.4 ± 2.6 K +at ν0 = 310 MHz [30, 318]. For the observation of 21-cm signal, we can write: +ν = 1420.4/(1 + z) MHz. In the equation (4.2), the factor of (ν/78 MHz)β can be +defined as a fraction of excess radio background, Ar. Depending on the origin, Ar +can have different values— we discuss about this more in next sections. Therefore, +in the final analysis, we vary the value of excess radiation fraction. +4.4.1 +Excess radiation during the cosmic dawn +In this work, we use the EDGES signal in the presence of excess radio radiation to +constrain the strength of PMFs. Some of the processes which we have discussed +responsible for the excess radio background can occur at earlier redshift (z ∼ 17) +[315–317]. Also, one of the interesting proposals in the Ref. [304] is to argue that +such a possibility can exist at the time of cosmic dawn, and it can help to explain +the EDGES signal. Here, authors show that the EDGES absorption signal can be +explained by having only 10 percent of the observed radio background by ARCADE +2. In Ref. [319, 320], the authors claim that thermal emission from the axion +quark nugget dark matter model can explain the EDGES signal, and it can also +contribute a fraction of the radiation excess observed by ARCADE 2. At present, +there exist several theoretical models to explain this excess at the time of cosmic +dawn. The stimulated emission from Bose stars can give a large contribution to +the radio background and explain the EDGES and ARCADE 2 observations [321]. +The radio emission from accreting Pop III black holes can produce the EDGES like +signal by increasing background radiation temperature [322]. In other scenarios, +the EDGES anomaly can be explained by axion-photon conversion in the presence +of intergalactic magnetic fields [323] or by radiative decays of standard model +neutrino induced by magnetic fields [324]. Radio excess can also be explained by +the cusp region of superconducting cosmic strings [325]. In ref. [326], authors +69 +PMFs & Excess Radio Background +Chapter 4 + +21 cm Line Astronomy and Constraining New Physics +consider radiative decays of relic neutrino and show that it can potentially explain +the ARCADE 2 excess together with the EDGES signal. Depending on the origin, +the excess fraction of radio radiation can have a different value. We discuss the +constraints on excess radiation later. Considering the above possibilities of having +early excess radiation, we believe that it is important to analyze constraints on +the primordial magnetic field in the presence of such radiation. +4.4.2 +Phenomenological model for excess radiation +As discussed in subsection (4.4.1), the possibility of an excess radio radiation back- +ground over the CMBR can not be denied. For the excess radio background, we +consider the phenomenological model following the Ref. [310–314]. Here, Authors +consider a uniform redshift-independent synchrotron-like radiation, motivated by +the ARCADE 2 and LWA 1 observations. This model can explain the EDGES +anomaly in addition to enhancement of cosmic down power spectrum. Accord- +ingly, from equation (4.2) and following the Refs. [310–314], +TR = T0 (1 + z) +� +1 + Ar +� +νobs +78 MHz +�β � +, +(4.3) +where, T0 = 2.725 K is the present day CMB temperature and β = −2.6 is the +spectral index. Here, Ar is the amplitude defined relative to the CMB at reference +frequency of 78 MHz. For the 21 cm signal νobs is 1420.4/(1 + z) MHz. Authors of +the Ref. [310], put a limit on the excess radiation background to 1.9 < Ar < 418 +at reference frequency of 78 MHz by considering the effect of an uniform radiation +excess on the 21 cm signal from the cosmic dawn, dark ages and reionization. +Authors consider a synchrotron-like spectrum with spectral index −2.6 . The case +with Ar ∼ 418 corresponds to the LWA 1 limit on Ar at the reference frequency +of 78 MHz [30, 310]. The stringent constraint on excess radiation comes from the +Low-Frequency Array (LOFAR) to Ar < 182 (95 percent CL) and Ar < 259 (99 +percent CL) for a spectral index of −2.6 [313]. +Chapter 4 +PMFs & Excess Radio Background +70 + +21 cm Line Astronomy and Constraining New Physics +4.5 +Impact on the thermal and ionization history +due to primordial magnetic fields +In the presence of decaying magnetic fields, the gas temperature can increase. +Tgas can even increase above the background radiation and can erase the 21 cm +absorption signal reported by EDGES [50, 254, 259, 303]. Therefore, present-day +PMFs strength can be constrained by the EDGES observation in the presence of +excess radiation reported by ARCADE 2 and LWA 1 [5, 27, 30, 304, 310, 327]. In +the presence of turbulent decay and ambipolar diffusion, the thermal evolution of +the gas with the redshift can be written as [254, 259, 302, 303, 328], +dTgas +dz += 2 Tgas +1 + z + +Γc +(1 + z) H (Tgas − TCMB) +− +2 +3 ntot(1 + z) H (Γturb + Γambi) , +(4.4) +Here, fHe = 0.079 and TCMB = T0 (1 + z) is the cosmic microwave background +(CMB) temperature. At early times, Tgas remains in equilibrium with CMB tem- +perature due to Compton scattering. However, the gas temperature will not be +strongly affected by the comparatively small amount of energy in the non-thermal +radio radiation. Therefore, Tgas and Tα can be assumed independent of the excess +radiation [304]. The change in the free electron fraction (xe) with redshift is given +by equation (2.9) with E = 0 . Heating rate per unit volume due to the ambipolar +diffusion (Γambi) and turbulence decay (Γturb) is given by [254, 259], +Γambi = +(1 − xe) +γ xe (MH Nb)2 +|(∇ × B) × B|2 +16 π2 +, +(4.5) +Γturb = +1.5 m [ln(1 + ti/td)]m +[ln(1 + ti/td) + 1.5 ln{(1 + zi)/(1 + z)}]m+1H EB , +(4.6) +here, m = 2(nB + 3)/(nB + 5), zi = 1088 is the redshift when heating starts due +the magnetic fields (recombination epoch), γ = 1.9 × 1014 (Tgas/K)0.375cm3/g/s is +71 +PMFs & Excess Radio Background +Chapter 4 + +21 cm Line Astronomy and Constraining New Physics +the coupling coefficient, MH is the mass of hydrogen atom and Nb is the number +density of baryons. td = 1/ +� +kd VA(kd, z) +� +is the decay time for the turbulence. For +matter dominated era, ti = 2/ +� +3 H(zi) +� +and VA(kd, z) = B(kd, z)/ +� +4 π ρb(z) +�1/2 is +the Alfv´en wave velocity. B(kd, z) is the magnetic field strength smoothed over the +scale of kd at redshift z. kd is constrained by the damping wavenumber of Alfv´en +wave. PMFs with wavenumber (k) larger than kd, are strongly damped by the +radiative-viscosity [259, 303, 329–332]. Moreover, EB = B2/(8π) is the magnetic +field energy density, +dEB +dz += 4 EB +1 + z + +1 +H (1 + z) ( Γturb + Γambi ) . +(4.7) +Here, we assume that PMFs are isotropic and homogeneous Gaussian random +magnetic field, whose power spectrum is given by the following equation [50, 259, +262, 333] +⟨ ˜Bi(k) ˜B∗ +j(q)⟩ = (2π)3 +2 +δ3 +D(k − q) +� +δij − kikj +k2 +� +PB(k) , +(4.8) +here, PB(k) is the magnetic power spectrum, k = |k| is the comoving wave number +and δD is the Dirac delta function. Here, we consider a power-law spectrum of the +magnetic fields in the Fourier space for k < kd [50], +PB(k) = +(2π)2 +Γ +� +(nB + 3)/2 +� B2 +0 +� +k +Mpc−1 +�nB +Mpc3 . +(4.9) +Here, nB is the spectral index. In particular, nB = 2 for white noise [265], nB = 4 +for the Batchelor spectrum [334] and nB = −2.9 for nearly scale invariant spec- +trum [259]. As discussed above, magnetic fields are strongly damped by the large +radiative-viscosity for wavenumber larger than kd before recombination. There- +fore, we consider a sharp cut-off for power spectrum of PMFs: PB(k) = 0 for +k ≥ kd [50]. Following the Ref. [50], we take the time evolution of the Alfv´en +wave damping scale: kd(z) = kd,i f(z) and f(zi) = 1. Here, kd,i is the damping +Chapter 4 +PMFs & Excess Radio Background +72 + +21 cm Line Astronomy and Constraining New Physics +wavenumber at recombination epoch, +kd,i = 2π Mpc−1 +� +1.32 × 10−3 +� B0 +nG +�2 � 0.02 +Ωbh2 +� �Ωmh2 +0.15 +�1/2 �− +1 +nB+5 +. +(4.10) +Here, to smooth the magnetic field amplitude over the inverse length scale of kd,i , +we choose the Gaussian window function in Fourier space (k) as [49, 50, 335], +B2 +kd,i = +� ∞ +0 +d3k +(2π)3 e +−k2� +2π +kd,i +�2 +PB(k) = B2 +0 +� +kd,i +2π Mpc−1 +�nB+3 +. +(4.11) +The magnetic field strength smoothed over the scale of 1 Mpc, +B2 +1 Mpc = +� +(dk/2π)3 exp[−(k/Mpc−1)2] PB(k) = B2 +0 . +Lorentz force and the magnetic energy density can be calculated as [50], +|(∇ × B) × B|2 = +� +k1,k2 +k2 +1 PB(k1) PB(k2) f 2nB+8(z) (1 + z)10 , +(4.12) +here +� +k1,k2[· · · ] = +� � +d3k1/(2π)3 × d3k2/(2π)3 [· · · ], and +EB = 1 +8π +� +d3k +(2π)3 PB(k) f nB+3(z) (1 + z)4 . +(4.13) +We can get the redshift evolution of the function f(z), by substituting equation +(4.13) in equation (4.7). +4.6 +Impact on the thermal and ionization history +due to background radiation +Heating of IGM due to background radio radiation during cosmic dawn era has +been discussed in chapter (2). After inclusion of heating due to excess radio radi- +73 +PMFs & Excess Radio Background +Chapter 4 + +21 cm Line Astronomy and Constraining New Physics +ation and x-ray, the equation (4.4) will modify, +dTgas +dz += dTgas +dz +����� +[eq.(4.4)] ++ dTgas +dz +����� +x−ray +− +ΓR +(1 + z) (1 + fHe + xe) , +(4.14) +where, dTgas/dz +�� +[eq.(4.4)] stands for the gas temperature evolution represented in +equation (4.4). +To include the x-ray heating of the gas, we consider the tanh +parameterization [51, 74, 75]. In the presence of x-ray radiation, the ionization +fraction evolution will also change. For the present case, we consider the fiducial +model, for x-ray heating and ionization fraction evolution, motivated by Ref. [51]. +The heating effects of both the VDKZ18 (the last term in equation 4.14— discussed +in chapter 2) and x-ray are shown in plots (4.1, 4.2, 4.4, 4.7 & 4.9). +4.7 +Result and discussion +We consider the following values for the cosmological parameters: Ωm = 0.31, +Ωb = 0.048, h = 0.68, σ8 = 0.82 and ns = 0.97 [48]. To study the gas temperature +evolution with redshift in the presence of primordial magnetic field dissipation, +we solve the coupled equations (2.9 with E = 0), (4.4) and (4.7). +To get the +Lorentz force term in equation (4.5), we solve the equation (4.12). Similarly, to +get the magnetic field energy density in equation (4.6), we solve the equation +(4.13). To get the evolution of the f(z) with redshift, df(z)/dz, we substitute +equation (4.13) in equation (4.7) with initial condition f(zi) = 1 . To obtain upper +constraint on PMFs strength, we solve the equation (1.19) with equations (4.4), +(2.9 with E = 0) and (4.7) for T21 ≃ −300 mK or −500 mK by varying B0, nB +and Ar. For infinite Lyα coupling TS ≃ Tgas, therefore, TS solely depends on the +gas temperature. While, for finite Lyα coupling, TS depends on both the gas and +background radiation temperature. +In figures (4.1), (4.2) & (4.3), we plot the gas temperature vs. redshift for +different values of present-day strengths of PMFs (B0) and excess radio background +Chapter 4 +PMFs & Excess Radio Background +74 + +21 cm Line Astronomy and Constraining New Physics +100 +101 +102 + 10 + 20 + 30 + 40 +Tgas (K) +z +No heating +VDKZ18, Ar = 0 +VDKZ18, Ar = 100 +VDKZ18, Ar = 418 +x-ray +VDKZ18 + x-ray, Ar = 0 +VDKZ18 + x-ray, Ar = 100 +Figure 4.1: The gas temperature evolution with redshift. +The solid blue lines +represent the case when there is no x-ray, VDKZ18 or magnetic heating. VDKZ18 +corresponds to the heat transfer from the background radiation to gas mediated +by Lyα. The shaded region represents the EDGES observation redshift range, +15 ≤ z ≤ 20 . In this figure, we consider only VDKZ18 and x-ray heating with +excess radiation (Ar). +75 +PMFs & Excess Radio Background +Chapter 4 + +21 cm Line Astronomy and Constraining New Physics +100 +101 +102 + 10 + 20 + 30 + 40 + 50 +Tgas (K) +z +No heating +VDKZ18, Ar = 0, B0 = 0 nG +B0=1×10-1 nG +VDKZ18, Ar = 0, B0=1×10-1 nG +VDKZ18, Ar=100, B0=1×10-1 nG +VDKZ18+x-ray, Ar=100, B0=1×10-1 nG +B0=3×10-1 nG +VDKZ18, Ar=100, B0=3×10-1 nG +x-ray, B0=3×10-1 nG +VDKZ18+x-ray, Ar=100, B0=3×10-1 nG +Figure 4.2: The caption is same as in figure (4.1), except here, we include different +combination of VDKZ18, x-ray and magnetic heating, and spectral index is fixed +to −2.99 . +100 +101 +102 +103 +104 +101 +102 +103 +Tgas (K) +z +No heating +B0 = 3×10-1 nG, nB = -2.99 +B0 = 3×10-1 nG, nB = -2.50 +B0 = 3×10-1 nG, nB = -1.99 +B0 = 1×10-1 nG, nB = -2.50 +B0 = 1×10-1 nG, nB = -1.99 +Figure 4.3: The caption is same as in figure (4.1), except here, we vary the spectral +index and plot magnetic heating of the gas. +Chapter 4 +PMFs & Excess Radio Background +76 + +21 cm Line Astronomy and Constraining New Physics +fraction (Ar). The solid blue lines represent the case when there is no heating of +the IGM gas, i.e. +no x-ray, VDKZ18 or magnetic heating. +The pink shaded +band in the figure shows the EDGES redshift range, 15 ≤ z ≤ 20, for the 21 cm +absorption signal. In plot (4.1), we consider only VDKZ18 and x-ray heating. +The orange dashed line describes the heating due to VDKZ18 only while keeping +Ar = 0. Next, we increase the value of Ar from 0 to 100. This case is described +by the dashed-green line in plot (4.1), which shows a significant rise in the gas +temperature due to the excess radiation fraction. Further, if one increases the Ar +to its LWA 1 limit, i.e. Ar = 418, the gas temperature does not change significantly +from Ar = 100 case, as shown by the solid magenta curve. It happens because +ΓR ∝ (TR/TS − 1) ∼ TR/TS, equation (2.15). As we increase Ar, TR/TS increases +slowly. For example, at z = 17, TR/TS is 6.5 for Ar = 0, 51.4 for Ar = 100 and +54.9 for Ar = 418. Here, we can see that, even increasing Ar to ∼ 4 times (100 to +418), TR/TS increases by only 6.8 percent. Therefore, increasing further Ar will +not affect gas temperature significantly. To analyse the role of x-ray heating, we +have first considered the heating due to x-ray only, depicted by the red dashed +line. The inclusion of VDKZ18 for Ar = 0 further increases the gas temperature +slightly, as shown by the black dashed line. +In this case of inclusion of x-ray +heating, if we increase the value of Ar to 100, there is a significant increase in the +gas temperature as shown by the solid green line. We find the contribution due to +x-ray heating dominates for redshift values z ≲ 15. +In plot (4.1), we compare the contribution of VDKZ18 and x-ray heating. In +plot (4.2), we compare the contributions of VDKZ18, x-ray and magnetic heating +while keeping the spectral-index, nB = −2.99 for a nearly scale-invariant magnetic +field spectrum. While in figure (4.3), we vary the magnetic spectral index (nB) +and plot the magnetic heating of the gas. +In plot (4.2), we have included the effect of primordial magnetic fields on the +IGM gas evolution. The solid blue line represents the case when there is no heating, +and the dashed-black curve shows the case of VDKZ18 with no magnetic fields +77 +PMFs & Excess Radio Background +Chapter 4 + +21 cm Line Astronomy and Constraining New Physics +and x-ray for Ar = 0. The double dot-dashed green curve represents the case +when there is only the magnetic heating with a magnetic field strength of B0 = +1×10−1nG. Next, we include the case of VDKZ18 for Ar = 0 in the pure magnetic +heating scenario, as shown by the red dashed curve. Now, if we increase Ar from 0 +to 100, the gas temperature rises significantly in the shaded region as shown by the +dash-dotted red curve in figure (4.2). Now the further addition of x-ray heating is +shown by the cyan plot, which shows significant heating in the shaded region. Next, +for more analysis, we increase the magnetic field strength from B0 = 1×10−1 nG to +B0 = 3×10−1 nG and study cases with VDKZ18 and x-ray as before. The magenta +dashed line depicts the case with only magnetic heating. The green dashed line +shows the case of VDKZ18 with Ar = 100. The orange curve shows the case with +magnetic and x-ray heating only. Here, as expected, the gas temperature decreases +after the inclusion of the x-ray effect with the magnetic fields. It happens because +the ionization fraction increases by x-ray radiation. Ambipolar diffusion evolves as +Γambi ∝ (1−xe)/xe; therefore, as ionization fraction increases, ambipolar diffusion +of the magnetic field decreases. +Thus, the heating due to magnetic fields also +decreases. +Therefore, including the x-ray contribution with the magnetic field +decreases the magnetic field diffusion. Hence, the gas temperature decreases (this +effect also occurs for B0 = 1 × 10−1 nG, but it is not visible in the plot). The +black dot-dashed line includes all the three effects: magnetic and x-ray heating +together with VDKZ18 for Ar = 100 and B0 = 3 × 10−1 nG. Here, the addition of +the VDKZ18 heating for Ar = 100 increases the gas temperature above the solid +orange line. It is also lower than the magenta dashed line because of the inclusion +of the x-ray contribution. At the smaller redshift, x-ray heating dominates over +all other heating mechanisms, and all lines merge. +In figure (4.3), we plot the magnetic heating of the gas for the different spectral +index (nB) and B0. The solid lines, except the blue one, represent the magnetic +heating for B0 = 3 × 10−1 nG, while double dot-dashed lines are for B0 = 1 × +10−1 nG. Increasing the spectral index, the magnetic heating due to ambipolar +Chapter 4 +PMFs & Excess Radio Background +78 + +21 cm Line Astronomy and Constraining New Physics +diffusion and turbulent decay increases as Γambi ∝ +� +1/Γ[(nB + 3)/2] +�2 and Γturb ∝ +1/Γ[(nB + 3)/2] (by ignoring the logarithmic and integral dependencies). +For +example, if one changes nB from its value −2.99 to −1 then 1/Γ[(nB+3)/2] changes +from 5 × 10−3 to 1. +Therefore, by increasing nB from −2.99 to −1, magnetic +heating enhances significantly. To get T21 (equation 1.19) around −500 mK or +−300 mK at z = 17.2, one needs to ensure that even by increasing nB, that +the factor xHI (1 − TR/TS) remains same. Thus from equations (4.5), (4.6) and +(4.9) when we increase nB, we have to decrease B0 so that the magnetic heating +contribution to the gas remains the same. Therefore, by increasing nB, the upper +bound on B0 will become more stringent. Here, we also include the collisional +ionization of the gas in equation (2.9), as this term is important only when gas +temperature is ≳ 1.58 × 105 K. Otherwise this term is exponentially suppressed +as ∝ exp[−(13.6 eV)/Tgas] [259, 336, 337]. In plot (4.3), the gas temperature rises +by increasing B0, as more magnetic energy is getting injected into thermal energy +of the gas via Γambi ∝ E2 +B and Γturb ∝ EB. However, for redshift z ≲ 100, the gas +temperature starts decreasing as the cooling effect due to expansion of the Universe +become dominant, as can be seen in equations (4.4) & (4.7) (it also depends on +the strength and spectral index of the magnetic field). Since, with the expansion +of the Universe, magnetic energy density (EB) also dilutes, the contributions from +Γambi and Γturb decreases as can be seen from equations (4.5), (4.6) and (4.7). +In figure (4.4), we plot the spin (dashed lines) and gas (solid lines) tempera- +ture. For Ar = 0, i.e. TR = TCMB, we get Tgas ≃ TS as seen by the overlapping +dashed and solid blue lines in the shaded region. xα and xc are ∝ 1/TR as can be +seen from equations (1.8) and (1.9). Therefore, the coupling between the gas and +spin temperature decreases by increasing Ar. As discussed before, increasing the +value of Ar above ∼ 100, the spin temperature increases, but the increment in gas +temperature becomes insignificant, and the TR/TS ratio increases slowly. There- +fore, as xα and xc decreases, the difference between the gas and spin temperature +increases, as shown in the plot (4.4). Increasing the values of Ar from 100 (green +79 +PMFs & Excess Radio Background +Chapter 4 + +21 cm Line Astronomy and Constraining New Physics +100 +101 +102 + 10 + 20 + 30 + 40 + B0 = 3×10-1 nG, nB = -2.99 +T (K) +z +VDKZ18, Ar = 0 +VDKZ18, Ar = 100 +VDKZ18, Ar = 418 +VDKZ18+x-ray, Ar = 100 +Figure 4.4: This figure shows the gas (solid lines) and spin (dashed lines) temper- +ature evolution, The shaded region corresponds to the redshift 15 ≤ z ≤ 20 — the +redshift range for EDGES reported signal. +-1000 +-800 +-600 +-400 +-200 + 0 + 10 + 15 + 20 + 25 + 30 + B0 = 3×10-1 nG, nB = -2.99 +T21 (mK) +z +VDKZ18, Ar = 0 +VDKZ18, Ar = 100 +VDKZ18, Ar = 418 +VDKZ18+x-ray, Ar = 100 +Figure 4.5: This figure shows the 21 cm differential brightness temperature with +redshift for same cases in plot (4.4). +Chapter 4 +PMFs & Excess Radio Background +80 + +21 cm Line Astronomy and Constraining New Physics +101 +102 +Ar +10 +15 +10 +14 +10 +13 +10 +12 +10 +11 +10 +10 +10 +9 +10 +8 +B0 (Gauss) +-1.0 +-1.25 +-1.50 +-1.75 +-2.0 +-2.25 +-2.50 +-2.75 +-2.99 +-2.99 +2.99 +2.75 +2.50 +2.25 +2.00 +1.75 +1.50 +1.25 +1.00 +nB +2.99 +2.75 +2.50 +2.25 +2.00 +1.75 +1.50 +1.25 +1.00 +Figure 4.6: In this figure, we study upper bounds on present-day magnetic field +strength (B0) with excess radiation fraction (Ar) for different values of the spectral +index, nB. The green-yellow and red-grey colour schemes represent the cases when +T21|z=17.2 ≃ −500 mK and −300 mK, respectively. For T21|z=17.2 ≃ −300 mK case +the value of nB written with blue coloured text , while for −500 mK case it is +written with black coloured text. Here, we consider TS ≃ Tgas and do not take into +account the x-ray and VDKZ18 effects. +lines) to 418 (black lines), the difference between gas and spin temperatures in- +creases. Figure (4.5), shows the plots for 21 cm differential brightness temperature +vs. redshift, for all the cases discussed in plot (4.4). As we increase the Ar from 0 +to 100 the |T21| increases. By increasing Ar from 100 to 418, values of T21 does not +change significantly. Further, including x-ray heating and magnetic heating (for +B0 = 3 × 10−1 nG and nB = −2.99) the gas temperature rises and |T21| decreases. +In figures (4.6) and (4.7), we plot the maximally allowed values of B0 versus +radiation excess (Ar) for different spectral indexes. The colour-bars represent the +81 +PMFs & Excess Radio Background +Chapter 4 + +21 cm Line Astronomy and Constraining New Physics +101 +102 +Ar +10 +15 +10 +14 +10 +13 +10 +12 +10 +11 +10 +10 +10 +9 +10 +8 +B0 (Gauss) +-1.0 +-1.25 +-1.50 +-1.75 +-2.0 +-2.25 +-2.50 +-2.75 +-2.99 +-2.99 +2.99 +2.75 +2.50 +2.25 +2.00 +1.75 +1.50 +1.25 +1.00 +nB +2.99 +2.75 +2.50 +2.25 +2.00 +1.75 +1.50 +1.25 +1.00 +Figure 4.7: The caption is same as in figure (4.6), except here, we consider the +effects of VDKZ18 and x-ray heating on the gas due to first stars after z ≲ 35 and +consider finite Lyα coupling. +Chapter 4 +PMFs & Excess Radio Background +82 + +21 cm Line Astronomy and Constraining New Physics +variation of the magnetic field spectral index. In the plots, the spectral index +varies from its nearly scale-invariant value (-2.99) to -1. Here, we consider both +the EDGES best fit and upper constraint on the 21 cm absorption signal for +constraining B0. The green-yellow colour scheme represents the case with T21|z=17.2 +≃ −500 mK, while the red-grey colour scheme represents the case with T21|z=17.2 ≃ +−300 mK. Numerical values of nB for the different colour bands are written with +different colour. +For T21|z=17.2 ≃ −300 mK case the value of nB written with +blue coloured text , while for T21|z=17.2 ≃ −500 mK case it is written with black +coloured text. The colour-bars are common for both the plots. +In figure (4.6), we consider infinite Lyα coupling (xα ≫ xc, 1), i.e. TS ≃ Tgas. +Here, we do not consider the x-ray and VDKZ18 effects on the gas and thus the +21 cm signal T21 ∝ (1 − TR/Tgas). +As we increase Ar, the amplitude of |T21| +increases, and we get more window to increase the gas temperature. In this plot, +we consider heating only due to the decaying magnetohydrodynamics. Therefore, +we can increase B0 as we increase Ar. As discussed earlier, by decreasing nB, the +amplitude of the magnetic field power spectrum also decreases, resulting in less +magnetic energy dissipation into the gas kinetic energy. Thus by reducing values of +nB from -1 to -2.99, we get more window to increase B0. Next, when one increases +T21 from -500 mK to -300 mK, the allowed value of B0 also increases. This is shown +by the red-grey colour scheme in figures (4.6) and (4.7). In figure (4.7), we consider +the effects of VDKZ18 and x-ray on IGM gas evolution due to first stars after z ≲ +35 and consider finite Lyα coupling. As discussed earlier, Tgas ̸= TS for Ar > 0 and +the difference between gas and spin temperature increases as Ar increases. Thus, +in the presence of first star’s effects, the upper bound on the present-day strength +of PMFs modifies. Following the Refs. [51, 74, 75], we consider WF coupling +coefficient, xα = 2Aα(z) × (T0/TR). Here, Aα(z) = Aα(1 + tanh[(zα0 − z)/∆zα]), +the step height Aα = 100, pivot redshift zα0 = 17 and duration ∆zα = 2. The +collisional coupling coefficient, xc = T10/TR×(NH kHH +10 )/A10. After the inclusion of +x-ray and VDKZ18 heating effects, the gas temperature remains > 10 K. Therefore, +83 +PMFs & Excess Radio Background +Chapter 4 + +21 cm Line Astronomy and Constraining New Physics +we can take kHH +10 +≈ 3.1 ×10−11 (Tgas/K)0.357 exp(−32 K/Tgas) cm3/sec for 10 K < +Tgas < 103 K. As illustrated in plot (4.1), (4.2), (4.4) & (4.5), increasing excess +radiation fraction Ar above ∼ 100, the TR/TS remains nearly constant and this +also mean that T21 remain unchanged. Consequently one can not increase the +value of B0 and one gets nearly flat profile for B0 for Ar ≳ 100 in figure (4.7). +In figures (4.8) & (4.9), we plot the maximally allowed values of B0 vs nB +for various values of Ar. The colour-bars represent the variation in Ar. In the +plots, Ar varies from 5 to LWA 1 limit ∼ 418. We consider both the EDGES +best fit and upper constraint on 21 cm absorption signal for constraining B0. +The green-yellow scheme represent the case with T21|z=17.2 ≃ −500 mK, while +the red-grey colour scheme represent the case T21|z=17.2 ≃ −300 mK. Numerical +values of Ar for the different colour bands are written in different colours. For +T21|z=17.2 ≃ −300 mK case the value of Ar written with blue coloured text , while +for T21|z=17.2 ≃ −500 mK case it is written with black coloured text. The spectral +index ranges from -2.99 to -1. The red dashed line represents the Planck 2015 +upper constraint on the present-day magnetic field strength with spectral index in +both plots. This constraint has been taken from Refs. [49, 50]. +In plot (4.8), we consider TS ≃ Tgas and we do not take into account the x- +ray and VDKZ18 effects on IGM gas evolution. The zoomed inset in the figure +shows the contour plot when T21|z=17.2 ≃ −300 mK. Here, considering T21|z=17.2 ≃ +−300 mK, for nB < −2.98 the Ar ≳ 200 is excluded similarly for nB < −2.96 +the Ar ≳ 280 is excluded by Planck 2015 upper constraint on B0. Likewise, for +T21|z=17.2 ≃ −500 mK, for nB < −2.97 the Ar ≳ 280 is excluded. For spectral index +-2.9 and excess radiation fraction 418, we get the upper constraint on B0 to be +∼ 1 nG and 1.3 nG by requiring T21|z=17.2 ≃ −500 mK (EDGES best fit constraint) +and −300 mK (EDGES upper constraint), respectively. While for nB = −1, these +bound change to 1.1 × 10−3 nG and 1.6 × 10−3 nG for T21|z=17.2 ≃ −500 mK +and −300 mK, respectively. In plot (4.9), we include both the VDKZ18 and x- +ray effect and consider finite Lyα coupling. As discusses earlier, for Ar ≳ 100, +Chapter 4 +PMFs & Excess Radio Background +84 + +21 cm Line Astronomy and Constraining New Physics +3.0 +2.5 +2.0 +1.5 +1.0 +nB +10 +15 +10 +14 +10 +13 +10 +12 +10 +11 +10 +10 +10 +9 +10 +8 +B0 (Gauss) +418 +418 +280 +280 +200 +120 +60 +10 +5 +5 +10 +60 +120 +200 +280 +418 +Ar +5 +10 +60 +120 +200 +280 +418 +Figure 4.8: In this figure, we study upper bounds on the present-day magnetic +field strength (B0) with spectral index (nB) for different values of excess radiation +fraction (Ar). The green-yellow and red-grey colour schemes represent the cases +when T21|z=17.2 ≃ −500 mK and −300 mK, respectively. For T21|z=17.2 ≃ −300 mK +case the value of nB written with blue coloured text , while for −500 mK case it +is written with black coloured text. The red dashed line depicts the Planck 2015 +upper constraint on the present-day magnetic field strength [49, 50]. Here, we +consider TS ≃ Tgas and do not take into account the x-ray and VDKZ18 effects. +85 +PMFs & Excess Radio Background +Chapter 4 + +21 cm Line Astronomy and Constraining New Physics +3.0 +2.5 +2.0 +1.5 +1.0 +nB +10 +15 +10 +14 +10 +13 +10 +12 +10 +11 +10 +10 +10 +9 +10 +8 +B0 (Gauss) +- 418 +60 +- 418 +60 +10 +5 +5 +10 +60 +120 +200 +280 +418 +Ar +5 +10 +60 +120 +200 +280 +418 +Figure 4.9: The caption is same as in figure (4.8), except here, we consider the +heating effects of VDKZ18 and x-ray on IGM gas due to first stars after z ≲ 35 +and consider finite Lyα coupling. The colour-bars are common for both plots. +Chapter 4 +PMFs & Excess Radio Background +86 + +21 cm Line Astronomy and Constraining New Physics +TR/TS ratio remain nearly constant. Thus, in the plot (4.9), we can see that for +Ar ≳ 100, the upper bound on B0 is not changing significantly— the plots are +merged for Ar ≳ 100. These plots have been shown by the zoomed inset. The +right upper zoomed inset is shown for T21 ≃ −300 mK, while left lower zoomed +inset is shown for green-yellow contour plots when T21 ≃ −500 mK. Hence, further +increasing Ar > 100 will not change significantly the upper bound on B0. As +illustrated in figure (4.4), TS > Tgas for Ar > 0, and T21 ∝ (1−TR/TS). Therefore, +to get T21 ≃ −300 mK or −500 mK, we need to lower B0 compared to previous +scenario— figure (4.8). Hence, we get the more stringent upper bound on present- +day magnetic field strength in figure (4.9). +For spectral index -2.9 and excess +radiation fraction 418, we get the upper constraint on B0 to be ≲ 1.7×10−1 nG and +1.2 × 10−1 nG by requiring T21|z=17.2 ≃ −300 mK and −500 mK, respectively. For +nB = −1, we get B0 ≲ 6.9×10−5 nG and 3.7×10−5 nG by requiring EDGES upper +and best fit constraint on 21 cm differential brightness temperature. Decreasing +the values of Ar, the upper constraint on B0 becomes more stringent. For example, +when Ar = 5, we get upper bound on present day magnetic field strength to be +≲ 1.4 × 10−1 nG for spectral index -2.99, and for spectral index nB = −1 we get +B0 ≲ 3.8 × 10−6 nG by requiring EDGES best fit constraint on T21. The upper +bounds are also well below the Planck 2015 constraint [49]. +4.8 +Conclusions +In the present work, we study the upper constraint on the strength of the pri- +mordial magnetic fields for different spectral index using the bound of EDGES +observation on T21, in the presence of uniform redshift-independent synchrotron +like radiation reported by ARCADE 2 and LWA 1 [27, 30, 304, 310]. We have +considered excess radiation fraction up to the LWA 1 limit (i.e. Ar ∼ 418) at +the reference frequency of 78 MHz [30, 310]. To get the upper constraint on B0, +we have used both the EDGES upper and best-fit constraints on T21. We have +87 +PMFs & Excess Radio Background +Chapter 4 + +21 cm Line Astronomy and Constraining New Physics +considered two scenarios: First, infinite Lyα coupling (i.e. xα ≫ xc, 1) without +the effects of x-ray and VDKZ18 on IGM gas evolution. In another scenario, we +consider the finite Lyα coupling with x-ray and VDKZ18 effects. The following +summarises our results for T21 = −500 mK: +In the first scenario, for Ar = 418, we get B0 ≲ 3.7 nG for spectral index -2.99, +while for nB = −1 we get B0 ≲ 1.1 × 10−3 nG. When Ar = 5, upper constraint on +present-day magnetic field strength varies from B0 ≲ 2.9×10−1 nG to 1.8×10−5 nG +by varying nB from -2.99 to -1, respectively. +In the second scenario, the upper bounds on B0 will modify [34, 51]. +For +Ar = 418, we get the upper constraint on magnetic field to be B0(nB = −2.99) ≲ +4.9 × 10−1 nG and B0(nB = −1) ≲ 3.7 × 10−5 nG. While for Ar = 5, we get upper +bound on present day magnetic field strength to be ≲ 1.4 × 10−1 nG for spectral +index -2.99, and for spectral index -1 we get B0 ≲ 3.8 × 10−6 nG. +We would like to note that these upper bounds on B0 that we have reported +here are also consistent with the Planck observations [49, 338]. +Chapter 4 +PMFs & Excess Radio Background +88 + +“Who sees the future? Let us have free scope for +all directions of research” +Ludwig Eduard Boltzmann, “Lectures on Gas +Theory” translated by Stephen G. Brush +5 +Primordial Magnetic Fields and +Baryon-Dark matter Interaction +In the previous chapter (4), we have analysed the upper bound on present-day +strength of PMFs in the light of EDGES observation and excess radio background +reported by ARCADE 2 and LWA 1 observations [25]. As discussed earlier in +chapter (1), to explain EDGES observation one requires that either the background +radio radiation should be grater than ∼104 K in the absence of any non-standard +mechanism for the evolution of the gas temperature or the gas temperature should +be less than 3.2 K for the standard evolution of CMB temperature at the centre +of the “U” profile for the best fitting amplitude [5]. The first possibility has been +investigated by authors of the Ref. [339–342]. In the second scenario, IGM gas +can be cooled by emitting the photons between the Ly-limit to Ly-γ wavelengths +[343, 344]. There are very few mechanisms to cool the gas. Since the dark matter + +21 cm Line Astronomy and Constraining New Physics +is colder than the gas, effective cooling of the gas can be obtained by elastic +scattering between the dark matter and baryon particles [73, 345, 346]. A new +kind of interaction between dark matter and baryons was proposed by the authors +of reference [345, 347] to explain the EDGES absorptional signal. The authors +consider a non-standard “Coulomb-like” interaction: σ = ˆσ v−4; v is the relative +velocity between the dark matter and baryons and ˆσ is the strength of baryon-dark +matter interaction cross-section [73, 345–348]. Here, the interaction between dark +matter and baryons does not depend on whether the baryons are free or bound +within atoms [345]. The cooling of the gas, by transferring energy to the dark +matter, is tightly constrained because of constraints on the dark matter mass and +cross-section by cosmological and astrophysical observations [73, 345, 349, 350]. +In the present chapter, we reanalyse the constraints on PMFs in the presence of +baryon-dark matter interaction proposed by the authors of reference [345]. In the +presence of baryon-dark matter interaction the bounds on magnetic field, baryon +dark matter cross-section strength (ˆσ) and dark matter mass (MDM) can strongly +influence each other. +This requires to rework the bounds on ˆσ , MDM and B0 +which can explain the observed absorption signal by EDGES collaboration. The +upper limit on the magnetic field strength can modify in presence of baryon-dark +matter interaction cross-section. +In the presence of a strong magnetic field, a +large baryon-dark matter interaction cross-section is required to balance magnetic +heating of gas to explain the EDGES signal as compared to a weak magnetic field. +Subsequently, the strong magnetic-fields can even erase the 21 cm signal— this +gives an upper bound on the strength of magnetic-fields, dark matter mass and +baryon-dark matter cross-section. +In order to explain the EDGES absorption signal, the gas temperature needs to +be cooler than the ΛCDM prediction. During the Cosmic dawn era, the Universe +was at its coldest phase, and the relative velocity between the dark matter and +baryon was very small, O(10−6). Also, the temperature of the dark matter was +colder than the baryon temperature during this period, so an interaction of the +Chapter 5 +PMFs & Baryon-Dark matter Interaction +90 + +21 cm Line Astronomy and Constraining New Physics +baryon with dark matter can cool the gas temperature. Since the relative velocity +is small, scattering cross section of the type σ = ˆσ v−4 can enhance the interaction +rate and cool the gas sufficiently to explain EDGES absorption dip [345, 346, 348]. +In this chapter, we consider magnetic heating of the gas and dark matter via +ambipolar and turbulent decay. Here, we take cosmological parameters Ωb, Ωm, +and h as Ωb = 0.04859, Ωm = 0.315 and h = 0.68 [48]. +5.1 +Baryon-dark matter interaction in presence +of magnetic fields +In this section, we discuss the effects of magnetic fields on the gas temperature in +the presence of baryon-dark matter interaction. The gas temperature evolves as +discussed in the chapter (4), except here, the cooling rate (dQgas/dt) will add due +to the energy transfer from gas to dark matter [254, 347], +dTgas +dz += 2 Tgas +1 + z + +ΓC +(1 + z)H (Tgas − TCMB) +− +2 +3 ntot(1 + z) H (Γturb + Γambi) + +2 +3 (1 + z) H +dQgas +dt +. +(5.1) +The cooling rate (dQgas/dt) depends on the temperature difference and relative +velocity between dark matter and baryons, +dQgas +dt += +2 Mb ρDM ˆσ e−r2/2 +√ +2 π (Mb + MDM)2 u3 +th +� +Tgas − TDM +� +− µ ρDM +ρM +v D(v) , +(5.2) +here, Mb ≈ MH is the baryon mass and can be taken as mass of hydrogen atom. +ρDM and ρM are the dark matter and total matter energy density, respectively. +Moreover, r = v/uth, v is the relative motion between baryons and dark matter +while u2 +th = Tgas/Mb + TDM/MDM . Here, TDM is the dark matter temperature +and µ = Mb MDM/(Mb + MDM) is the reduced mass. The first term in equation +(5.2), arises due to the temperature difference between dark matter and gas. As +91 +PMFs & Baryon-Dark matter Interaction +Chapter 5 + +21 cm Line Astronomy and Constraining New Physics +TDM < Tgas, the first term is positive. It implies that the energy of gas is being +transferred to dark matter with time. The second term in equation (5.2), comes +due to the friction between two fluids caused by velocity difference— drag term, +and it is given by µ (ρDM/ρM) v D(v), +D(v) ≡ +ρM ˆσ +Mb + MDM +1 +v2 F(r) , +(5.3) +here, r = v/uth and the function F(r) is defined as, +F(r) ≡ erf +� r +√ +2 +� +− +� +2 +π r e−r2/2 , +(5.4) +here, erf() is the Gauss error function. When the relative velocity between dark +matter and baryons is zero, i.e. r = 0, one gets F(0) = 0. In this case there will +not be any drag heating of gas and dark matter. As r → ∞, F(r) → 1. For any +value of r ≥ 0, one finds that F(r) ≥ 0. Therefore, the last term in equation (5.2) +always remains negative. It implies that the energy of gas always increases due to +the drag. In equation (5.2), one can check that the heating gets maximize due to +drag as MDM → Mb. The dark matter temperature evolution can be written as, +dTDM +dz += 2 +TDM +(1 + z) + +2 +3 (1 + z) H +dQDM +dt +, +(5.5) +here, first term represents the cooling of the dark matter due to expansion of +the Universe. Heat transfer rate for dark matter (dQDM/dt) can be obtained by +interchanging b ↔ DM and Tgas ↔ TDM in equation (5.2). As drag term (5.3) +remains symmetric under the transformation b ↔ DM, it heats the dark matter +also. We can also check that total energy density of the system is conserved [347], +NDM +dQDM +dt ++ Nb +dQgas +dt +− ρDM ρb +ρM +v D(v) = 0 , +(5.6) +here, NDM and Nb are number density of dark matter and baryons. As the relative +motion between dark matter and baryons is damped due to friction between both +Chapter 5 +PMFs & Baryon-Dark matter Interaction +92 + +21 cm Line Astronomy and Constraining New Physics +fluids and expansion of the Universe, one can write the evolution of relative motion +as, +dv +dz = +v +1 + z + +D(v) +(1 + z) H . +(5.7) +Temperature evolutions of the gas and dark matter require free electron fraction. +It is given by equation (2.9) with E = 0 . As it has been confirmed in Ref. [50], that +cooling due to effects like Lyα emission, Bremsstrahlung and recombination does +not have that much effects on the dynamics of the gas and dark matter, therefore, +we have not considered these effects in the present work. +5.2 +Results and Discussion +Solving coupled equations (2.9 with E = 0 , 4.7, 5.1, 5.5 and 5.7) with initial +conditions Tgas(1010) ≃ TCMB(1010), TDM(1010) ∼ 0 K, xe(1010) = 0.057 and +B(z) = B0 (1+z)2|z=1010 is the initial magnetic field strength, we get the tempera- +ture evolution of the dark matter and gas for different dark matter masses, strength +of baryon-dark matter interaction cross-sections and magnetic field’s strengths. +Figures (5.1), (5.2) and (5.3) show the evolution of the gas and dark matter tem- +perature with redshift (z). The solid blue line in all these figures correspond to +gas temperature when both the magnetic field and baryon-dark matter interaction +are zero. In this case, gas temperature falls as Tgas ∝ (1 + z)2 after z ∼ 200 and +reaches 6.8 K at z = 17. +In figure (5.1), temperature evolution of the gas and dark matter is given for +different strength of PMFs at constant ˆσ = 10−41 cm2 and MDM = 10−1 GeV. For +both the cases B0 = 10−5 G and 10−6 G, gas temperature falls down due to Hubble +expansion and baryon-dark matter interaction till z ∼ 30 and ∼ 20, respectively, +then temperature rises due to magnetic heating. We note that, TDM also increases +due to the energy transfer from gas to dark matter depending on ˆσ and MDM. +93 +PMFs & Baryon-Dark matter Interaction +Chapter 5 + +21 cm Line Astronomy and Constraining New Physics +10 +100 +1000 +0.1 +1 +10 +100 +1000 +Redshift +( z ) +Temperature +( in Kelvin ) +σ=10-41 cm2, md =0.1 GeV +T d , B 0 = 10 - 6 G +T gas , B 0 = 10 - 6 G +T d , B 0 = 10 - 5 G +T gas , B 0 = 10 - 5 G +T gas , σ = 0 , B 0 = 0 +Figure 5.1: This figure shows the temperature evolutions of baryon and dark +matter in the presence of PMFs and baryon-dark matter interaction. Blue line +corresponds to temperature evolution of gas in the absence of both magnetic heat- +ing and baryon-dark matter interaction. The red (green) solid lines represents the +variation of the gas temperature and the dotted red (green) line shows the vari- +ation of the dark matter temperature in presence of PMFs and the baryon-dark +matter interaction. In this plot we vary the strength of PMFs, and keep ˆσ & +dark matter mass constant to 10−41 cm2 & 10−1 GeV, respectively. In all figures, +notation for the mass of dark matter is written with md. While in the text, it is +written as MDM. +Chapter 5 +PMFs & Baryon-Dark matter Interaction +94 + +21 cm Line Astronomy and Constraining New Physics +10 +100 +1000 +0.1 +1 +10 +100 +1000 +Redshift +( z ) +Temperature +( in Kelvin ) +B0=10-6 G, md =0.1 GeV +T d , σ = 10 - 42 cm 2 +T gas , σ = 10 - 42 cm 2 +T d , σ = 10 - 41 cm 2 +T gas , σ = 10 - 41 cm 2 +T gas , σ = 0 , B 0 = 0 +Figure 5.2: The caption is same as in figure (5.1), except here, we only vary the +strength of baryon-dark matter cross-section, and keep B0 & dark matter mass +constant to 10−6 G & 10−1 GeV, respectively. +10 +100 +1000 +0.1 +1 +10 +100 +1000 +Redshift +( z ) +Temperature +( in Kelvin ) +B0=10-6 G, σ=10-41 cm2 +T d , m d = 1 GeV +T gas , m d = 1 GeV +T d , m d = 0.1 GeV +T gas , m d = 0.1 GeV +T gas , σ = 0 , B 0 = 0 +Figure 5.3: The caption is same as in figure (5.1), except here, we only vary the +dark matter mass, and keep B0 & ˆσ constant to 10−6 G & 10−41 cm2, respectively. +95 +PMFs & Baryon-Dark matter Interaction +Chapter 5 + +21 cm Line Astronomy and Constraining New Physics +Larger the strength of magnetic fields, earlier the heating begins. For example, +heating for the case with B0 = 10−5 G starts earlier compared to the case with +B0 = 10−6 G in figure (5.1). Although TDM at z ∼ 1010 is taken to be zero, it +increases due to the energy transfer from baryons to dark matter. By increasing +B0, magnetic-heating of the gas rises, subsequently, the value of TDM also rises. It +can be seen in figure (5.1), temperature of dark matter for B0 = 10−5 G is larger +compared to B0 = 10−6 G. +Figure (5.2) shows the temperature evolution of gas and dark matter for dif- +ferent strength of baryon-dark matter interaction cross-section when B0 = 10−6 G +and MDM = 10−1 GeV are fixed. Larger the ˆσ, more heat transfers from gas to +dark matter and cools the gas efficiently. For the green lines ˆσ = 10−42 cm2. As we +increase ˆσ to 10−41 cm2, the gas temperature decreases— shown by red solid line. +It decreases because the energy transfer from gas to dark matter becomes more +efficient by increasing interaction between dark matter and baryons. It results in +more heating of dark matter— shown by red dashed line. +For B0 = 10−6 G and ˆσ = 10−41 cm2, temperature evolution for different dark +matter mass is shown in Figure (5.3). As we increase the dark matter mass from +10−1 GeV to 1 GeV, temperature of both the dark matter and gas increases, and +it becomes more efficient for large dark matter mass [347]. +This drag heating +is important when mass of dark matter is around ∼ 1 GeV [347]. When MDM +approaches to 1 GeV, in addition to magnetic heating of the gas, the heating +due to drag term also becomes effective . +Therefore, the gas temperature for +MDM = 1 GeV is higher than MDM = 10−1 GeV. +5.2.1 +Correlation between dark matter mass and baryon- +dark matter cross section +In this subsection, we analyse the effect of B0, MDM and ˆσ on gas and dark matter +temperature. In Fig. (5.4), we study constraints on MDM and ˆσ for T21 ≃ −500 mK +Chapter 5 +PMFs & Baryon-Dark matter Interaction +96 + +21 cm Line Astronomy and Constraining New Physics +10-15 +10-14 +10-13 +10-6 +10-5 +10-4 +10-3 +10-2 +10-1 +100 +Planck 2015 excluded region +CMB-S4 excluded region +σ̂ (GeV -2) +md (GeV) +B0 =3.48×10-6 G +B0 =2.67×10-6 G +B0 = 1.0 ×10-6 G +B0 = 1.0 ×10-9 G +Figure 5.4: The figure shows the minimal cross-section required to get T21 ≃ +−500 mK (solid lines) and T21 ≃ −300 mK (dashed line) at z = 17 as a function of +mass for different strengths of PMFs. Here, we assume no x-ray heating of gas, and +spin temperature is completely coupled to gas temperature, i.e. Tgas ≃ TS. The +solid (dashed) magenta, black, blue and red line correspond to B0 = 3.48×10−6 G, +2.67×10−6 G, 10−6 G and 10−9 G respectively. The CMB-S4 (forecast) and Planck +2015 constraints on ˆσ and MDM with 95% C.L. have been taken from the Refs. +[51, 52]. The green and gold regions are excluded by Planck 2015 and CMB-S4 +forecast respectively. (1 GeV−2 = 3.89 × 10−28 cm2) +97 +PMFs & Baryon-Dark matter Interaction +Chapter 5 + +21 cm Line Astronomy and Constraining New Physics +(Tgas ≃ 3.26 K) and −300 mK (Tgas ≃ 5.2 K). Here, we have taken xα ≫ 1 +to plot T21 profiles. Thus, from equation (1.7) one can get TS ≈ Tgas as xc is +already ≪ 1 at required redshift due to the small number density of hydrogen, +free electrons and protons. +Subsequently, one can calculate T21 from equation +(1.19). In Fig. (5.4), we consider cases B0 = 3.48 × 10−6 G, 2.67 × 10−6 G, 10−6 G +and 10−9 G and solve equations (2.9 with E = 0) and (4.7, 5.1, 5.5 & 5.7) for +Tgas ≃ 3.26 and 5.2 K at z = 17 to get ˆσ vs MDM plots. The solid and dashed +lines represent the cases when T21 ≃ −500 mK and −300 mK, respectively. The +gold and green regions respectively show the CMB-S4 (forecast) and Planck 2015 +upper constraint on ˆσ − MDM with 95% C.L. [51, 52]. The magenta, black, blue +and red lines corresponds to B0 = 3.48 × 10−6 G, 2.67 × 10−6 G, 10−6 G and +10−9 G. As we increase the magnetic field strength from 10−9 G to ∼ 10−6 G, +larger value of ˆσ is required for MDM ∈ {10−6, 1} GeV to maintain T21 ≃ −500 or +−300 mK at z=17. To get EDGES upper limit on T21 (i.e. −300 mK), required +ˆσ is smaller compared to the case when T21 = −500 mK. This is because we need +to transfer less energy from gas to the dark matter to obtain EDGES upper limit +on T21. We get the upper limit on PMFs strength to 2.67 × 10−6 G by CMB- +S4 (forecast) constraint on ˆσ − MDM and maintaining T21 ≃ −300 mK at z=17. +For B0 = 2.67 × 10−6 G, MDM ≳ 10−2 GeV is excluded by CMB-S4 forecast for +T21 ≃ −300 mK. By Planck 2015 constraint on ˆσ − MDM, the allowed maximum +strength of PMFs is 3.48×10−6 G by requiring EDGES upper constraint on T21 at +z=17. For B0 = 3.48 × 10−6 G, mass of dark matter ≳ 1 × 10−2 GeV is excluded. +Similarly, for the B0 = 10−6 G, MDM ≳ 8 × 10−1 GeV is excluded by CMB-S4 +forecast. +When the dark matter mass approaches mass of hydrogen, the drag +term in equation (5.3) also starts to contribute significantly in heating of the gas +in addition to the magnetic heating. Therefore, higher mass of dark matter is +excluded for higher magnetic field as shown by figure (5.4). As discussed in [347], +when MDM ∼ 1 GeV, the drag term heat up both the gas and dark matter in such +a way that we can not obtain Tgas = 3.26 K at z = 17 as required for the EDGES +Chapter 5 +PMFs & Baryon-Dark matter Interaction +98 + +21 cm Line Astronomy and Constraining New Physics +15 +16 +17 +18 +19 +20 +- 0.5 +- 0.4 +- 0.3 +- 0.2 +Redshift +( z ) +T 21 in Kelvin +md =0.1 GeV, σ=6.22×10-15 GeV-2 +Standard +EDGES +B 0 =1.65 ×10 - 6 G +B 0 =1.35 ×10 - 6 G +B 0 =10 - 6 G +Figure 5.5: 21 cm differential brightness temperature (assuming infinite Lyα cou- +pling) vs redshift when their is no x-ray heating. The dotted black (orange) colour +represents standard ΛCDM (EDGES) predictions for the global T21 signal. Green, +red and blue solid curves correspond to B0 = 1×10−6, 1.35×10−6 and 1.65×10−6 G +respectively. Here, MDM = 10−1 GeV and ˆσ = 6.22 × 10−15 GeV−2. +signal. There is a independent bound on the primordial magnetic field from CMB +of the order of ≲ nG [287, 288]. This constraint, in our analysis, restricts value of +ˆσ . Here, we note that in our analysis further decreasing value of B0 below 10−9 G, +does not change our result in significant way. +5.2.2 +Effect of primordial magnetic fields on the global 21 +cm signal +We have discussed above that, with increase in the strength of the magnetic field +the temperature of the gas increases for a fix MDM and ˆσ. +In figures (5.5) & +(5.6), we plot 21 cm differential brightness temperature with redshift for different +magnetic field strengths. These figures are obtained by keeping MDM = 10−1 GeV +99 +PMFs & Baryon-Dark matter Interaction +Chapter 5 + +21 cm Line Astronomy and Constraining New Physics +-500 +-400 +-300 +-200 +-100 + 0 + 10 + 15 + 20 + 25 + 30 +T21 (mK) +z +B0= 2.0 ×10-6 G +B0= 1.0 ×10-6 G +B0= 8.0 ×10-7 G +B0= 6.0 ×10-7 G +Figure 5.6: T21 plot with redshift when x-ray heating and finite Lyα coupling +are considered [51]. Black, blue, green and red solid curves correspond to B0 = +2 × 10−6, 1 × 10−6, 8 × 10−7 and 6 × 10−7 G respectively. The magenta dashed +line is corresponds to the EDGES upper bound on T21 : −300 mK. The values of +MDM and ˆσ are same as considered in figure (5.5). +Chapter 5 +PMFs & Baryon-Dark matter Interaction +100 + +21 cm Line Astronomy and Constraining New Physics +and ˆσ = 6.22×10−15 GeV−2 constant. In figure (5.5), to plot T21 we assume infinite +Lyα coupling (xα → ∞ ⇒ TS ≃ Tgas) and do not include the x-ray heating. For +B0 = 1 × 10−6 G, the 21 cm line absorption signal reported by EDGES (i.e. +−500 mK) can be explained. In figure (5.6), we include the x-ray heating and +consider finite Lyα coupling (xα) [51, 74–76]. As we decrease B0 from 2 × 10−6 G, +the minimum value of T21 profile decreases. For the case when B0 = 1 × 10−6 G +(blue solid line), minimum of T21 profile is well below the EDGES upper limit on +T21 (i.e. −300 mK— magenta dashed line). In figure (5.5), when there is infinite +Lyα coupling, T21 = −300 mK corresponds to B0 = 1.35 × 10−6 G. Thus, we need +to lower B0 values when the finite Lyα coupling is considered to get desired value +of T21. As shown in figures (5.5) and (5.6), brightness temperature is suppressed by +the increase of the strength of the magnetic field and it can even erase the standard +21 cm signal when the magnetic field strength increases above ∼ 1 × 10−6 G. This +sets the upper limit on the strength of the magnetic field for MDM = 10−1 GeV +and ˆσ = 6.22 × 10−15 GeV−2. +5.3 +Conclusions +Magnetic fields in [50, 259] have shown to heat the gas during the cosmic dawn +era by the ambipolar diffusion and the turbulence decay. Since, it could erase +the observed 21 cm absorption signal, one can calculate the upper bound on the +magnetic field. One of the promising mechanisms to explain the absorption signal +of the 21 cm line is to have interaction between the dark matter and baryons +[73, 345]. In this work, we have shown that in the presence of such an interaction +the upper bound on the strength of magnetic fields can significantly be altered. +The magnetic-energy converted to the thermal energy and it heats both the gas and +dark matter when ˆσ is non-zero. This is an extra heating effect of dark matter in +addition to the drag heating. The drag term heats the dark matter and baryons; +but in the lower range of dark matter mass (≪ 1 GeV) it becomes small. To +101 +PMFs & Baryon-Dark matter Interaction +Chapter 5 + +21 cm Line Astronomy and Constraining New Physics +explain the observed anomaly in the 21 cm signal by the EDGES, a large baryon- +dark matter scattering cross-section is required to balance the magnetic heating +effect. +An earlier saturation occurs in baryon-dark matter cross-section in the +presence of the strong magnetic fields. We have also explored the millicharged dark +matter scenario. In this case, we are not able to reproduce the EDGES signal by +considering the upper bound on ˆσ − MDM by Planck 2015 and CMB-S4 (forecast) +[51, 52]. Recently, the similar results about “millicharged” and “Coulomb-like” +dark matters also have been obtained in reference [351]. They find that 100% +millicharged dark matter scenario can not reproduce the EDGES result for any +parameter space. The inclusion of PMFs will further increase the gas temperature +and reduce the amplitude of 21-cm absorptional signal. Therefore, it will further +worsen the situation for millicharged dark matter scenario. +Considering upper bound on ˆσ − MDM by Planck 2015 [52] and EDGES upper +constraint on T21 (−300 mK) at z = 17 [5], we found upper bound on the magnetic +field strength: B0 = 3.48 × 10−6 G, while considering CMB-S4 forecast constraint +[51] we get B0 = 2.67 × 10−6 G for the dark matter mass ≲ 10−2 GeV. +Chapter 5 +PMFs & Baryon-Dark matter Interaction +102 + +“I seem to have been only like a boy playing on +the seashore, and diverting myself in now and +then finding a smoother pebble or a prettier shell +than ordinary, whilst the great ocean of truth lay +all undiscovered before me.” +Isaac Newton, “Memoirs of Newton” (1855), +Vol II By David Brewster +6 +Summary and Future outlook +6.1 +Summary +The 21 cm signal is shown to be a prestigious probe in the cosmological laboratory +to provide robust bounds on the physics of the early and late time Universe. The +signal can give a good insight into the period when the galaxies and first stars +were formed. In the thesis, I have analysed bounds on the present-day strength +of primordial magnetic fields, sterile neutrino lifetime & mixing angle with active +neutrinos, and primordial black hole dark matter fraction using the global 21 cm +signal during the cosmic dawn era. The 21 cm line corresponds to the wavelength of +hyperfine transition between 1S singlet and triplet states of the neutral hydrogen +atom. +The corresponding frequency for the 21 cm line is 1420.4 MHz. +For a +transition at redshift z, the frequency can be mapped for a present-day observed +frequency as 1420.4/(1 + z) . + +21 cm Line Astronomy and Constraining New Physics +In the ΛCDM framework of cosmology, the evolution of the gas temperature +and ionization fraction are well-established during the cosmic dawn era [153]. The +addition of any exotic source of energy can significantly impact the ionization +and thermal history of the Universe. +The change in the gas temperature can +significantly modify the absorption feature in the global 21 cm signal during cosmic +dawn [3]. This can provide constraints on the properties of such exotic sources of +energy injection. +The EDGES collaboration has reported the 21 cm differential brightness tem- +perature: T21 = −500+200 +−500 mK with 99 percent confidence limit centred at 78 MHz +or redshift z = 17.2 [5]. By considering TS = Tgas, the observed brightness temper- +ature translates to gas temperature as Tgas(z = 17.2) = 3.26+1.94 +−1.58 K. In the ΛCDM +framework, the gas temperature at redshift z = 17.2 remains around 7 K. This +corresponds to differential brightness temperature T21(z = 17.2) ≃ −220 mK— +equation (1.19) for TS ≃ Tgas. +To resolve the tension between the theoretical +prediction based on ΛCDM model and EDGES observation, one requires to in- +crease the ratio of TR/TS in equation (1.19) over theoretical predictions in redshift +range 15 ≤ z ≤ 20. This can be achieved either by increasing the background +radiation or decrease the gas temperature. Both possibilities have been studied +by several authors; for example, see the Refs. [25–27, 289, 304, 315–317, 319– +321, 325, 326, 345, 346, 352–354]. +However, such mechanisms to increase the +background radio radiation or cooling the gas are debatable issues. One of such +mechanisms to cool gas is baryon dark matter interaction [345]. This approach has +been questioned by several authors [51, 73, 340, 349, 355–358]. Here, it is to be +noted that the authors do not consider heating of the gas by decaying or annihi- +lating dark matter. Injection of electrons and photons by decaying or annihilating +dark matter into IGM can heat the gas more than cooling of the gas [156, 157]. +Moreover, the EDGES measurement has been also questioned in several articles [6– +8, 77, 78]. Recently, SARAS 3 observation reported that the EDGES observation +is not of an astrophysical origin and it is rejected with the 95.3 percent confidence +Chapter 6 +Summary and Future outlook +104 + +21 cm Line Astronomy and Constraining New Physics +level [6]. In the Ref. [77], the authors have questioned the fitting parameters for +the foreground emission and data. There is a possibility that the absorption fea- +ture in the EDGES observation can be a ground screen artifact [7]. The absorption +amplitude may modify depending on modelling of foreground [8, 78]. In Ref. [359], +the authors perform the Bayesian comparison of fitting models for EDGES data +and argue that the highest evidence models favour an amplitude of |T21| < 209 mK. +In the light of these controversies, it is require to verify the EDGES result by other +observations. The future updated version of the hydrogen Epoch of Reionization +Array (HERA)a, Thirty Meter Telescope (TMT)b, JWST, etc., will be able to +probe the cosmic dawn era more precisely. The following summarizes the results +reported in the thesis: +6.1.1 +Bounds on dark matter candidates +About 85 per cent of the total matter content in the Universe is dominated by +dark matter. In the last decades, many dark matter models have been proposed +to explain various astrophysical observations. However, the microscopic nature of +dark matter is still unknown. During my doctoral research, I have considered sterile +neutrinos and primordial black holes as dark matter candidates and constrain their +properties using the absorption feature in 21 cm differential brightness temperature +during the cosmic dawn era. +As discussed earlier, here, we have taken 21 cm +differential brightness temperature such that it does not change from its standard +theoretical value (∼ −220 mK) by more than a factor of 1/4 (i.e. −150 mK) or +1/2 (i.e. −100 mK) at redshift 17.2 . +Sterile Neutrino Dark Matter +In the warm dark matter models, one of the theoretically well-motivated candidates +is KeV mass sterile neutrinos. We have constrained the sterile neutrino dark matter +ahttp://reionization.org/ +bhttp://tmt.org/ +105 +Summary and Future outlook +Chapter 6 + +21 cm Line Astronomy and Constraining New Physics +lifetime and mixing angle with active neutrino as a function of sterile neutrino +mass [9]. Here, we have considered the two scenarios to get the bounds: First, +IGM evolution without the heat transfer from the background radiation to gas +mediated by Lyα photons (VDKZ18 effect). Next, we have considered additional +VDKZ18 heating effects on the IGM gas. The following summarises our results +for T21 = −150 mK : +In the first scenario, the lower bound on the sterile neutrino lifetime varies +from 8.3 × 1027 sec to 9.4 × 1025 sec by varying sterile neutrino mass from 2 KeV +to 50 KeV. While the upper bound on the mixing angle varies from 6.8 × 10−9 to +6.1 × 10−14 by varying sterile neutrino mass from 2 KeV to 50 KeV. +In the second scenario, the lower bound on the sterile neutrino lifetime varies +from 1.5 × 1028 sec to 1.7 × 1026 sec by varying sterile neutrino mass from 2 KeV +to 50 KeV. While the upper bound on the mixing angle varies from 3.8 × 10−9 to +3.42 × 10−14 by varying sterile neutrino mass from 2 KeV to 50 KeV. +Primordial Black Hole Dark Matter +Spinning primordial black holes can substantially affect the ionization and thermal +history of the Universe. Subsequently, it can modify the 21 cm absorption signal +during cosmic dawn era by injecting energy due to Hawking evaporation. We study +the upper projected bounds on the fraction of dark matter in the form of PBHs +as a function of mass and spin. Our projected constraints are stringent compared +to DSNB, INTEGRAL observation of the 511 KeV line, IGRB, Planck, Leo T +and COMPTEL. In the near future, AMEGO collaboration will be able to probe +some parameter space in our considered mass range of PBHs. In the present work, +we have considered the monochromatic mass distribution of PBHs. The allowed +parameter space can also be explored for different PBHs mass distributions such +as log-normal, power-law, critical collapse, etc. [251]. We find the most robust +lower projected constraint on the mass of PBHs, which is allowed to constitute +the entire dark matter, to 1.5 × 1017 g, 1.9 × 1017 g, 3.9 × 1017 g and 6.7 × 1017 g +Chapter 6 +Summary and Future outlook +106 + +21 cm Line Astronomy and Constraining New Physics +for PBH spins 0, 0.5, 0.9 and 0.9999, respectively. The lower bound on MPBH +for ΩPBH = ΩDM, for extremal spinning PBHs is nearly four times larger than +non-spinning ones [16]. +6.1.2 +Primordial Magnetic Fields +Observations suggest that the magnetic fields are ubiquitous in the Universe— from +the length scale of planets and stars to the cluster of galaxies [17–20]. The origin +and evolution of PMFs are one of the outstanding problems of modern cosmology +(Ref. [23, 24] and references cited therein). Decaying PMFs can inject magnetic +energy into the thermal energy of the IGM and heat the gas. As discussed earlier, +the EDGES collaboration reported an absorption profile for the global 21 cm signal +with an amplitude of −500+200 +−500 mK in the redshift range 15 − 20. To explain the +EDGES anomaly, one requires to enhance the background radio radiation above +the CMB radiation or lower the gas temperature below 3.2 K at redshift ∼ 17. We +have explored the upper bounds on the present-day strength of the PMFs in both +scenarios by considering different models [25, 26]. +In the Presence of Excess Radio Radiation +As discussed, one requires to enhance the background radiation above the CMBR +to explain the EDGES anomaly. For excess radiation fraction to be LWA 1 limit, +we have reported upper bounds on the present-day PMFs strength, B0 on the scale +of 1 Mpc. The following summarises our results for T21 = −500 mK (EDGES best +fit result): +We have reported B0 ≲ 3.7 nG for spectral index nB = −2.99 for excess +radiation fraction to be LWA 1 limit. While for nB = −1, the upper bound gets +more stringent: B0 ≲ 1.1 × 10−3 nG. We also discuss the effects of first stars on +IGM gas evolution and the allowed value of B0. By decreasing excess radiation +fraction below the LWA 1 limit, we get a more stringent bound on B0 [25]. +107 +Summary and Future outlook +Chapter 6 + +21 cm Line Astronomy and Constraining New Physics +In the Presence of Baryon-Dark Matter Interaction +One of the alternatives to explain the EDGES anomaly is by cooling the gas below +3.2 K. Since the dark matter is colder than the gas, adequate cooling of the gas +can be obtained by introducing the baryon-dark matter interaction beyond the +ΛCDM model. The introduction of baryon-dark matter interaction relaxes the +upper bound on B0 by transferring energy of the gas to the dark matter using +drag between gas and dark matter. Considering upper bound on ˆσ−md by Planck +2015 and EDGES upper constraint on T21 (−300 mK) at z = 17, we found upper +bound on the present-day strength of PMFs: B0 = 3.48×10−6 G, while considering +CMB-S4 forecast constraint we get B0 = 2.67 × 10−6 G for the dark matter mass +≲ 10−2 GeV. We have also discussed the bounds on ˆσ − md by considering Planck +2018 upper bound on B0 ∼ 10−9 G for EDGES best fit and upper bound on T21 +[26]. +Chapter 6 +Summary and Future outlook +108 + +A +A.1 +Spin temperature of hydrogen +In the presence of collisions, rate of change in the population of singlet state [2], +dnS +dt = −nS P C +ST + nT P C +TS . +(A.1) +In the steady state, the transition coefficients from equation (A.1): nT/nS = +P C +ST/P C +TS . In the presence of collisions, the spin temperature will be kinetic tem- +perature of gas only. Therefore, from equation (1.5), +P C +ST = 3 exp +� +− TTS +Tgas +� +× P C +TS ≃ 3 +� +1 − TTS +Tgas +� +× P C +TS . +(A.2) +As discussed in the section (1.2), Tgas, Tα ≫ TTS : exp [−TTS/Tgas] ≃ 1 − TTS/Tgas . +Similarly, for the Lyα radiation, Tgas and P C +TS will be replaced by Tα and P α +TS , + +21 cm Line Astronomy and Constraining New Physics +respectively, in equation (A.2), +P α +ST ≃ 3 +� +1 − TTS +Tα +� +× P α +TS . +(A.3) +In the hydrogen atom, there can be spontaneous and induced emissions by back- +ground radiation also, +P R +TS = A10 + B10 IR +ν , +(A.4) +here, B10 IR +ν is the induced emission due to background radiation and IR +ν is the +specific intensity for 21 cm transition. Here, A10 and B10 are Einstein coefficients +and their relation is given by A10 = 2 ν2 +TS TTS B10 . For the background radiation, +in the Rayleigh-Jeans limit from equation (1.11): IR +ν = 2 ν2 +TS TR . Therefore, from +equation (A.4), +P R +TS = +� +1 + TR +TTS +� +A10 . +(A.5) +The induced transition from singlet to triplet due to background radiation [2], +P R +ST = B01 IR +ν = 3 B10 IR +ν = 3 A10 +TR +TTS +. +(A.6) +Using equations (A.5) and (A.6), +P R +ST +P R +TS +≃ 3 +� +1 − TTS +TR +� +. +(A.7) +In the detailed balance between the population of 1S singlet and triplet states +(dnS/dt = 0), by solving the equation (1.6) with the use of equations (1.5), (A.2), +(A.3) and (A.7), we get, +� +1 − TTS +TS +� += +� +1 − TTS +TR +� ++ xα +� +1 − TTS +Tα +� ++ xc +� +1 − TTS +Tgas +� +1 + xα + xc +, +(A.8) +Chapter A +Appendix +110 + +21 cm Line Astronomy and Constraining New Physics +here, xα = P α +TS/P R +TS and xc = P C +TS/P R +TS . Solving the equation (A.8), we get [2, 3], +T −1 +S += T −1 +R + xα T −1 +α ++ xc T −1 +gas +1 + xα + xc +. +(A.9) +A.2 +Emergent brightness temperature +Solving differential equation (1.13) with initial conditions: when, l = 0 → τν = 0 +and Iν = Iν0 (figure 1.4), +Iν = Sν (1 − e−τν) + Iν0 e−τν . +(A.10) +Here, using equation (1.11), Iν = 2 ν2 T ′ +R is the final/emergent specific intensity of +light— of frequency ν. Sν = 2 ν2 Texc is the specific intensity due to the medium +having an excitation temperature, Texc, at a frequency of ν . +Iν0 = 2 ν2 TR is +the initial specific intensity of the light. As a result, we find the final/emergent +brightness temperature as [2, 3], +T ′ +R = Texc (1 − e−τν) + TR e−τν . +(A.11) +A.3 +Optical depth of hydrogen medium +The radiative transfer equation in the presence of emission and absorption of a +light with travelled distance dl in the medium, +dIν +dl = TTS +4 π φ(ν) [ nT A10 + nT B10 Iν − nS B01 Iν ] , +(A.12) +here, TTS = 2 π νTS, and φ(ν) represents line profile of the light beam. TTS/(4 π) +represents the energy of light beam per unit solid angle. The first term in the +bracket is due to the spontaneous emission from the triplet to the singlet state, +and it is proportional to the population density of the triplet state. The second +111 +Appendix +Chapter A + +21 cm Line Astronomy and Constraining New Physics +and third terms in the bracket are due to the stimulated/induced emission and +absorption, respectively. Comparing equations (A.12) and (1.12), we get, +αν = TTS +4 π φ(ν) [ nS B01 − nT B10 ] . +(A.13) +To get the optical depth of hydrogen medium, we can integrate equation (A.13) +over dl (equation 1.14), +τν = +3 A10 +32 π ν2 +TS +× TTS +TS +× nHI +� +φ(ν)dl , +(A.14) +here, we have used the relations: A10 = 2 ν2 +TS TTS B10 and B01 = 3 B10 . As the +neutral hydrogen number density: nHI = nS + nT, the singlet state population +density can be approximated by nS ≃ nHI/4 — from equation (1.5). The ratio +nT/nS , has given by equation (1.5). By solving the integral in equation (A.14) +for a line profile φ(ν) = 1/∆ν with the Doppler shift in the frequency due to the +moving medium with a proper velocity v along the line of sight in the comoving +coordinate (∆r = (1 + z) ∆l ); we find the optical depth for hydrogen medium as +[3], +τν = +3 nHI +32 π ν3 +TS +× TTS +TS +× A10 +H × +�H/(1 + z) +∂v/∂r +� +. +(A.15) +Here, ∂v/∂r is the proper velocity gradient along the line of sight, and it can be +taken as H/(1+z) for high redshift or in the absence of peculiar velocity. Here, nHI +can be written as xHI nH, and xHI is the neutral hydrogen fraction. The hydrogen +number density can be expressed in the form of dimensionless baryon energy den- +sity: nH ≃ 8.5 × 10−6 (1 + δb) Ωb h2 (1 + z)3 cm−3 . Here, δb = (ρb − ¯ρb)/¯ρb is the +baryon density contrast. ρb and ¯ρb are total and average baryon energy density, +respectively. For the matter dominated era, we can take H = H0 +√Ωm (1 + z)3/2 . +Here, H0 and Ωm are present-day values of Hubble parameter and dimensionless +matter energy density parameter, respectively. After some manipulation, we get +Chapter A +Appendix +112 + +21 cm Line Astronomy and Constraining New Physics +the final expression for optical depth of hydrogen medium for 21 cm line [3, 68–71], +τν ≃ 27 xHI (1 + δb) (1 + z) +�mK +TS +� � 0.15 +Ωm h2 +1 + z +10 +�1/2 �Ωb h2 +0.023 +� +. +(A.16) +For a global 21 cm signal we can take 1 + δb as ∼ 1. +113 +Appendix +Chapter A + + +References +[1] H. I. Ewen and E. M. Purcell. Observation of a Line in the Galactic Radio +Spectrum: Radiation from Galactic Hydrogen at 1,420 Mc./sec. Nature, 168 +(4270):356, 1951. DOI: 10.1038/168356a0. +[2] G. B. Field. Excitation of the hydrogen 21-cm line. Proceedings of the IRE, +46(1):240–250, 1958. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='PICASSO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='Project in CAnada to Search for Super-symmetric Objects ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='FIRAS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='Far Infrared Absolute Spectrophotometer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='COUPP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='Chicagoland Observatory for Underground Particle Physics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='AMEGO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='All-sky Medium Energy Gamma-ray Observatory ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='PMFs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='Primordial Magnetic Fields ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='MHD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='MagnetoHydroDynamics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='ARCADE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='Absolute Radiometer for Cosmology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Astrophysics and Dif- fuse Emission LWA Long Wavelength Array HESS High Energy Stereoscopic System Contents List of Acronyms iii 1 Introduction 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 Evolution of our Universe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 Big-Bang nucleosynthesis .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 Recombination and photon decoupling .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3 Through the dark ages to the present day .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 21 cm line as a probe during end of darkness .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 21 cm differential brightness temperature .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 Evolution of the global 21 cm signal .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3 21 cm line as a probe of new physics .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 Sterile neutrino dark matter — Chapter 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 Primordial black hole dark matter — Chapter 3 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3 Primordial magnetic fields — Chapter 4 & 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 21 2 Sterile Neutrino Dark Matter 23 v vi 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 Sterile neutrinos as dark matter .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 Existing bounds on sterile neutrinos .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 28 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3 Radiative decay of sterile neutrinos .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 29 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4 Impact on the thermal and ionization history .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5 Bounds on the sterile neutrinos .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 33 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6 Summary .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 40 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7 Additional study .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 41 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 Bounds in light of varying T21 and redshift .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 41 3 Primordial Black Hole Dark Matter 45 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 Primordial black holes as dark matter .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 47 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 Signature of Primordial Black Holes .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 48 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 Existing bounds on Primordial Black Holes .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 49 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3 Impact on the thermal and ionization history .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 49 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4 Results and Discussion .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 52 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5 Conclusions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 59 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6 Additional study .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 60 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 Bounds in light of varying T21 and redshift .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 60 4 PMFs & Excess Radio Background 63 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 Generation of primordial magnetic fields .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 64 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 Existing bounds on primordial magnetic fields .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 66 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3 Evolution of PMFs after recombination .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 66 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4 Background excess radio radiation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 67 vii 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 Excess radiation during the cosmic dawn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 69 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 Phenomenological model for excess radiation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 70 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5 Impact on the thermal and ionization history due to primordial magnetic fields .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 71 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6 Impact on the thermal and ionization history due to background radiation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 73 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7 Result and discussion .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 74 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='8 Conclusions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 87 5 PMFs & Baryon-Dark matter Interaction 89 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 Baryon-dark matter interaction in presence of mag- netic fields .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 91 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 Results and Discussion .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 93 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 Correlation between dark matter mass and baryon- dark matter cross section .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 96 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 Effect of primordial magnetic fields on the global 21 cm signal .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 99 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3 Conclusions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 101 6 Summary and Future outlook 103 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 Summary .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 103 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 Bounds on dark matter candidates .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 105 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 Primordial Magnetic Fields .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 107 A Appendix 109 viii A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 Spin temperature of hydrogen .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 109 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 Emergent brightness temperature .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 111 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3 Optical depth of hydrogen medium .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 111 References 115 “It might seem limited to impose our human per- ception to try to deduce the grandest cosmic code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' But we are the product of the universe and I think it can be argued that the entire cosmic code is im- printed in us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Just as our genes carry the mem- ory of our biological ancestors, our logic carries the memory of our cosmological ancestry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We are not just imposing human-centric notions on a cosmos independent of us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We are progeny of the cosmos and our ability to understand it is an inheritance.” Janna Levin, How the Universe Got its Spots (2002) 1 Introduction In the 21st century, our knowledge of the Universe has proliferated— thanks to the tremendous progress of observational instruments in the last three decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Espe- cially the precision cosmology has grown remarkably in the past three decades as a result of an ample amount of high-quality Cosmic Microwave Background (CMB) data, in addition to the data and comprehensive studies of supernovae, stars and nearby galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' It is the greatest triumph of precision cosmology that we now know the age of our Universe, and it is only the tip of the iceberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As another example, we know that observable/baryonic matter in the Universe is only about 5 percent and the leftover energy component consists the dark matter (∼ 26 percent) and dark energy (∼ 69 percent) based on the ΛCDM model of cosmology— the standard model of cosmology [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, Λ represents the dark energy, and CDM represents the cold dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The ΛCDM model, together with the cosmo- ahttp://lambda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='gov/product/cobe/ 21 cm Line Astronomy and Constraining New Physics Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1: The graphic represents the evolution of precision cosmology over three decades— comparison between CMB temperature maps reported by each satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Left to right: Cosmic Background Explorer (COBE)a— launched in 1989, WMAP— launched in 2001 and Planck— launched in 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Image credits: NASA/JPL-Caltech/ESA, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='gov/mission pages/planck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' logical inflation, can provide a complete picture of the evolution of our Universe from the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The CMB data from Wilkinson Microwave Anisotropy Probe (WMAP)b played a crucial role in establishing the ΛCDM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' It is also sup- ported by the Planckc observations [48, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The ΛCDM model is widely accepted now, and there are various good reasons to believe this model: N-body simulations of structure formation based on the ΛCDM framework can explain the observed large scale structure of the Universe [54], it can also explain the CMB anisotropies & polarization [48, 53, 55–57] and accelerating expansion of the Universe caused by cosmological constant Λ [48, 58, 59]d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In addition to this, the predictions for the helium and deuterium fractions by the standard Big-Bang Nucleosynthesis (BBN) for ΛCDM cosmology agree very well with observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' bhttps://map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='gov/ chttps://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='int/Science Exploration/Space Science/Planck dSaul Perlmutter with Brian P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Schmidt and Adam G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Riess received the Nobel Prize in Physics for 2011 “for the discovery of the accelerating expansion of the Universe through obser- vations of distant supernovae.” Chapter 1 Introduction 2 COBE WMAP Planck21 cm Line Astronomy and Constraining New Physics Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2: The evolution of the Universe from it’s beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Image credits: European Space Agency (ESA)e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 Evolution of our Universe Before going into 21 cm cosmology, we briefly review cosmic history from the beginning of our Universe to the present day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The figure (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2), shows a schematic picture of the evolution of our Universe from the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The best current widely agreed model of the origin and evolution of our Universe is the Big-Bang model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' According to this model, our Universe came into existence with a Big-Bang about 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='8 billion years ago [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Observations of CMB also support this theory [48, 60]f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' According to our best present-day understanding, the early Universe had an exponential expansion after the Big-Bang— it is known as the inflationary epochg [61, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' There are several reasons to believe the inflation model: It can solve the three technical problems of the Big-Bang model— the horizon problem, flatness problem and the magnetic monopole problem [61, 62]: ehttps://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='int/ fResults from the COBE were honoured with the Nobel Prize in Physics 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' gAlan H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Guth, Andrei D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Linde and Alexei A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Starobinsky received 2014 KAVLI prize in Astrophysics “for pioneering the theory of cosmic inflation.” 3 Introduction Chapter 1 10-32 seconds 1 second 100 seconds 380 000 years 300-500 million years Billions of years 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='8 billion years ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='Beginning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='of the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='Universe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='Light and matter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='Dark ages ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='Inflation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='Formation of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='Light and matter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='First stars ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='Galaxy evolution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='The present Universe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='Accelerated expansion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='light and matter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='are coupled ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='separate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='Atoms start feeling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='The first stars and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='of the Universe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='Dark matter evolves ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='Protons and electrons ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='the gravity of the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='galaxies form in the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='independently: it starts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='form atoms ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='cosmic web of dark ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='densest knots of the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='clumping and forming ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='matter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='Light starts travelling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='cosmic web ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='a web of structures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='freely: it will become the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='Cosmic Microwave ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='Background (CMB) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='Tiny fluctuations: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='Frequent collisions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='As the Universe expands,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Last scattering of The Universe is dark as Light from first stars and Light can interact the seeds of future particles collide less light off electrons stars and galaxies are again with electrons structures and light frequently yet to form → Polarisation → Polarisation Gravitational waves?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' the Universe21 cm Line Astronomy and Constraining New Physics The Big-Bang model fails to explain why causally disconnected regions ap- pear homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The observation shows that the CMB temperature is uniform up to a scale of ∆T/T ≈ 10−5 even when observed in opposite di- rections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, ∆T is the temperature difference between the two regions of the sky, and T is the average temperature over the whole sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Assuming the standard Big-Bang model, opposite directions were so far separated that they always have been acausal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Then, why does CMB appear so uniform?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' It is known as the horizon problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The second one is the flatness problem: The present-day total energy den- sity of the Universe is equal to the critical energy density of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Any departure from the critical density will result in the curvature of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The observation shows dimensionless curvature energy density of the Universe Ωk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='001 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='002 [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' It implies a flat Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' A slight deviation of total energy density from critical energy density would have re- sulted in extreme effects on the flatness of the Universe over the cosmic time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, a flat universe like ours requires extreme fine-tuning conditions in the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' It is known as the flatness problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The Grand Unified Theories (GUT) predict the existence of magnetic monopoles as at a very high temperature as the electromagnetic, weak and strong forces are not fundamental forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, there can exist many stable magnetic monopoles in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' No monopoles have been observed yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' It is known as the monopole problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' These problems of the Big-Bang model can be circumvented by introducing the cosmic inflation model [61, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Additionally, the inflation can give an idea of the origin of the observed structures in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The quantum fluctuations, prior to inflation, embedded in the initial energy density might have grown to astronom- ical scales over the cosmic time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Later, the dense regions might have condensed into structures like stars, galaxies and clusters of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The inflation epoch ends Chapter 1 Introduction 4 21 cm Line Astronomy and Constraining New Physics when inflation potential steepens, and the inflation field acquires kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Then inflation sector energy creates the standard model particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' This process is known as reheating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As the Universe expands continuously, it cools down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Then, Baryogenesis (excess of baryons over antibaryons)h, electroweak phase transition (100 GeV) and QCD phase transition (150 MeV) takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The table (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1), rep- resents the time scale, redshift and temperature for various events in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The decoupling and freeze-out of various species can be understood by comparing the rate of interaction (Γ) and Hubble expansion (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' If tΓ ≪ tH, then particle interactions dominates over expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, t∗ ≡ 1/∗ is the time scale for corre- sponding rate (∗ ≡ Γ or H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, local thermal equilibrium can be reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As Universe cools down, the value of tΓ increases faster than tH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' At tΓ ∼ tH particles starts to decouple from thermal equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Different species decouple at different times as tΓ varies from species to species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' If the mass (m) of particles becomes larger than their temperature (T), the distribution function is exponen- tially suppressed, ∝ e−m/T and particles freeze out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For example, the cross-section for weak interaction is σ ∼ G2 F T 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' GF = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='17 × 10−5 GeV−2 is Fermi constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' It implies Γ/H ∼ (T/MeV)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For example, the neutrinos interact through weak interaction only and they decouple around T ∼ 1 MeV from primordial plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 Big-Bang nucleosynthesis When plasma cools down below ∼ 100 KeV, around three minutes after the be- ginning of the Universe, Big-Bang nucleosynthesis takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In this phase, light elements were formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The neutrons and protons start to form deuterium via the process, n + p ↔ D + γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1) hThe exact time and mechanism for Baryogenesis are not exactly known yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 5 Introduction Chapter 1 21 cm Line Astronomy and Constraining New Physics Event time redshift Temperature Inflation 10−36 sec Baryogenesis ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Electroweak phase transition 20 ps 1015 100 GeV QCD phase transition 20 µs 1012 150 MeV Dark matter freeze-out ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Neutrino Decoupling 1 sec 6 × 109 1 MeV Electron-positron annihilation 6 sec 2 × 109 500 KeV Big-Bang nucleosynthesis 3 minute 4 × 108 100 KeV Matter-radiation equality 60 Kyr 3400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='75 eV Recombination 260−380 Kyr 1400 − 1100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='33 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='26 eV Photon decoupling ∼ 380 Kyr ∼ 1100 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='27 eV First stars formation ∼ 100 Myr ∼ 30 ∼ 7 meV Reionization ∼ 400 Myr ∼ 11 ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6 meV Dark energy-matter equality 9 Gyr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='33 meV Present 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='8 Gyr 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='24 meV Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1: Approximate time scale, redshift and temperature for various events in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Table credit: Daniel Baumann, “Lecture notes on cosmology: Part III Mathematical Tripos.” Chapter 1 Introduction 6 21 cm Line Astronomy and Constraining New Physics Now, these formed nuclei can form the heavier nuclei via the process, D + p ↔ He3 + γ and D + He3 ↔ He4 + p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2) The number density ratio of these elements can be found easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For example: in equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1), µn + µp = µD as µγ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, µ is the chemical potential for the corresponding species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' It implies the number densities ratio to be, � nD nn np � eq = 3 4 � mD mn mp 2 π T �3/2 e−(mD−mn−mp)/T , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3) here, T is the plasma temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' mD, mn and mp are masses of deuterium, neutron and proton, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 Recombination and photon decoupling Within the ΛCDM cosmology, free electrons and protons cool sufficiently after ∼ 3 × 105 years of Big-Bang to form neutral hydrogen atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Recombination occurs around redshift 1100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' During this epoch, electrons and protons combine to form hydrogen atoms via the process, e− + p ↔ H + γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4) When the plasma temperature was above 1 eV, there were still free electrons and protons in the plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Photons remain tightly coupled to electrons due to Comp- ton scattering, and electrons were coupled to protons due to Coulomb scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In turn, there was only a small density of neutral hydrogen atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' When the plasma temperature decreased sufficiently, electrons and protons combined and formed hydrogen atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Subsequently, the free electron density fell rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As the number density of free electrons decreased adequately, the mean free path of photons increased sharply;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' and photons decoupled from plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As discussed in 7 Introduction Chapter 1 21 cm Line Astronomy and Constraining New Physics the end of section (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1), one can estimate the photon decoupling redshift by rela- tion, tΓγ(zdec) ∼ H(zdec) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, Γγ(zdec) = ne(zdec) σT is the photon interaction rate or photon mean free path at the time of decoupling, zdec is the redshift of photon decoupling from plasma and σT is Thomson cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The electron number density (ne) can be found by using the Saha equation for the process in equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' After solving the relation, we can find zdec ∼ 1100 or cor- responding time to 380,000 yr after Big-Bang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We can also estimate that the free electron fraction in the plasma remains only about one percent— the plasma becomes mostly transparent for photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' This time is known as the surface of last-scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' After decoupling, these photons stream freely and are known as CMB radiation (CMBR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3 Through the dark ages to the present day After photon decoupling from baryonic matter, there were no luminous objects— this epoch is known as the dark ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The Universe was predominantly neutral during this era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' This period of darkness ensued until the first luminous object was not formed in the Universe for about a hundred million years after the Big-Bang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' During this era, overdensity was growing in the dark matter perturbations already.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Later, these overdensities reached a critical value and collapsed to form dark matter halos— a gravitationally bound structure [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The first generation of luminous objects sprung up around redshift 30 inside dark matter halos— this period is known as the Cosmic Dawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As of now, it is not clear that these objects were either quasars or stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As the first stars formed in very different circumstances, they probably were very different from our nearby stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' After the formation of the first luminous objects, their radiation start to ionize the gas in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' This era is known as the epoch of reionization (EoR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Three-year WMAP obser- vations of CMB suggest that reionization starts around redshift 11 and ends by ∼ 7 [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Planck observations suggest instantaneous reionization with mid-point Chapter 1 Introduction 8 21 cm Line Astronomy and Constraining New Physics redshift of reionization 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='79 [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Supernovae observations suggest that the Universe enters into an accelerated expansion phase around redshift ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5 [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' This accelerating expansion can not be explained only by matter in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' To explain, one requires the existence of dark energy [59, 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Then, we reach the present-day after 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='8 billion years from the Big-Bang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The first complexity in the physics, after the dark ages, emerged with the event of the formation of the first luminous objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As of now, this era is not observed due to the lack of our instrumental capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The recently launched James Webb Space Telescope (JWST)i will be able to probe the Universe back to redshift ∼ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' One of the best pre-eminent and promising methods to probe the cosmic dawn era is the observation of the redshifted radiation from the hyperfine transition in the ground state of the neutral hydrogen atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The low-frequency radio telescopes, sensitive to a frequency of as low as 40 MHz, can help to explore this era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 21 cm line as a probe during end of darkness The 21 cm signal appears to be a treasure trove to provide an insight into the period when the first luminous objects were formed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' hereafter we will refer these objects as first stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The 21 cm line has been actively used to trace the neutral hydrogen in Milky Way for more than seven decades since its first observation in 1951 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' It was first suggested by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' van de Hulst in 1945 that a 21 cm line might be observable in the galactic radiation spectrum [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' However, probing the neutral hydrogen during and pre cosmic dawn via the 21 cm signal is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' These periods are observed in the form of absorption/emission by the neutral hydrogen medium relative to the CMBR or background radiation at a reference wavelength of 21 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' It is referred as the 21 cm differential brightness temperature— we will discuss it later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The 21 cm line corresponds to the wavelength for hyperfine transition between ihttps://jwst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='gov/ 9 Introduction Chapter 1 21 cm Line Astronomy and Constraining New Physics 1S singlet and triplet states of the neutral hydrogen atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The corresponding frequency for the 21 cm line is 1420.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For a transition at redshift z, the frequency can be mapped for a present-day observed frequency as 1420.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4/(1 + z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Hydrogen is the dominating fraction in the Inter-Galactic-Medium (IGM) during cosmic dawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, it is convenient and advantageous to study IGM using the 21 cm signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The transition probability for the hyperfine state is once in ∼ 107 years in the absence of any external sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The presence of any exotic source of energy can significantly affect the hyperfine transition, thus spin temperature of the hydrogen gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The spin temperature (TS) is characterized by the number density ratio in 1S singlet and triplet states of the neutral hydrogen atom, nT nS = gT gS × exp � −2πνTS TS � , νTS = 1420.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4 MHz ≃ 1/(21 cm) , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5) here, nT and nS are the population of triplet and singlet states, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Hy- perfine splitting suppresses the singlet and lifts the triplet state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' gT = 3 and gS = 1 are the statistical or spin degeneracies of triplet and singlet states, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the cosmological scenarios, there are three processes that can affect the spin temperature: background radio radiation, Lyα radiation from the first stars and + , , ν = 1420 MHz λ = 21 cm 1s 2S1/2 Singlet Triplet Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3: A schematic diagram for hyperfine transition in ground state of neutral hydrogen atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Chapter 1 Introduction 10 21 cm Line Astronomy and Constraining New Physics collisions of a hydrogen atom with another hydrogen atoms, residual electrons or protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the presence of all these three effect, we can write the rate of change in the population density of singlet state, dnS dt = −nS(P R ST + P α ST + P C ST) + nT(P R TS + P α TS + P C TS) , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6) here, PST and PTS are excitation and de-excitation coefficients, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' R, α and C superscripts represent the excitation/de-excitation due to background radio radiationj, Lyα radiation from first stars and collisions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the detailed balance between the population of 1S singlet and triplet states, by solving the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6)— see appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1, one can find the spin temperature as [2, 3], T −1 S = T −1 R + xα T −1 α + xc T −1 gas 1 + xα + xc , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7) here, Tα and TR is the colour temperature of Lyα radiation from first stars and background radio radiation temperature, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Tgas is the gas temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' It refers to the temperature of either neutral species, ions, electrons or protons— all remain in thermal equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Before the first luminous objects formation, there was no Lyα radiation implying xα & Tα = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' After the first luminous objects formation, their Lyα photons started repeatedly scatter with the gas, and brought the Lyα radiation into a local thermal equilibrium with the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, during the cosmic dawn era the colour temperature can be taken as gas temperature, Tα ≃ Tgas [2, 3, 66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' xα = P α TS/P R TS is the Lyα coupling coefficient due to Wouthuysen- Field effect [2, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, P R TS = (1 + TR/TTS) A10, TTS = 2 π νTS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='068 K and A10 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='85 × 10−15 sec−1 is the Einstein coefficient for spontaneous emission from triplet to singlet state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For the all presented scenarios in the thesis: TR ≳ 49 K ≫ TTS at required redshift z ∼ 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Thus, one can approximate P R TS ≃ A10 ×(TR/TTS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' P α TS = 4 Pα/27 and Pα is the rate of scattering of Lyα photons [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' xc = P C TS/P R TS jP R TS includes both the induced emission due to background radio radiation and spontaneous emission— equation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 11 Introduction Chapter 1 21 cm Line Astronomy and Constraining New Physics is the collisional coupling coefficient due to scattering between hydrogen atoms or scattering of hydrogen atoms with other species such as electrons and protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Hence, the Lyα and collisional coupling coefficients [3], xα = P α TS P R TS = 4 Pα 27 A10 × TTS TR , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='8) xc = P C TS P R TS = P C TS A10 × TTS TR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9) Here, the de-excitation coefficient due to collisions in gas: P C TS = nHI kHH 10 +ne kHe 10 + np kHp 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' nHI, ne and np are the number density of neutral hydrogen, electrons and protons in the medium, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' kHH 10 is the rate of scattering between hydrogen atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' kHe 10 is the rate of scattering between hydrogen atoms and electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' kHp 10 is the rate of scattering between hydrogen atoms and protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For a more detailed review, see the review article by Pritchard and Loeb [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 21 cm differential brightness temperature Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4: A schematic diagram for the change in brightness temperature of a light when it passes through a medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Chapter 1 Introduction 12 21 cm Line Astronomy and Constraining New Physics As discussed above, the 21 cm signal is observed in the form of differential brightness temperature during the cosmic dawn era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' If a light with initial intensity (Iν0) & brightness temperature (TR) passes through a medium having optical depth (τν) & excitation temperature (Texc), there can be an absorption or emission by the medium resulting in a different final/emergent intensity (I′ ν) and brightness temperature (T ′ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The divergence of the emergent brightness temperature (T ′ R) from the initial brightness temperature (TR) is known as the differential brightness temperature (observed temperature by antennas), δTB = T ′ R − TR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='10) In observation, we measure the specific intensity of radiation at some frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As discussed above, the initial frequency ν of light at redshift z changes with time due to the expansion of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For present-day, it will modify to ν/(1 + z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Accordingly, the frequency of 1420.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4 MHz of a light originated in the redshift range z = 15 − 10 will suppress to O(105 Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' While the CMB peak occurs around a frequency of O(108 Hz)— this is much higher than the 21 cm line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, we can approximate the blackbody spectrum as the Rayleigh-Jeans limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In this limit the observed specific intensity of radiation at a frequency ν, Iν = 4 π ν3 exp(2 π ν/T) − 1 2πν/T ≪ 1 −−−−−−→ Iν ≡ 2 ν2 T , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='11) T is the brightness temperature of the blackbody.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The emergent brightness tem- perature, T ′ R in equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='10), is a combination of TR and Texc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We can find T ′ R by solving the equation of radiative transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' If a light passes through a medium— figure (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4), the change in its intensity (dIν) due to the absorption or emission with travelled distance (dl), dIν dl = jν − ανIν , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='12) 13 Introduction Chapter 1 21 cm Line Astronomy and Constraining New Physics where, jν is emission of light by spontaneous, stimulated emission, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' αν is the absorption coefficient of medium at frequency ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, we follow the review articles by Pritchard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [3] and Furlanetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Writing equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='12) as, dIν dτν = Sν − Iν , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='13) here, dτν = αν dl and Sν = jν/αν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, τν = � αν dl , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='14) is the optical depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Optical depth is a function of the absorption of light by the medium with travelled distance in the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' By solving the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='12) and using equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='11), we can find the T ′ R— see the appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2, T ′ R = Texc (1 − e−τν) + TR e−τν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='15) The differential brightness temperature, by equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='10), δTB = (Texc − TR) × (1 − e−τν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For the expending Universe, the temperature of radiation is ∝ (1 + z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Thus, the redshifted differential brightness temperature for present-day, δTB = Texc − TR 1 + z × (1 − e−τν) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='16) In our case, the medium is hydrogen gas and the Texc for the 21 cm line is TS— defined in equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The τν is ≪ 1 for neutral hydrogen gas— optically thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Hereafter, we will write δTB as T21 for the 21 cm line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, the 21 cm differential brightness temperature [3], T21 ≃ TS − TR 1 + z × τν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='17) The optical depth can be found by solving the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='14) for a hydrogen Chapter 1 Introduction 14 21 cm Line Astronomy and Constraining New Physics medium and a line profile— see appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3, τν ≃ 27 xHI (1 + z) �mK TS � � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='15 Ωm h2 1 + z 10 �1/2 �Ωb h2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='023 � , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='18) here, xHI = nHI/nH is the fraction of neutral hydrogen in the Universe, and nH is the total number density of hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Ωm = ρM/ρcr and Ωb = ρb/ρcr are the dimensionless energy density parameters for total matter and baryons in the Universe, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' ρM and ρb are the energy density for total matter and baryons, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' ρcr = 3 H2/(8 π GN) is the critical energy density and GN is the gravitational constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' h = H0/(100 Km sec−1 Mpc−1) and H0 is the present- day value of Hubble parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Substituting the value of τν from equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='18) into equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='17), we get the final expression for the global 21 cm differential brightness temperature [3, 68–71], T21 ≃ 27 xHI � 1 − TR TS � � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='15 Ωm h2 1 + z 10 �1/2 �Ωb h2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='023 � mK .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='19) Depending on the ratio TR/TS, there can be three scenarios for 21 cm signal: If TS = TR then T21 = 0 and there will not be any signal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' for the case when TS > TR, emission spectra can be observed, and when TS < TR, it leaves an imprint of absorption spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 Evolution of the global 21 cm signal Usually, in the ΛCDM model of cosmology, the contribution in the background radiation is assumed to be solely by the CMB radiation, TR ≡ TCMB ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' TCMB is the CMBR temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, in this subsection, we discuss the evolution of the global 21 cm signal when only CMBR is present as background radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' At the end of recombination, the baryon number density of the Universe is dom- kThe position and amplitude of the second dip from the left (between redshift 30 − 15) may modify depending on models of first-stars formation or x-ray heating of the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 15 Introduction Chapter 1 21 cm Line Astronomy and Constraining New Physics Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5: The figures represents the evolution of fluctuation in the 21 cm signal (above) and global 21 cm signal (below) when the background radiation is CMBRk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Image credits: Pritchard & Loeb, Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=', 75, 086901, (2012) [3, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' inated mainly by the neutral hydrogen, a small fraction of helium, residual free electrons and protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' After recombination (z ∼ 1100) down to z ∼ 200, the resid- ual free electrons undergo Compton scattering and maintain thermal equilibrium between electrons and CMBR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The free electrons remain in thermal equilibrium with other gas components implying Tgas ∼ TCMB [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Using equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='19), we can find that T21 = 0 , and the 21 cm signal is not present during this era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' From z ∼ 200 until 40, the number density of free electrons decreases significantly and this makes the Compton scattering insufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As a result, the gas decouples from CMBR, and its temperature falls adiabatically: Tgas ∝ (1+z)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The gas tem- perature falls below CMBR implying an early 21 cm absorption signal— known as the collisional absorption signal [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' During this period, collisions among the gas components dominate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' xc ≫ 1, which implies TS ∼ Tgas — equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Nevertheless, this signal is not observed yet due to the poor sensitivity of present- day available radio antennas as the sensitivity of antennas falls dramatically below Chapter 1 Introduction 16 10 million 100 million 250 million 500 million 1 billion Time after Big Bang [Years] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='30 Redshift= 160 80 40 20 15 14 13 12 11 10 9 8 7 50 First galaxies form [mK] 0 Reionization begins Reionization ends Brightness [ 50 Dark Ages 100 Heating begins Cosmictime 150 0 20 40 60 80 100 120 140 160 180 200 Frequency [MHz]21 cm Line Astronomy and Constraining New Physics ∼ 50 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' After z ∼ 40 to the formation of the first starl, number density and temperature of the gas are very small, hence, xc → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, T21 ∼ 0 and no signal is present there [3, 73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' After the first star formation, gas temperature cou- ples again to the spin temperature due to Lyα radiation emitted from the first stars by Wouthuysen-Field (WF) effect [2, 4, 66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, xα ≫ 1, xc and absorption spectra can be seen— equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' After z ∼ 15, the gas temperature starts to rise due to x-ray radiation emitted from the first starsm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Consequently, the temperature of gas rises above CMB temperature and the emission spectra can be seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As the reionization ends, neutral hydrogen fraction becomes very small and no signal is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The small fraction of neutral hydrogen were left only in dense regions of collapsed structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' These regions can be analysed by 21 cm forest— an analogy to Lyα forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3 21 cm line as a probe of new physics As shown in the figure (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6), the 21 cm signal can probe a large volume of the history of our Universe— pink region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Currently, we are not able to probe the high redshift Universe (z ≳ 30) as the sensitivity of presently available radio antennas becomes very low below ∼ 50 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We expect that the future advanced technology for the 21 cm signal observation will be able to probe the Universe above the redshift 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the thesis, we focus on the 21 cm signal between the redshift range of 30 to 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' After z ∼ 200 gas temperature falls adiabatically and reaches to ≃ 7 K at z = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2, while the CMB temperature reaches to ≃ 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' From the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='19), this implies a value of absorptional amplitude of T21 to ∼ −220 mK in absence of any heating effects on the IGM gas due to first stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, to calculate T21, we have taken xHI to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The xHI can be written as 1 − xe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In our case, at lThe redshift of first stars formation is not well known and it could be around 35 to 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' mIt is also not very clear when x-ray heating begins to dominate the temperature of the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We use the fiducial models for x-ray heating considered in references [51, 74–76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 17 Introduction Chapter 1 21 cm Line Astronomy and Constraining New Physics z ∼ 17 the ionization fraction, xe ≲ O(10−3) implying xHI ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, xe = ne/nH is the ionization fraction and ne is the number density of residual free electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The presence of any exotic source of energy can inject energy into IGM and heat the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' This in turn can modify the absorption amplitude in the global 21 cm signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' This feature can provide a robust bound on the properties of such sources of energy injection into IGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the thesis, the following four works has been Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6: The CMB observations can only probe the thin outer shell (z ∼ 1100), and the observation of large scale structures can probe a small fraction of volume near the centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We expect that the future advanced technology for the 21 cm signal observation will be able to probe the entire pink region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the thesis, we focus on the 21 cm signal from the redshift 30 to 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Image credits: With the permission of Josh Dillon [32];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' originally reproduced from Tegmark & Zaldarriaga (2009) [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Chapter 1 Introduction 18 Modern z= 12 Z 50 Z= 110021 cm Line Astronomy and Constraining New Physics considered: sterile neutrinos and primordial black holes as dark matter candidates and constrain their properties in the light of the global 21 cm signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Another two works discussed in the thesis are related to the constraining strength of primordial magnetic fields that might have been generated in the early Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In 2018, the Experiment to Detect the Global Epoch of Reionization Signature (EDGES)n collaboration reported an absorption profile for the 21 cm signal in the redshift range 15 − 20 [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The EDGES collaboration reported T21 to be −500+200 −500 mK in the redshift range 15−20 centred at 78±1 MHz and in symmetric “U” shaped form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' This absorption amplitude is nearly two times smaller than predicted by theoretical models based on ΛCDM framework (∼ −220 mK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' It is argued that to explain the EDGES observation, for the best fitting amplitude at the centre of the “U” profile, either the cosmic background radiation temperature TCMB ≳ 104 K for the standard Tgas evolution or Tgas ≲ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 K in the absence of any non-standard evolution of the TCMB [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Recently, many articles have questioned the EDGES measurement [6–8, 77, 78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For example, in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [77], the authors have questioned the fitting parameters for the foreground emission and data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' There is a possibility that the absorption feature in the EDGES observation can be a ground screen artifact [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The absorption amplitude may modify depending on the modelling of the foreground [8, 78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In a recent article [6], authors claimed that the EDGES observation might not be of an astrophysical origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We revisit the EDGES observation and controversies over it in the chapter (6) also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the light of these controversies, in the recent two articles (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 & 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2), we do not consider the absorption amplitude reported by the EDGES collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In these articles, we take 21 cm differential brightness temperature such that it does not change, from its standard theoretical value (∼ −220 mK), by a factor of more than 1/4 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' −150 mK) or 1/2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' −100 mK) at redshift 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' While in the older two articles (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3), we have considered the absorption amplitude reported by the nhttps://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='haystack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='edu/astronomy/astronomy-projects/edges-experiment-to-detect- the-global-eor-signature/ 19 Introduction Chapter 1 21 cm Line Astronomy and Constraining New Physics EDGES collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 Sterile neutrino dark matter — Chapter 2 In the warm dark matter models, one of the theoretically well-motivated candidates is KeV mass sterile neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Sterile neutrinos are radiatively unstable and can inject photon energy into the IGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The injection of energy into the IGM can modify the temperature and ionization history of the IGM gas thus absorption amplitude of 21 cm signal during cosmic dawn era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore one can constraint the lifetime of sterile neutrinos and the mixing angle of sterile neutrinos with active neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The article has been published as: Pravin Kumar Natwariya and Alekha C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Nayak, “Bounds on sterile neutrino lifetime and mixing angle with active neutrinos by global 21 cm signal”, Physics Letters B 827 (2022) 136955.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 Primordial black hole dark matter — Chapter 3 Primordial black holes (PBHs) have attracted much interest in recent years and have been a part of intense studies for more than five decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As PBHs are massive, interact only gravitationally and are formed in the very early Universe, they can be considered as a potential candidate for non-particle dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Hawking evaporation of PBHs can inject energy into the IGM and therefore be constrained by the absorption feature in the global 21 cm signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The mass and spin are fundamental properties of a black hole, and they can substantially affect the evaporation rate of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In this work, we derive an upper bound on the dark matter fraction in the form of the primordial black holes with a non-zero spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The article has been published as: Pravin Kumar Natwariya, Alekha C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Nayak and Tripurari Srivastava, “Constraining spinning primordial black holes with global 21-cm signal”, Mon Not R Astron Soc 510, 4236–4241 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Chapter 1 Introduction 20 21 cm Line Astronomy and Constraining New Physics 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3 Primordial magnetic fields — Chapter 4 & 5 Observations suggest that the magnetic fields (MFs) are ubiquitous in the Universe– from the length scale of planets and stars to the cluster of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The origin and evolution of PMFs are one of the outstanding problems of cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Decaying PMFs can inject magnetic energy into thermal energy of the IGM and heat the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As briefly mentioned earlier, one requires to cool the IGM gas during cosmic dawn below the standard evolution or increase the radio background at required redshift to explain the EDGES observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, we explore the upper bounds on the present-day strength of the PMFs in both the scenarios by considering different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The articles have been published as: Pravin Kumar Natwariya, “Constraint on Primordial Magnetic Fields In the Light of ARCADE 2 and EDGES Observations”, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' C 81 (2021) 5, 394.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Jitesh R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Bhatt, Pravin Kumar Natwariya, Alekha C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Nayak and Arun Ku- mar Pandey, “Baryon-Dark matter interaction in presence of magnetic fields in light of EDGES signal”, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' C 80 (2020) 4, 334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Chapter 6 summarises the main results of the thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We also discuss pos- sibilities of further extensions and future scopes of the results obtained in the thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 21 Introduction Chapter 1 “Would you tell me, please, which way I ought to go from here?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' ‘That depends a good deal on where you want to get to,’ said the Cat” Lewis Carroll, Alice in Wonderland “It is the nature of all greatness not to be exact” Edmund Burke, speech “On American Taxation” 2 Sterile Neutrino Dark Matter Despite the searching for decades, the nature of dark matter is still unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' It is one of the biggest mysteries in particle physics and cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Although ΛCDM model of cosmology is highly successful in explaining Big-Bang nucleosynthesis, CMB anisotropies and large scale structures of the Universe, it faces challenges at a smaller length scale, ≲ 1 Mpc (for a detailed review see [79] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' These problems include the missing satellite or dwarf galaxy problem [80, 81], the too-big-to-fail problem [82, 83] and the core-cusp problem [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the simulations, the cold dark matter scenario clusters hierarchically and predicts a large number of satellite galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' However, the observations show less number of satellite galaxies [80, 81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For example, the Milky Way size halo simulations show around 500 satellites, while observations show a far less number of satellite galaxies [81, 85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Subsequently, the missing satellite creates a new problem also: The simulation of Galactic size haloes predicts a larger number of big satellites 21 cm Line Astronomy and Constraining New Physics that are so massive that there is no way not to host visible stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, these massive satellites should be visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In contrast, the observations show no such satellites consistent with the simulations [82, 83, 86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' N-body simulations of cold dark matter also show the cuspy profile for dark matter density at the halo centre, while the observation of rotation curves suggest the flat profile [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the light of these problems, alternatives to the cold dark matter model have been proposed, for e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' self-interacting dark matter [87–90], fuzzy cold dark matter [91, 92], warm dark matter (WDM) [93–97], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The difference between cold, warm and hot dark matter can be characterized in the form of their thermal velocities, v = � (3 T/m) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, v, T and m are the speed, temperature and mass of the particle, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' One can see that a larger speed implies a higher temperature for a fixed mass of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Roughly, if their speed is less than ten percent of the light speed (v ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1), they can be considered cold dark matter candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' If v is ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1, they can be considered hot dark matter candidates [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The WDM lies in between the hot and warm dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The WDM behaves similar to CDM on large length scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' This scale can be characterized in the form of “free-streaming length”— the other important concept to differentiate between hot, cold or warm dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Typically, the free-streaming length can be estimated by how far a particle has travelled from beginning to matter-radiation equality [99], λfs = � teq 0 v a dt , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1) here, teq is the matter-radiation equality time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For a length scale larger than λfs, WDM behaves as CDM— i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' it makes structures hierarchically above λfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' While below the length scale λfs, there is a possibility that WDM may create structures “top-down”— i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' small structures may emerge via the fragmentations of large structures [95, 98, 99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The free-streaming length can be found as [99], λfs ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4 �mWDM KeV �−4/3 �ΩWDM h2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='135 �1/3 Mpc/h , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2) Chapter 2 Sterile Neutrino Dark Matter 24 21 cm Line Astronomy and Constraining New Physics here, mWDM is the mass of WDM particle and ΩWDM is the dimensionless energy density parameter for WDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The free-streaming scale is inversely proportional to mass of particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' It implies that the size of formed-first-structures will increase for a smaller particle mass— the numbers of small-length-scale structures will suppress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For example, if one considers the mass of the WDM particle to be 10 KeV, then the free-streaming scale will be ∼ 2 × 101 Kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, one can overcome the missing satellite problem by considering an adequate mass of WDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the hot dark matter scenario, the free-streaming length typically is so large that density fluctuations below cluster scale would get washed up, and formed-first-structures would have been the size of superclusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Later, their fragments might have formed the clusters, then galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' While the observation shows that galaxies formed first, then emerged as clusters and then superclusters due to their mutual gravitational attraction [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As discussed above, the nature of dark matter has significant effects on structure formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The WDM can also solve the angular momentum problem— galaxies have smaller specific angular momenta in CDM simulation compared to observations [100, 101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Additionally, by including the baryonic feedback with WDM can address the too-big-to-fail and core-cusp problems also [102–104].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The two popular candidates for WDM are sterile neutrinos and gravitinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The presence of sterile neutrino warm dark matter having KeV mass can also explain the recently observed unexpected and unidentified emission line around 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5 KeV in x-ray spectra of nearby galaxies and clusters [97, 105–108].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In this chapter, we consider sterile neutrino and study its lifetime and mixing angle with active neutrinos [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 Sterile neutrinos as dark matter Sterile neutrino with KeV mass is one of the exciting and well-motivated candidates for WDM (Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [35, 109, 110] and Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The standard model of particle physics considers the neutrinos as massless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' However, experiments and theoretical 25 Sterile Neutrino Dark Matter Chapter 2 21 cm Line Astronomy and Constraining New Physics models questioned the standard model of particle physics over the past years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' One well-studied example is neutrino oscillation [111–114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' To explain the observations of neutrino oscillations, one has to extend the standard model to introduce the massive neutrinos (for more details, see the reviews [115, 116])a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' There are three flavours of active neutrinos— electron, muon and tau neutrino, but absolute value of their masses are not very well known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Nevertheless, the square mass difference between different flavours has been constrained by various oscillation experiments, such as solar, atmospheric, reactor and accelerator (see the Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [121] and reviews [122, 123]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the standard model of particle physics, all particles get their mass via Higgs Mechanism, but neutrinos remain massless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' One of the mechanisms via which neutrinos can get their mass is the Seesaw mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As of now, active neutrinos have been observed with only left-handed chirality [124].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' To give mass to neutrinos, we also require the right-handed counterpart of active neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The right-handed neutrinos can have mass from a few eV to GUT scale [124].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the Seesaw mechanism, sterile neutrino naturally appears as an eigenstate of the neutrino mass matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Introducing a new Yukawa interaction with new Weyl fermions N β [123], LY ⊃ −yαβ(i σ2 H∗) LαN β + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3) here, α and β are summed over e, µ, τ and 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=', n , respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' n is the number of fields of N β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' i σ2 H∗ and Lα = (να, eα)T are SU(2)L doublet and carry opposite hypercharges: +1/2 and -1/2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, their combination is total singlet, implying N β to be total singlet also [123].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' When Higgs field (H) acquires vacuum expectation value (v), the neutrino mass term can be written as, Lmass ⊃ −M αβ D να N β + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4) aTakaaki Kajita with Arthur B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' McDonald received the Nobel Prize in Physics for 2015 “for the discovery of neutrino oscillations, which shows that neutrinos have mass” [117–120].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Chapter 2 Sterile Neutrino Dark Matter 26 21 cm Line Astronomy and Constraining New Physics the Dirac mass term M αβ D ≡ yαβ v/ √ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Since N β does not have any strong, electromagnetic or weak coupling, it is called the sterile;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' and it can be considered a dark matter candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' να has weak coupling with standard model particles, and it is called active neutrino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As sterile neutrinos are singlet, in principle we can write a Majorana mass term for N β: Lmass = −(1/2) M αβ M N α N β + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' From equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4), Lmass ⊃ −1 2 nT M n ≡ −1 2 nT � � 0 MD MT D MM � � n + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5) here, n = (νe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='ντ, N 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='N n)T, MD = M αβ D and MM = M αβ M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Assuming ||MM|| ≫ ||MD||, as MM is not protected by any symmetry and MD can not be larger than electroweak scale because it will require Yukawa coupling ≫ 1, the eigenvalues of mass matrix: mν 1 = O (M 2 D/MM) and mN = O(MM) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We get the light neutrino mass to mν ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 eV by taking ||MD|| ∼ 100 GeV and ||MM|| ∼ 1014 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The sterile neutrinos are stable— have a larger lifetime compared to the age of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, they can make an excellent candidate for the warm dark matter if they also have mass in the KeV range [122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' One of the minimal extensions of the standard model of particle physics, where neutrino mass and KeV sterile neutrinos in the context of dark matter are widely explored via the Seesaw mechanism, is the Neutrino Minimal Standard Model (νMSM) [122–125].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In this model, we can lower one of the eigenvalues of MM to get KeV scale sterile neutrino while keeping others super-heavy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the basis (νa, νs, N);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' N represents the heavier sterile states, the mass matrix [123], M = � � � � � � � � � � � � � � 0 0 0 M1 s M11 D M12 D 0 0 0 M2 s M21 D M22 D 0 0 0 M3 s M31 D M32 D M1 s M2 s M3 s µs 0 0 M11 D M21 D M31 D 0 M1 M 0 M12 D M22 D M32 D 0 0 M2 M � � � � � � � � � � � � � � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6) 27 Sterile Neutrino Dark Matter Chapter 2 21 cm Line Astronomy and Constraining New Physics here, νs and νa are sterile and active neutrinos, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Applying the Seesaw mechanism, we can find the mass of neutrinos: Mν ≃ −MD M −1 M M T D−Ms µ−1 s M T s ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' and the mass of light sterile neutrino: ms ≃ µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The νMSM model is a minimal extension of the standard model of particle physics— with only three additional sterile neutrinos up to the Planck scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' One having KeV scale mass— can account for dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The other two heavier sterile neutrinos can account for the observed light neutrino masses by the Seesaw mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' They can also explain the baryon asymmetry in the Universe through oscillation-induced leptogenesis if they are nearly degenerate in the mass range 150 MeV−100 GeV [109, 125].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' More details about KeV sterile neutrino models can be found in the review article by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Merle [126].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 Existing bounds on sterile neutrinos The possibility of KeV mass range sterile neutrinos as a WDM candidate can be explored and constrained by the observation of large scale structures in the Universe [94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' There are several model-dependent mechanisms that can produce sterile neutrinos in the early Universe [35, 127, 128].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In recent years, various tech- niques have been proposed to probe the unexplored sterile neutrino dark matter parameter space, for example, by mapping of x-ray intensity at different redshift [129], by observing KeV energy photons using instruments onboard Transient High Energy Sky and Early Universe Surveyor (THESEUS) mission (for the details of instruments sensitivity of THESEUS, see the Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [130]), by exploring the im- prints of sterile neutrino on solar neutrino fluxes [131], by testing the hypothesis of decaying-sterile-neutrino [132, 133], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The lower bound on the mass of ster- ile neutrinos can be obtained by the Pauli exclusion principle [35, 123].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' These bounds depends on momentum distribution and the dwarf galaxy used for astro- nomical data [35, 123].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The authors of the ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [134], finds the lower bound on the mass of non-resonantly produced sterile neutrino to be > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7 KeV when all the Chapter 2 Sterile Neutrino Dark Matter 28 21 cm Line Astronomy and Constraining New Physics dark matter is composed of sterile neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Additionally, the parameter space of sterile neutrino dark matter has been constrained by various observations and theoretical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The observations from Nuclear Spectroscopic Telescope Array (NuSTAR)b did not find any sign of anomalous x-ray lines for sterile neutrino mass range 10 − 40 KeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The future updated version of NuSTAR will be able to probe for sterile neutrino mass range 6 − 10 KeV [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the context of EDGES signal, authors of the reference [135], put a constraint on the Dodelson-Widrow sterile neutrinos mass to 63+19 −35 KeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The WMAP, Lyα forest and x-ray observa- tions constrain the sterile neutrino mass in the range from ∼ 2 KeV to ∼ 50 KeV [136–140].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The authors of the Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [141, 142] compare the observed satellite galaxy with conferred from WDM simulations of Galaxy-sized halo and constrain the mass of sterile neutrino ≳ 2 KeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Further, individual bounds on the sterile neutrino parameter space can be found in the Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [39, 143–151].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3 Radiative decay of sterile neutrinos Sterile neutrinos with KeV mass can decay to active neutrinos via two channels: νs → νa νa ¯νa and νs → νa γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In this work, we study the effect of radiative decay of sterile neutrinos on the thermal and ionization history of the Universe, and constrain the sterile neutrino decay time and mixing angle with active neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The decay of sterile neutrino to active neutrino via the radiative process can inject the photon energy into IGM and modify the absorption amplitude of the 21 cm signal during cosmic dawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Hence, we can constrain the sterile neutrino decay time and mixing angle with the active neutrino using the 21 cm absorption signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In this process, half of the total energy of a sterile neutrino (mνs/2) is carried away by a photon and remaining by an active neutrino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The decay width of sterile neutrino bhttps://heasarc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='gov/docs/nustar/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='html 29 Sterile Neutrino Dark Matter Chapter 2 21 cm Line Astronomy and Constraining New Physics for radiative process can be written as ([35, 152] and reference cited therein), Γνs = Γνs→νaγ = 9 α G2 F 1024 π4 sin2(2 θ) m5 νs , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7) here, θ ≡ � i=e, µ, τ |θi|2 is the total mixing angle between sterile and active neu- trinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7), GF and α are the Fermi and fine structure constant, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' mνs stands for the mass of the sterile neutrino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The mixing angle θ ≪ 1, therefore sin2(2 θ) ≃ 4 sin2(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We can write the decay width as [35, 152], Γνs = τ −1 νs ≃ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='52 × 10−22 sin2(θ) � mνs KeV �5 � 1 sec � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='8) here, τνs is the lifetime or decay time of sterile neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For sterile neutrinos to be dark matter candidate, their lifetime must be larger than age of the Universe, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4 × 1017 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Using this fact and equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='8), one can estimate the upper bound on the total mixing angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4 Impact on the thermal and ionization history Evolution of the ionization fraction with redshift in the presence of energy injection by decaying sterile neutrinos [153–159], dxe dz = P H (1 + z) × � nHx2 e αB(Tgas) − (1 − xe) βB(Tgas) e−Eα/Tgas� − 1 H (1 + z) � 1 E0 − 1 − P Eα � (1 − xe) E 3 nH , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9) where xe = ne/nH is the ionization fraction, ne is the free electron number density and nH is the total hydrogen number density in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' αB and βB are the case-B recombination coefficient and photo-ionization rate, respectively [153, 154, 156].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' E0 = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6 eV and Eα = (3/4) E0 are ground state binding energy and Lyα transition energy for the hydrogen atom, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' P is the Peebles coefficient Chapter 2 Sterile Neutrino Dark Matter 30 21 cm Line Astronomy and Constraining New Physics [156, 157, 160], P = 1 + KH ΛH nH (1 − xe) 1 + KH (ΛH + βH) nH (1 − xe) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='10) here, KH = π2/(E3 α H) and ΛH = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='22/sec account for the redshifting of Lyα photon due to expansion of the Universe and the 2S-1S level two photon decay rate of the hydrogen atom, respectively [161].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The last term in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9), describes the additional effect of sterile neutrinos decay on the ionization fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' E ≡ E(z, mνs) is the energy deposition rate per unit volume into IGM gas due to decaying sterile neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' It can be written as [156, 157, 162], E(z, mνs) = FS fabs(z, mνs) × ρνs,o τνs (1 + z)3 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='11) here, τνs is the lifetime of sterile neutrino to decay in a active neutrino and a photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' FS is the fraction of the sterile neutrinos that are decaying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We consider that all sterile neutrinos are decaying, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' FS = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' ρνs,0 = mνs nνs,0 is the present day energy density of sterile neutrino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' nνs,0 is the present day number density of sterile neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For the present work, we consider that all the dark-matter is composed of sterile neutrinos, ρνs,0 ≡ ρDM,0 , and ρDM,0 is the present day dark-matter energy density [35, 128, 162, 163].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' fabs(z, mνs) is the energy deposition efficiency into IGM by decaying sterile neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The energy deposition happens due to only radiative decay of sterile neutrino as active neutrinos interact very weakly with matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, we consider only radiative decay of sterile neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' fabs(z, mνs) depends on the redshift and mass of sterile neutrino [162].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The mass of decaying particles enters only through fabs(z, mνs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the presence of energy deposition into IGM, the gas temperature evolution with redshift [153–157, 159], dTgas dz = 2 Tgas (1 + z)+ ΓC (1 + z) H (Tgas − TCMB) − 2 3 H (1 + z) × (1 + 2 xe) E 3 ntot , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='12) 31 Sterile Neutrino Dark Matter Chapter 2 21 cm Line Astronomy and Constraining New Physics here, ntot = nH (1 + fHe + xe) is the total number density of gas, fHe = nHe/nH is the helium fraction, nHe is the helium number density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The first term in this equation comes due to the expansion of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The matter temperature falls with redshift adiabatically: ∝ (1 + z)2 when Compton scattering (second term) becomes insufficient (z ≲ 200) and τνs → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The Compton scattering rate is defined as, ΓC = 8 σT arT 4 CMB xe 3 (1 + fHe + xe) me , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='13) where, σT, ar and me are the Thomson scattering cross-section, Stefan-Boltzmann radiation constant and mass of electron, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Above the redshift z ∼ 200, the gas remains in thermal equilibrium with photons due to Compton scattering as ΓC ≫ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' At z = 200, one can find that ΓC ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4×10−14 sec−1 when E = 0, while, H = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6×10−15 sec−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As ΓC ∝ (1+z)4 and H ∝ (1+z)3/2 for matter dominated era, the Compton scattering rate will dominate over H as one increase z above 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, the gas and CMB share same temperature above z ∼ 200 — as second term dominates over the first term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Below z ∼ 200, the Compton scattering rate becomes smaller compared to H resulting in an adiabatic evolution of the gas when there is no last term present in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The last term corresponds to the energy deposition into IGM due to radiative decay of sterile neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Following the Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [156, 157, 164, 165], we consider the ‘SSCK’ approximation— in which (1 − xe)/3 fraction of deposited energy goes into ionization, nearly same amount goes into excitation, and remaining (1 + 2xe)/3 fraction goes into IGM heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We also discuss the projected bounds on sterile neutrinos after the inclusion of the process of gas heating in the cosmic dawn era by CMBR using Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [34], in subsequent discussion we call this process VDKZ18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, the energy transfer between gas and CMBR is mediated by Lyα photons from the first stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The authors claim that it can increase the gas temperature by the order of (∼ 10%) at z ∼ 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, it is to be noted that we do not include the x-ray heating of the gas due to the uncertainty of known physics of the first stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For a fix value of Chapter 2 Sterile Neutrino Dark Matter 32 21 cm Line Astronomy and Constraining New Physics T21 at a redshift, if we include the x-ray heating of the gas, the projected bounds becomes stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Including the heating due to VDKZ18 effect, equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='12) will modify as, dTgas dz = dTgas dz ����� [eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='12)] − ΓR (1 + z) (1 + fHe + Xe) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='14) where, dTgas/dz �� [eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='12)] represents the temperature evolution in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='12), and heating rate due to energy transfer from CMB photons to the thermal energy of gas by Lyα photons, ΓR = xHI A10 2 H xR �TR TS − 1 � T10 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='15) here, A10 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='86 × 10−15 sec−1 is the Einstein coefficient for spontaneous-emission from triplet state to singlet state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' xR = 1/τ21 × [1 − exp(−τ21)] and τ21 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 × 10−2 xHI [(1 + z)/20]1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5 (10 K/TS) is the 21 cm optical depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' T10 = 2πν10 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='0682 K and xHI ≃ 1 − xe is the neutral hydrogen fraction in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5 Bounds on the sterile neutrinos As described in the (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3), we get an absorption profile in the 21 cm signal around redshift z ∼ 17 with an amplitude of T21 ∼ −220 mK in the theoretical models based on ΛCDM framework of cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We take 21 cm differential brightness temperature such that it does not change, from its standard value (∼ −220 mK), by more than about a factor of 1/4 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' −150 mK) or 1/2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' −100 mK) at redshift 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We solve the coupled equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='12) for different mass and lifetime of sterile neutrino to get xHI and Tgas at redshift z = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' To get any absorption signal in redshift range 15−20, the gas temperature should be less than CMB temperature in shaded region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' By requiring T21 ≃ −150 mK or −100 mK at z=17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2, we can put the projected constraints on the lifetime of sterile neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 33 Sterile Neutrino Dark Matter Chapter 2 21 cm Line Astronomy and Constraining New Physics 10-1 100 101 102 103 104 101 102 103 TCMB Tgas (K) z τνs = 2x1026 sec τνs = 6x1026 sec τνs = 1x1027 sec (a) 200 150 100 50 0 50 100 5 10 15 20 25 30 T21 (mK) z τνs = 2x1026 sec τνs = 6x1026 sec τνs = 1x1027 sec (b) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1: The gas temperature evolution with redshift in the presence of decaying sterile neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The red dashed line represents the CMB temperature evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The black solid line depicts the Tgas when there is no sterile neutrino decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The shaded region corresponds to EDGES absorption signal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 15 ≤ z ≤ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In these figures, we keep mass of sterile neutrino fix to 10 KeV and vary lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In figure (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1b), we plot evolution of 21 cm differential brightness temperature as a function of redshift for the cases represented in figure (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Subsequently, using equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='8), we can also put projected constraints on the mixing angle of sterile neutrinos with active neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the figures (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1a), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2a) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3a), we plot the gas temperature evolution as a function of redshift for different mass and lifetime of sterile neutrino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The red dashed line in all plots represents the CMB temperature evolution with redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The black solid line represents the gas temperature evolution when there is no effect of decaying sterile neutrino on the IGM gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The shaded pink region corresponds to redshift range 15 ≤ z ≤ 20 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We obtain these results by considering fabs(z, mνs) from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [162].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In figures (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1b), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2b) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3b), we plot the evolution of the 21 cm differential brightness temperature as a function of redshift for the scenarios discussed in figures (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1a), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2a) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3a), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We consider the tanh parametrization model for the Wouthuysen-Field coupling coefficient (xα) to get Chapter 2 Sterile Neutrino Dark Matter 34 21 cm Line Astronomy and Constraining New Physics 10-1 100 101 102 103 104 101 102 103 TCMB Tgas (K) z mνs = 2 KeV mνs = 6 KeV mνs = 10 KeV mνs = 25 KeV (a) 200 150 100 50 0 50 100 5 10 15 20 25 30 T21 (mK) z mνs = 2 KeV mνs = 6 KeV mνs = 10 KeV mνs = 25 KeV (b) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2: The figure caption is same as in figure (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1), except here, we consider τνs constant to 6 × 1026 sec and vary mass of sterile neutrino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 10-1 100 101 102 103 104 101 102 103 TCMB Tgas (K) z τνs = 2x1026 sec τνs = 6x1026 sec τνs = 1x1027 sec (a) 200 150 100 50 0 50 100 5 10 15 20 25 30 T21 (mK) z τνs = 2x1026 sec τνs = 6x1026 sec τνs = 1x1027 sec (b) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3: The figure caption is same as in figure (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1), except here, we keep fabs(z, mνs) = 1/2 and vary lifetime of sterile neutrino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 35 Sterile Neutrino Dark Matter Chapter 2 21 cm Line Astronomy and Constraining New Physics T21 profiles [51, 74, 75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the shaded region of the figures, the spin temperature can be approximated as gas temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, when the gas temperature is lower than CMB temperature, we get the absorption profile, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' T21 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' When the gas temperature rises above the CMB temperature, T21 becomes positive, and we see an emission profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In all figures (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1b), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2b) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3b), above the redshift z ≳ 25, xα, xc < 1, therefore, the spin temperature is dominated by CMB temperature, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' T21 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' To get the absorption profile at z ∼ 17, one has to keep Tgas < TCMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In figure (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1a), we keep the mass of sterile neutrino (mνs) fix to 10 KeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The violet solid line depicts the gas temperature evolution when lifetime of sterile neutrino is 2 × 1026 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As we increase the lifetime of sterile neutrino from 2 × 1026 sec to 1 × 1027 sec, the gas temperature decreases— shown by green and cyan curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' It happens because by increasing the τνs, the radiative decay of sterile neutrinos decreases and the number of photons injected into IGM also decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, we get less heating of IGM by increasing the τνs, and it results in a smaller amplitude (larger dip) of T21 shown by figure (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In plot (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2a), lifetime of sterile neutrino is fixed to 6 × 1026 sec and the values of mνs varies from 2 KeV (violet solid line) to 25 KeV (yellow solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' If one increases the sterile neutrino mass from 2 KeV (violet line) to 6 KeV (green line), the heating of IGM decreases significantly in the shaded region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' It happens because ρνs = mνsnνs, nνs is the number density of sterile neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore at a particular redshift, when one increases mνs the number density of sterile neutrino decreases, and we get less photon injunction, produced from decaying sterile neutrinos, into the IGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Hence, one gets less heating of IGM when the mass of sterile neutrino increases, and it results in a smaller amplitude (larger dip) of T21 shown by figure (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' If one considers the immediate and complete absorption of the photon energy into IGM, then energy deposition efficiency, fabs = 1/2 — half of the total energy of sterile neutrino will be carried away by active neutrino [162, 166].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The mass Chapter 2 Sterile Neutrino Dark Matter 36 21 cm Line Astronomy and Constraining New Physics of the sterile neutrino in the equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='12), enters through only fabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, the energy deposition rate, equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='11), will depend only on the lifetime of sterile neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' This case has been depicted in figure (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3a) for the different values of τνs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In this case, as expected, the heating of IGM increases more compared to the cases in figure (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The corresponding profiles for 21 cm signal are shown in figure (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In figure (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3a), the gas temperature for τνs = 2×1026 sec is higher than the CMB temperature in the shaded region– (violet line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, we get a emission profile for T21 in figure (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3b) for τνs = 2 × 1026 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For τνs = 6 × 1026 sec, at redshift ∼ 17, the gas temperature is comparable to the CMB temperature, therefore we do not see any absorption/emission in the 21 cm signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Above the redshift ∼ 17, temperature of gas is lower than CMB, therefore, we see a small absorption in the profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Below the redshift ∼ 17, the temperature of the gas is higher than CMB, therefore, we see a emission profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For the case with τνs = 1 × 1027 sec, in the shaded region, temperature of the gas is smaller than the CMB (cyan line), therefore, we get an absorption profile for the 21 cm signal— figure (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In figure (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4), we plot the lower projected constraints on lifetime as a function of mνs by requiring T21 such that it does not suppress the standard theoretical value of T21(z = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2) ≈ −220 mK more than about a factor of 1/4 or 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Considering T21 < −150 mK, will further strengthen our projected bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The red coloured curves depict the lower projected constraints on τνs when T21 ≃ −150 mK, while the black coloured curves represent the lower projected constraint on τνs when T21 ≃ −100 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' To get the dashed line, we do not take into account the VDKZ18 heating of the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For the dotted line we consider VDKZ18 heating of the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Inclusion of VDKZ18, gives more stringent projected constraint on τνs as gas temperature rises due to the energy transfer from CMB photons mediated by Lyα photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In figure (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5), we obtained the upper projected constraint on mixing angle of sterile neutrinos with active neutrinos as a function of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For reference, we have also plotted the x-ray constraint on mixing angle as function 37 Sterile Neutrino Dark Matter Chapter 2 21 cm Line Astronomy and Constraining New Physics of mνs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The constraint is obtained by assuming solely radiative decay of sterile neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' x-ray constraint comes from the fact that no such x-rays have been seen in observations [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The red and black coloured curves depict the upper projected constraint on mixing angle when T21 ≃ −150 mK and -100 mK, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' To get the dashed curves, we do not take into account the VDKZ18 heating of the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For the dotted line we have included the VDKZ18 heating of the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, it is to noted that these bounds do not depend on dark-matter clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, the bounds are free of astrophysical parameters such as density profile or mass function of dark-matter halos, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' To obtain these bounds, we do not consider any non- standard cooling mechanism to cool the IGM or any source of radio photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The results in figure (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5) are comparable with the x-ray constraint for the higher mass of sterile neutrinos, while we get more stronger bounds for lower mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 1025 1026 1027 1028 1029 101 τνs (sec) mνs (KeV) T21 ≃ -150 mK T21 ≃ -100 mK Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4: The figure represents lower projected bounds on the lifetime of sterile neutrinos as a function of mass of sterile neutrinos by keeping 21 cm differential brightness temperature, T21 ≃ −150 and −100 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The dotted (dashed) line represents the case when energy transfer from CMB photons to gas is included (excluded) [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Chapter 2 Sterile Neutrino Dark Matter 38 21 cm Line Astronomy and Constraining New Physics 10-14 10-13 10-12 10-11 10-10 10-9 10-8 10-7 10-6 2 4 6 8 10 20 30 40 50 NuSTAR 20 NuSTAR 22 x-ray Swift-XRT 22 XMM 21 sin2(θ) mνs (KeV) Z = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 T21 ≃ -150 mK T21 ≃ -100 mK Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5: The figure represents upper projected bounds on the mixing angle of sterile neutrinos with active neutrinos as a function of mass of sterile neutrinos by keeping 21 cm differential brightness temperature, T21 ≃ −150 and −100 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The dotted (dashed) line represents the case when energy transfer from CMB photons to gas is included (excluded) [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The shaded regions are excluded for corresponding observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The x-ray constraint on mixing angle (cyan shaded region) has been taken from the Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The red shaded region depicts the upper bounds on sin2(θ) from NuSTAR observations [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, we have also plotted the recently reported bounds (after publication of our article) on sin2(θ) by NuSTAR— represented by NuSTAR 22 [37] and by Swift-XRT— represented by Swift-XRT 22 [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The grey shaded region is excluded by XMM-Newton [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 39 Sterile Neutrino Dark Matter Chapter 2 21 cm Line Astronomy and Constraining New Physics 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6 Summary We have constrained the sterile neutrino dark matter lifetime and mixing angle with active neutrino as a function of sterile neutrino mass, such that energy in- jection from radiative decay of sterile neutrino does not change the standard 21 cm absorption signal (∼ −220 mK) more than about a factor of 1/4 (−150 mK) or 1/2 (−100 mK) at the redshift, z = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We have considered the two scenar- ios to get the bounds: First, IGM evolution without the heat transfer from the background radiation to gas mediated by Lyα photons (VDKZ18 effect).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Next, we have considered the VDKZ18 effect on the IGM gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The following summarises our results for T21 = −150 mK : In the first scenario, the lower bound on the sterile neutrino lifetime varies from 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3 × 1027 sec to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4 × 1025 sec by varying sterile neutrino mass from 2 KeV to 50 KeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The lifetime of sterile neutrino decrease when one increases the mass of the sterile neutrino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' It happens because ρνs = mνsnνs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' At a particular redshift, when one increases mνs, the nνs decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Consecutively, one gets less radiative decay of sterile neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, we get more window to increase the gas temperature, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' we can decrease the lifetime of sterile neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The upper bound on the mixing angle (sin2 θ) varies from 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='8×10−9 to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1×10−14 by varying sterile neutrino mass from 2 KeV to 50 KeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the second scenario, the lower bound on the sterile neutrino lifetime varies from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5 × 1028 sec to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7 × 1026 sec by varying sterile neutrino mass from 2 KeV to 50 KeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' While the upper bound on the mixing angle varies from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='8 × 10−9 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='42 × 10−14 by varying sterile neutrino mass from 2 KeV to 50 KeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We have also plotted the x-ray constraint to rule out some parameter space for mixing angle of the sterile neutrinos with active neutrinos [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Although we have considered that sterile neutrinos account for all the dark matter in the Universe, sterile-neutrino may account for only a fraction of the dark matter abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In this scenario, the bounds on the sterile neutrino lifetime and mixing angle with Chapter 2 Sterile Neutrino Dark Matter 40 21 cm Line Astronomy and Constraining New Physics active neutrino may modify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7 Additional study 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 Bounds in light of varying T21 and redshift We have also studied the projected constraints on τνs and sin2(θ) by varying the absorption amplitude of T21 between 0 mK (no signal) to −200 mK at z = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' If we increase the value of T21 above 0, it gives a emission signal instead of an absorption signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, we restrict the maximum value of T21 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Also, we do not take the values of T21 below ∼ −200 mK, as the sterile neutrino term in the temperature evolution equation becomes insignificant compared to adiabatic and Compton scattering term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the ΛCDM model without invoking any new physics, we get T21(z = 17) = −220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='215 for the cosmological parameters Ωm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='31, Ωb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='048 and h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='68 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For the demonstration purpose of this, we take mνs = 2 KeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' After inclusion of physics of decaying sterile neutrinos, we get T21(z = 17) = −220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='213 mK for τνs = 4 × 1032 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' If we increase the value of T21 only by 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7 × 10−2 percent (from T21 = −220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='213 mK to −220 mK), the value of τνs changes significantly from 4 × 1032 sec to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='78 × 1030 sec— decreases by more than a factor of hundred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, we do not consider bounds on the τνs and sin2(θ) near the maximal absorptional value of T21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' and vary the value of T21 from −200 to 0 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In figure (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6a), we have plotted the lower projected bounds on the lifetime of sterile neutrinos as a function of mass for various values of T21 at z = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In figure (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6b), we have plotted the upper projected bounds on the mixing angle of sterile neutrinos with active neutrinos as a function of mass for various values of T21 at z = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' NuSTAR 22 bound is reported in July 2022 [37] and Swift- XRT 22 is reported in August 22 [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The observational bounds indicate that the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='decay of sterile neutrinos will not significantly impact the thermal history of the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='41 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='Sterile Neutrino Dark Matter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='Chapter 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='21 cm Line Astronomy and Constraining New Physics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1022 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1023 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1024 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1025 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1026 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1027 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1028 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1029 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='τνs (sec) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='mνs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='(KeV) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='Z = 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='T21 ≃ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='0 mK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='T21 ≃ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='-50 mK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='T21 ≃ -100 mK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='T21 ≃ -150 mK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='T21 ≃ -200 mK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='10-14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='10-13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='10-12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='10-11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='10-10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='10-9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='10-8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='10-7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='10-6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='NuSTAR 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='NuSTAR 22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='x-ray ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='Swift-XRT 22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='XMM 21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='sin2(θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='mνs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='(KeV) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='Z = 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='T21 ≃ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='0 mK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='T21 ≃ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='-50 mK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='T21 ≃ -100 mK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='T21 ≃ -150 mK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='T21 ≃ -200 mK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6: Plot (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6a), shows lower projected bounds on the lifetime of sterile neutrinos as a function of mass, while plot (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6b), shows upper projected bounds on the mixing angle of sterile neutrinos with active neutrinos as a function of mass of sterile neutrinos for varying 21 cm differential brightness temperature (T21) at z = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In figure (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6b), the shaded regions are excluded for corresponding observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The cyan shaded region represents the x-ray constraint [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The red shaded region depicts the upper bounds on sin2(θ) from NuSTAR observations [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, we have also included the recently reported bounds (after publication of our article) on sin2(θ) by NuSTAR— represented by NuSTAR 22 [37] and by Swift- XRT— represented by Swift-XRT 22 [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The grey shaded region is excluded by XMM-Newton [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Chapter 2 Sterile Neutrino Dark Matter 42 21 cm Line Astronomy and Constraining New Physics 1024 1025 1026 1027 1028 1029 2 4 6 8 10 20 30 40 50 τνs (sec) mνs (KeV) T21 ≃ -150 mK Z = 15 Z = 17 Z = 19 (a) 10-14 10-13 10-12 10-11 10-10 10-9 10-8 10-7 2 4 6 8 10 20 30 40 50 sin2(θ) mνs (KeV) T21 ≃ -150 mK Z = 15 Z = 17 Z = 19 (b) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7: Plot (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7a), shows lower projected bounds on the lifetime of sterile neutrinos, while plot (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7b), shows upper projected bounds on the mixing angle of sterile neutrinos with active neutrinos by keeping T21 to −150 mK for different values of redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Universe as the parameter space is excluded more stringently by observations for a higher mass of sterile neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For example, XMM 21 excludes the values of sin2(θ) ≳ 2 × 10−12 for mνs ≃ 6 KeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' If one wants to exclude this parameter space using 21-cm signal, it requires to consider T21 < −200 mK— i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' no significant modification in the thermal and ionization history of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For further analysis with variation of redshift values, we have also added the plots for the case with different values of redshift keeping the value of T21 constant— presented in figure (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, we vary redshift between 15 to 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As it is shown in figure (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4), for fiducial models for Lyα coupling and x-ray heating, we can not take the spin temperature to be gas temperature above z ∼ 17 and also x-ray starts to dominate below z ∼ 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, we restrict ourselves about redshift 17 and take a range from 15 to 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, we do not see significant variation in the projected bounds of lifetime and mixing angle with variation of the values of redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In figure (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='8),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' we vary both the value of redshift and T21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' and plot the lower 43 Sterile Neutrino Dark Matter Chapter 2 21 cm Line Astronomy and Constraining New Physics 1024 1025 1026 1027 1028 1029 2 4 6 8 10 20 30 40 50 τνs (sec) mνs (KeV) Z = 15,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' T21 ≃ -178 mK Z = 17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' T21 ≃ -165 mK Z = 19,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' T21 ≃ -155 mK (a) 10-14 10-13 10-12 10-11 10-10 10-9 10-8 10-7 2 4 6 8 10 20 30 40 50 sin2(θ) mνs (KeV) Z = 15,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' T21 ≃ -178 mK Z = 17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' T21 ≃ -165 mK Z = 19,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' T21 ≃ -155 mK (b) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='8: Plot (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7a), shows lower projected bounds on the lifetime of sterile neutrinos, while plot (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7b), shows upper projected bounds on the mixing angle of sterile neutrinos with active neutrinos by keeping T21 such that it does not change more than a factor of 1/4 from the minimum possible amplitude based on ΛCDM model for corresponding values of redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' projected bounds on τνs in figure (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='8a) and upper projected bounds on sin2(θ) in figure (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='8b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, we choose the value of T21, such that, it does not change more than a factor of 1/4 from the minimum possible amplitude based on ΛCDM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For the cosmological parameters, given above, we get the minimum pos- sible amplitude of T Min 21 to be −236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7 mK at z = 15, −220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 mK at z = 17 and −206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 mK at z = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Chapter 2 Sterile Neutrino Dark Matter 44 “The universe doesn‘t allow perfection.” Stephen Hawking, A Brief History of Time 3 Primordial Black Hole Dark Matter Primordial black holes have attracted much interest in recent years and have been a part of intense studies for more than five decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The idea of the black hole goes back to the 18th century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In 1784, John Michell proposed that there could be such supermassive bodies that light could not pass them, or all light emitted would return towards them [10–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Later, in 1915 Albert Einstein developed the general theory of relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In 1916, Karl Schwarzschild found the solution of black holes by solving the Einstein field equations for a point mass [167].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Subsequently, in 1963, Roy Kerr found the solution of rotating black holes [168].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In 1965, the more general solution of a rotating and charged black hole was found [169].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' There is a possibility that a colossal number of black holes might have been formed in the very early Universe— known as primordial black holes (PBHs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' PBHs can be created by various mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' It was first suggested by Zel’dovich and Novikov that the presence of initial inhomogeneities in the Universe can form PBHs [170].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' There is 21 cm Line Astronomy and Constraining New Physics a possibility that for many regions in the space, gravitational energy of the initial density fluctuations can exceed the kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' These regions would have a gravitational collapse instead of the expanding with Universe creating collapsed objects with a minimum mass of ∼ 10−5 g [171–173].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' There are various mechanisms that can produce inhomogeneities in the early Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For example, such density fluctuations can be generated due to the vacuum strings produced during the grand unification phase transition [174].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Indeed, these fluctuations were present in the very early Universe, as evident from observations of structures in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The other explanations of PBHs formation include the collapse of cosmic string loops, collisions of bubbles, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The cosmic string loops can disappear in two ways: First, they can shrink into scalar and gauge particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Second, some loops with specific initial shapes may disappear by collapsing in size below their Schwarzschild radius and form black holes [175–178].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the article [179], the authors consider the formation of PBHs due to collapsing cosmic strings and argue that PBHs can significantly contribute to the dark matter density if their relic mass is larger than 103 mpl, here mpl is the Planck mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In another scenario, the collapse of the cusps neighbourhood of cosmic strings loops can also form a large number of spinning PBHs [180].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The collisions between the bubbles during various phase transitions in the Universe can also give rise to the formation of PBHs [181–183].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' PBHs can also be produced in various inflation models [184–186].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Depending on the formation time (t), PBHs can have a wide range of masses (in most of the cases roughly order of the Hubble horizon mass at the formation time) [187, 188], MPBH ∼ 1015 � t 10−23 sec � g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1) For example, PBHs with mass 1015 g might have formed at t ∼ 10−23 after big- bang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In another example, PBHs formed during the QCD phase transition (t ∼ 10−5 sec) might have a mass comparable to a solar mass [189].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' PBHs formed Chapter 3 Primordial Black Hole Dark Matter 46 21 cm Line Astronomy and Constraining New Physics around neutrino decoupling (t ∼ 1 sec) can have a mass about 105 M⊙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 Primordial black holes as dark matter In the last decades, many particle-dark matter models have been proposed to explain the various astrophysical observations, as discussed in chapter (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The laboratory experiments for direct detection of dark matter have not observed any signature yet, for example, DarkSide-50, LUX, XENON1T, PandaX-II, CRESST, PICO, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [190–195].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In this situation, it is desirable to look for alternative scenarios where dark matter may not be an elementary particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As PBHs are massive, interact only gravitationally and are formed in the very early Universe, they can be considered as a potential candidate for non-particle dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Recently, PBHs have gathered much attention in the scientific community after the black hole binary merger detection by Virgo and LIGO collaborations, and these events suggest that PBHs may constitute a fraction of dark matter [13– 15, 196–198].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' PBHs having a mass below ∼ 1022 g can explain all the dark matter in the Universe as they are not ruled out by microlensing constraints [199].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We will discuss other constraints on dark matter fractions in the form of PBHs with mass below ∼ 1022 g later in the sections (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' One can explain the existence of dark matter in the form of PBHs without considering physics beyond the standard model (BSM) of particle physics by considering standard model Higgs fluctuations during inflation as instability can occur in Higgs potential at a scale O(1011 GeV) [200].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the article [184], authors consider the double inflation model to explain the formation of PBHs between two inflations and argue that PBHs can be accounted for dark matter in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' PBHs as missing matter or dark matter in the context of galaxy formation has been explored in old literature also [201, 202].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Authors of the Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [185], consider the formation of PBH dark matter due to the mild-waterfall phase of hybrid inflation and discuss how the tail of the mass distribution of PBHs can explain the origin for the supermassive black holes 47 Primordial Black Hole Dark Matter Chapter 3 21 cm Line Astronomy and Constraining New Physics observed at galactic centres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' These massive back holes can also provide the seed for present-day observed structures [185, 203].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' A fraction/all of dark matter in the form of PBHs can produce the r-process nucleosynthesis— a process that is responsible for producing about half of the heavier nuclei than iron, in the mass range 10−14 M⊙ < MPBH < 10−8 M⊙ [204].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Black holes can lose their mass by the emission of energetic particles due to Hawking evaporation [205].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For non-rotating and non-charged black holes formed in the very early Universe, their evaporation time scale can be given by [187], τ(MPBH) ∼ �MPBH 1015 g �3 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2) Therefore, PBHs having mass larger than 1015 g can survive the Hawking evapo- ration and account for present-day dark matter density [206].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 Signature of Primordial Black Holes It is possible that a fraction of PBHs can grow to intermediate-mass black holes and explain the ultraluminous x-ray sources reported in various observations [185, 207– 209].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' There are several hints that indicate the presence of PBHs, such as dynam- ics and star clusters of ultra-faint-dwarf-galaxies, correlations between x-ray and infrared cosmic backgrounds, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (for a detailed review, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [210]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The pres- ence of evaporating PBHs can explain the galactic/extra-galactic γ-ray background radiation [211–214], short-duration γ-ray bursts [215, 216], and reionization by in- jection of energetic photons and e± radiations into IGM [217, 218].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The emission of nucleons by evaporating PBHs can explain the observed baryon number density if more baryons are produced compared to antibaryons— in a baryon-symmetric Universe [213].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Clustering between PBHs can provide the seeds for galaxy forma- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' PBHs evaporation can explain the observed point-like γ-ray sources [217].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The presence of massive PBHs can also serve as seeds for active galactic nuclei (AGN) [217].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Chapter 3 Primordial Black Hole Dark Matter 48 21 cm Line Astronomy and Constraining New Physics 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 Existing bounds on Primordial Black Holes The fraction of dark matter in the form of PBHs (fPBH ≡ ΩPBH/ΩDM) is con- strained from various astrophysical observations and theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, ΩPBH and ΩDM are the dimensionless density parameters for PBHs and dark mat- ter, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' PBHs with mass smaller than ∼ O(1015 g) may have evaporated as of now and can be constrained from the impact on big bang nucleosynthesis by evaporated particles, background radiation etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Higher mass PBHs can be con- strained by the effect on large-scale structures, gravitational wave and lensing, and impact on thermal and ionization history of the IGM (for details, see the recent re- views [187, 219, 220] and the references cited therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the context of the 21 cm signal, the upper bound on the fPBH can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [221–228].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Angular mo- mentum is a fundamental property of a black hole, and it can modify the Hawking evaporation drastically [40, 229–231].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the case of rotating PBHs, authors of the Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [41, 232] have reported the various types of bound on fPBH as a function of PBHs mass and spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Future collaboration, All-sky Medium Energy Gamma-ray Observatory (AMEGO)a will be able to constrain some parameter space for the rotating PBHs [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We discuss more bounds on the fraction of PBH dark matter in the result and discussion section (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In this chapter, we consider the rotating PBHs and constrain dark matter fraction in the form of PBHs as a function of their mass for various values of angular momentum in the light of global 21 cm signal [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3 Impact on the thermal and ionization history During the cosmic dawn era, the evolution of the gas temperature and ionization fraction of the Universe are well-known [153, 154].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The addition of any exotic source of energy during the cosmic dawn era can significantly impact the ionization ahttps://asd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='gov/amego/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='html 49 Primordial Black Hole Dark Matter Chapter 3 21 cm Line Astronomy and Constraining New Physics and thermal history of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, we can constrain the properties of such exotic sources from the observations during the cosmic dawn era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Evaporating PBHs can heat the gas and modify the free electron fraction in the IGM [46, 232].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Rotating PBHs can emit more particles into IGM and substantially affect the IGM evolution compared to non-rotating PBHs [214, 229, 233].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, it is important to study the properties of spinning PBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Black holes can get their spin depending on generation mechanisms, merger or accretion [234–244].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' PBHs with higher mass can have a lifetime larger/comparable than the age of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, they have enough time to accrete mass and spin up [245].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the present work, we consider the Hawking emission of PBHs into background radiations (photons and electron/positron) and provide the projected constraints on the fraction of dark matter in the form of PBHs (fPBH) as a function of mass and spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We analyse projected bounds on spinning PBHs such that 21 cm differential brightness temperature does not change by more than a factor of 1/4 from the ΛCDM model prediction (|T21| ∼ 220 mK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' A rotating black hole with angular momentum JPBH and having mass MPBH can be defined with a rotation parameter, a∗ = JPBH/(GN M 2 PBH) [233], where GN is the gravitational constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Rotating black hole with higher spin (a∗ → 1) injects more energy into IGM and evaporates faster than non-rotating ones [40, 229–231].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, we expect that the bounds on fPBH to be more stringent compared to non-rotating PBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The energy injection per unit volume per unit time due to e± and photons into IGM, for monochromatic mass distribution of PBHs, can be written as [232, 246], Γe± PBH(z, a∗) = 2 � � f e c (E − me, z) (E − me) � d2Ne dt dE � � nPBH dE , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3) Γγ PBH(z, a∗) = � � f γ c (E, z) E � d2Nγ dt dE � � nPBH dE .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4) Energy injection into IGM happens by three processes: heating, ionization, and excitation of the gas [247–249].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' f i c represents the energy deposition efficiency Chapter 3 Primordial Black Hole Dark Matter 50 21 cm Line Astronomy and Constraining New Physics into IGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, c stands for above-mentioned three channels and i ≡ (electron/ positron, photon) stands for different types of injected particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The factor of 2 in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3) accounts for the total contribution of electrons and positrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' nPBH = fPBH (ρDM/MPBH) is the number density of the PBHs, and ρDM is the dark matter energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' d2N i/(dt dE) ≡ d2N i/(dt dE) � E, MPBH, a∗ � represents the number of i particles emitted by black hole per unit time per unit energy [232, 233, 250, 251], d2N i dt dE = 1 2 π � dof Γi(E, MPBH, a∗) eE′/TPBH ± 1 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5) here, Γi is the greybody factor— defines the probability of emitted particle i from black hole to overcome its gravitational potential well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' dof represents the degree of freedom [251].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Moreover, E is the total energy of emitted particle i and E′ = E − nΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' While, n is the axial quantum number and Ω is the angular velocity at black hole horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We use the BlackHawk codeb to calculate the spectra due to photons, electrons and positrons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' we take both the primary and secondary Hawking evaporation spectra into account— i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' emitted final particle j per unit time and per unit energy [251, 252] d2N j dt dE = � i d2N i dt dE′′ dN i j dE dE′′ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6) here, dN i j is the hadronization table accounts for the transformation of the primary spectra into secondary spectra [251–253].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the presence of Hawking radiation, the thermal evolution of the gas can be written as[157, 254], dTgas dz = 2 Tgas 1 + z + Γc (1 + z) H (Tgas − TCMB) − 2 ΓPBH 3 ntot(1 + z) H , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7) bhttps://blackhawk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='hepforge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='org/ 51 Primordial Black Hole Dark Matter Chapter 3 21 cm Line Astronomy and Constraining New Physics here, ΓPBH = Γe± PBH + Γγ PBH is the total energy injection per unit time and per unit volume into IGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We consider the following numerical values of the cosmological parameters: h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='674, ΩM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='315, Ωb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='049 and TCMB|z=0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='725 K [48, 255].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' To compute the energy deposition efficiency, thermal and ionization history of the Universe, we use DarkHistoryc package with necessary modifications [249].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4 Results and Discussion We take 21 cm differential brightness temperature such that it does not change, from its ΛCDM value (∼ 220 mK), by more than a factor of 1/4 at redshift 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We solve the coupled equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9— replacing E with ΓPBH) for different mass, spin and fraction of PBH dark matter to get xHI and Tgas at redshift z = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' To get any absorption signal in redshift range 15 − 20, the gas temperature should be less than CMB temperature in shaded region— redshift range from 15 to 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' By requiring T21 ≃ −150 mK at z=17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2, we constrain the parameter space of PBH dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the present chapter, we do not consider x-ray heating of the gas due to the uncertainty in the physics of the first stars— as we discussed earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For a fix value of T21 at a redshift, if one includes the x-ray heating of gas, our projected upper constraints on PBH dark matter fraction becomes stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, it is to be noted that the gas temperature may increase due to the energy transfer from the background radiation to the thermal motions of the gas mediated by Lyα radiation from the first stars [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' However, again due to the uncertainty in physics of the first star formation, we do not include this effect also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The inclusion of this effect will also further strengthen our projected upper bounds on fPBH— similar to discussed in chapter (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In order to understand how spin, fraction and mass of PBH dark matter can affect the thermal evolution of the gas, we plot the figures (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The shaded region corresponds redshift range, 15 − 20 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The red chttps://darkhistory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='io/en/master/ Chapter 3 Primordial Black Hole Dark Matter 52 21 cm Line Astronomy and Constraining New Physics 10-1 100 101 102 103 104 101 102 TCMB Tgas (No PBH) Tgas (K) 1+Z MPBH = 1x1015 g, fPBH = 10-7 a* = 0 a* = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5 a* = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='99 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1: The gas temperature evolution with redshift for evaporating primordial black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The red dashed lines represent the CMB temperature evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The black dashed lines depicts the Tgas when there is no PBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The shaded region corresponds to the redshift 15 ≤ z ≤ 20 (EDGES observed signal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In this figure, we consider PBHs mass and fPBH to 1 × 1015 g and 10−7, respectively, and vary the spin of PBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 10-1 100 101 102 103 104 101 102 TCMB Tgas (No PBH) Tgas (K) 1+Z MPBH = 1x1015 g, a* = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5 fPBH = 1 x 10-6 fPBH = 1 x 10-7 fPBH = 1 x 10-8 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2: The caption is the same as in Figure (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1), except here, we keep MPBH = 1 × 1015 g and a∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5 constant and vary fPBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 53 Primordial Black Hole Dark Matter Chapter 3 21 cm Line Astronomy and Constraining New Physics 10-1 100 101 102 103 104 101 102 TCMB Tgas (No PBH) Tgas (K) 1+Z a* = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5, fPBH = 10-7 MPBH = 1 x 1015 g MPBH = 3 x 1015 g MPBH = 5 x 1015 g Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3: The caption is the same as in Figure (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1), except here, we vary the mass of PBHs and keep spin and fPBH to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5 and 10−7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' dashed curves in all plots depict the CMB temperature evolution, while the black dashed line represents the gas temperature when there are no evaporating PBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In Figure (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1), we keep mass to 1 × 1015 g and fPBH = 10−7, and vary the spin of PBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As expected, when we increase the spin of PBHs, the gas temperature rises significantly in the shaded region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The solid violet curve represents the case when the spin of PBHs is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Increasing the spin to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5 (solid green line), the gas temperature increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Further increasing a∗ to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='99 (solid cyan line), the gas temperature rises further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In Figure (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2), we keep MPBH = 1 × 1015 g, spin to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5 and vary fPBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In this plot, as we increase the fPBH from 10−8 (solid cyan line) to 10−6 (solid violet line), the IGM heating rises rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' If the gas temperature becomes larger than the CMB temperature in the shaded region, it can erase the 21 cm absorption signal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' instead, it may give an emission signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, at desired redshift (in our scenario z = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2), one has to keep Tgas < TCMB to get an absorption signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Increasing fPBH, for a given mass, the number density of PBHs increases resulting in more energy injection into IGM by Hawking evaporation of PBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, fPBH plays a significant role in deciding whether one gets an Chapter 3 Primordial Black Hole Dark Matter 54 21 cm Line Astronomy and Constraining New Physics absorption profile or emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In Figure (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3), we vary the mass of PBHs and keep spin and fPBH constants to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5 and 10−7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In this plot, as we increase the mass of PBHs from 1 × 1015 g (solid violet line) to 5 × 1015 g (solid cyan line), the gas temperature decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' It happens for two reasons: (i) Increasing the mass of PBHs leads to a decrease in the total power contributions from Hawking evapo- ration of PBHs [250].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (ii) Ignoring the integral dependency in equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4), Γe± PBH and Γγ PBH are proportional to nPBH = fPBH (ρDM/MPBH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For a fixed dark-matter energy density and fPBH, the number density of PBHs increases by decreasing the black hole mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Thus, energy injection into IGM per unit volume and time (ΓPBH) increases, and one gets more heating of the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 55 Primordial Black Hole Dark Matter Chapter 3 21 cm Line Astronomy and Constraining New Physics 10-10 10-9 10-8 10-7 10-6 10-5 1015 1016 IGRB, a* = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9 Planck, a*=0 COMPTEL, a*=0 fPBH MPBH (g) a* = 0 a* = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5 a* = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9 a* = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9999 (a) 10-5 10-4 10-3 10-2 10-1 100 1016 1017 1018 INTEGRAL, a* = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9 Leo T, a*=0 AMEGO, a*=0 (forecast) AMEGO, a*=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9999 (forecast) fPBH MPBH (g) a* = 0 a* = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5 a* = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9 a* = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9999 (b) Chapter 3 Primordial Black Hole Dark Matter 56 21 cm Line Astronomy and Constraining New Physics Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4 (previous page): The projected upper bounds on the dark fraction of matter in the form PBHs (fPBH = ΩPBH/ΩDM) as a function of PBHs mass for different values of a∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The shaded regions are excluded from our analysis for fPBH when a∗ = 0 (dotted black line), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5 (dot-dashed black line), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9 (dashed black line) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9999 (solid black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The dashed blue curve depicts the upper constraint on fPBH by observations of the diffuse Isotropic Gamma-Ray Background (IGRB) for a∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9 [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The double-dot-dashed blue curve represents the upper constraint on fPBH from Diffuse Supernova Neutrino Background (DSNB) searches at Super- Kamiokande, while the solid blue line represents the INTErnational Gamma-Ray Astrophysical Laboratory (INTEGRAL) observation of 511 KeV γ-ray lines at Galactic centre constraint on fPBH for a∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9 [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The double-dot-dashed ma- genta (red) line represents the AMEGO forecast for a∗ = 0 (a∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9999) [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Near future, AMEGO collaboration will be able to probe the parameter-space above the magenta (red) double-dot-dashed curve for a∗ = 0 (a∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The solid green line stands for 95% confidence level bound from INTEGRAL obser- vation of Galactic gamma-ray flux for non-spinning PBHs [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Solid cyan curve depicts the upper bound from observing the 511 KeV γ-ray lines at the Galactic centre by assuming all the PBHs within a 3 Kpc radius of the Galactic centre for non-spinning PBHs [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The magenta solid line represents the Planck constraint [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The red solid line depicts the dwarf galaxy Leo T constraint [46] and the green dashed line shows the COMPTEL bound [47] for non-spinning PBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In Figure (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4), we plot the upper projected bounds on the fraction of dark matter in the form of PBHs as a function of PBHs mass for different spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, we have considered that 21 cm differential brightness temperature, T21, remains −150 mK at redshift z = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We vary the mass of PBHs from 1015 g to 1018 g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The shaded regions in both the plots are excluded for the corresponding PBH spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The dashed blue curve represents the upper constraint on fPBH by observations 57 Primordial Black Hole Dark Matter Chapter 3 21 cm Line Astronomy and Constraining New Physics of the diffuse Isotropic Gamma-Ray Background (IGRB) [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The double-dot- dashed blue curve represents the upper constraint on fPBH from Diffuse Supernova Neutrino Background (DSNB) searches at Super-Kamiokande, while the solid blue line represents the INTErnational Gamma-Ray Astrophysical Laboratory (INTE- GRAL) observation of 511 KeV γ-ray line at Galactic centre constraint on fPBH for a∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9 [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For a∗ = 0, the observation at the Jiangmen Underground Neu- trino Observatory (JUNO) will be able to place a 20 times stronger bound on the upper allowed value of fPBH for MPBH = 1015 g compared to Super-Kamiokande [41, 256].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The double-dot-dashed magenta (red) line represents the AMEGO fore- cast for a∗ = 0 (a∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9999) [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the near future, AMEGO collaboration will be able to probe the parameter-space above the magenta (red) double-dot-dashed curve for a∗ = 0 (a∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Solid green line stands for 95% confidence level bound from INTEGRAL observation of Galactic γ-ray flux for non-spinning PBHs [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The solid cyan curve depicts the upper bound from the observation of 511 KeV γ-ray lines at the Galactic centre by assuming all the PBHs within a 3 Kpc radius of the Galactic centre for non-spinning PBHs [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For the comparison, we have also plotted the bounds from Planck [45], Leo T [46] and COMPTEL [47] observations for non-spinning PBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In Figure (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4a), fPBH varies from 1 × 10−10 to 1 × 10−5, while, in Figure (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4b), it varies from 1 × 10−5 to its maximum al- lowed value 1 (ΩPBH = ΩDM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In Figure (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4), as we increase the value of spin from 0 to its extremal value, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9999, the upper bounds become more stringent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' This is due to an increment in evaporation of PBHs, and it results in more energy injection into the IGM [233, 257, 258].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As discussed earlier, increasing the mass of PBHs, energy injection into IGM decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Subsequently, one gets more window to increase the gas temperature or fPBH, and the upper bound becomes weaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, in Figure (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4), the upper bound on fPBH weakens as we increase the mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Our upper projected constraint on fPBH for a∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9 is comparable to the INTEGRAL observation of 511 KeV γ-ray lines for PBHs mass larger than ∼ 8×1016 and becomes stronger for smaller PBH masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Also, compared to IGRB Chapter 3 Primordial Black Hole Dark Matter 58 21 cm Line Astronomy and Constraining New Physics [40] and DSNB [41], our projected bounds are stringent for the considered mass range of PBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We find the most robust lower projected constraint on the mass of PBHs, which is allowed to constitute the entire dark matter, to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5 × 1017 g, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9 × 1017 g, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9 × 1017 g and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7 × 1017 g for PBH spins 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9999, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The lower bound on MPBH for ΩPBH = ΩDM, for extremal spinning PBHs is nearly four times larger than non-spinning PBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5 Conclusions Spinning primordial black holes can substantially affect the ionization and thermal history of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Subsequently, it can modify the 21 cm absorption signal in the cosmic dawn era by injecting energy due to Hawking evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We study the upper projected bounds on the fraction of dark matter in the form of PBHs as a function of mass and spin, considering that the 21 cm differential brightness temperature does not change by more than a factor of 1/4 from the theoretical prediction based on the ΛCDM framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Our projected constraints are stringent compared to DSNB, INTEGRAL observation of the 511 KeV line, IGRB, Planck, Leo T and COMPTEL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the near future, AMEGO collaboration will be able to probe some parameter space in our considered mass range of PBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the present work, we have considered the monochromatic mass distribution of PBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The allowed parameter space can also be explored for different PBHs mass distributions such as log-normal, power-law, critical collapse, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [251].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, it is to be noted that we have not considered heating of IGM gas due to x-ray from the first stars in the vague of known physics of the first stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For a fix value of T21 at a redshift, if one includes the x-ray heating of the gas, the projected bounds becomes stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 59 Primordial Black Hole Dark Matter Chapter 3 21 cm Line Astronomy and Constraining New Physics 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6 Additional study 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 Bounds in light of varying T21 and redshift We also study upper projected bounds on the fraction of the dark matter in the from of PBHs by varying the amplitude of 21 cm differential brightness tempera- ture and redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In figure (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5), we have plotted upper bounds on the fraction of dark matter in the form of primordial black holes as a function of mass for various values of T21 at z = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' To understand that how bounds change on fPBH with T21, we consider two scenarios for the spin of PBHs: a∗ = 0 (figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5c) and a∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9 (figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In both of the plots, we notice that when we change the value of T21 from −200 mK to −150 mK the bound relaxes with a factor of ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' By changing the T21 from −150 mK to −100 mK bounds relax by a factor of ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3, and, going from −100 mK to −50 mK the bounds relax by a factor of ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' By changing the T21 from −50 mK to 0, bounds relax by a factor of ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' This similar pattern 10-10 10-8 10-6 10-4 10-2 100 1015 1016 1017 fPBH MPBH (g) a* = 0, Z = 17 T21 ≃ 0 mK T21 ≃ -50 mK T21 ≃ -100 mK T21 ≃ -150 mK T21 ≃ -200 mK (c) 10-10 10-8 10-6 10-4 10-2 100 1015 1016 1017 fPBH MPBH (g) a* = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9, Z = 17 T21 ≃ 0 mK T21 ≃ -50 mK T21 ≃ -100 mK T21 ≃ -150 mK T21 ≃ -200 mK (d) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5: The plots represent the upper projected bounds on the fraction of dark matter in the form of primordial black holes (fPBH) as a function of mass of PBHs (MPBH) for varying 21 cm differential brightness temperature (T21) at z = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Figure (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5c) represents the case when spin of PBHs: a∗ = 0, while, figure (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5d) represents the case with a∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Chapter 3 Primordial Black Hole Dark Matter 60 21 cm Line Astronomy and Constraining New Physics 10-10 10-8 10-6 10-4 10-2 100 1015 1016 1017 fPBH MPBH (g) T21 ≃ -150 mK, a* = 0 Z = 15 Z = 17 Z = 19 (a) 10-10 10-8 10-6 10-4 10-2 100 1015 1016 1017 fPBH MPBH (g) a* = 0 Z = 15, T21 ≃ -179 mK Z = 17, T21 ≃ -166 mK Z = 19, T21 ≃ -155 mK (b) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6: Figure (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6a) represents upper projected bounds on fPBH when T21 ≃ −150 mK for different values of redshift (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Figure (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6b) represents upper bounds on fPBH when T21 does not change more than a factor of 1/4 from the minimum possible amplitude based on ΛCDM model for corresponding values of redshift (T Min 21 (z = 15) ≃ −238 mK, T Min 21 (z = 17) ≃ −221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 mK and T Min 21 (z = 19) ≃ −207 mK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, the cosmological parameters are: h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='674, ΩM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='315, Ωb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='049 [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Both figures obtained for a∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 61 Primordial Black Hole Dark Matter Chapter 3 21 cm Line Astronomy and Constraining New Physics also occurs for the case of sterile neutrinos and the factors remain same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, we can also find the constraints on fPBH for other values of spin when the bound on fPBH is given for any value of T21 ∈ {0, −50, −100, −150, −200} mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In figure (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6a), similarly to sterile neutrino case, we see that bounds do not change significantly for a fix value of T21 at different values of redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In figure (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6b), upper bounds on fPBH are obtained such that T21 does not change more than a factor of 1/4 from the minimum possible amplitude based on ΛCDM model for corresponding values of redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In this chapter, we consider the following numerical values of the cosmological parameters: h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='674, ΩM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='315, Ωb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='049 and TCMB|z=0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='725 K [48, 255].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, we get minimum possible value of T21 based on ΛCDM model to −238 mK at z = 15, −221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 mK at z = 17 and −207 mK at z = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Chapter 3 Primordial Black Hole Dark Matter 62 “Astronomy and Pure Mathematics are the mag- netic poles toward which the compass of my mind ever turns.” Carl Friedrich Gauss, In Letter to Bolyai (30 Jun 1803) 4 Primordial Magnetic Fields and Excess Radio Background Observations suggest that the magnetic fields are ubiquitous in the Universe— from the length scale of planets and stars to the cluster of galaxies [17–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Fermia and High Energy Stereoscopic System (HESS)b gamma-ray observation suggests that even voids could host magnetic fields with strength O(10−16 G) with a typical coherent scale of Mpc [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Magnetic fields can also play a significant role in reionization, relic electron density and structure formation [259].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The presence of magnetic fields can substantially affect the evolution and dynamics of structures in the Universe as they can contribute to the total pressure against gravitational collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' This could modify the total matter power spectrum on small scales, ahttps://fermi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='gov/ bhttps://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='mpi-hd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='de/hfm/HESS/ 21 cm Line Astronomy and Constraining New Physics ≲ 1 Mpc [259–264].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The presence of magnetic fields during recombination can also have important consequences, such as it could have lead to the collapse of gas clouds after recombination, formation of first pre-galactic stars, quasars [265].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The Earth has a magnetic field of the order of O(G), and it is sustained for years by some dynamo mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Similarly, other astronomical objects near to Earth, such as Sun and other solar system planets, also show the presence of magnetic fields [266].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Our home galaxy Milky Way, other spiral galaxies and their interstellar medium (ISM) contain magnetic fields with the strength O(µG) [261, 264, 266– 268].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Moreover, galaxy clusters, intercluster medium, filaments, IGM, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=', also show the magnetic fields [19, 21, 269–271].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' These magnetic fields are likely to be seeded by primordial magnetic fields (PMFs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' These PMFs might have originated in the very early Universe, and subsequently amplified in the small scale structures by some mechanisms [21, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 Generation of primordial magnetic fields The origin and evolution of PMFs is one of the outstanding problems of modern cosmology (Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [23, 24] and references cited therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' It would be very difficult to explain the magnetic fields in the voids and high redshift galaxies with only late- time astrophysical processes without magnetic fields from the very early Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, these magnetic fields indeed may have a primordial origin [21, 272– 274].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' There are several theoretical models that can generate the magnetic field in the early Universe with a large coherent scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The two scenarios to generate PMFs that are vastly discussed in the literature are phase transitions in the early Universe and various models of inflation (for details, see the recent review [24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [275], the authors discuss how the inflation model can generate large scale, ∼ O(Mpc), magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The generated magnetic fields have a small strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' To amplify the field, one has to break the conformal invariance of the electromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The authors consider three mechanisms to break the con- Chapter 4 PMFs & Excess Radio Background 64 21 cm Line Astronomy and Constraining New Physics formal invariance: Coupling of the photon to the axions, gravitational field and massless-charged-nonconformally invariant scalar field [275].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Authors of the Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [276], extend the inflation model by introducing the coupling between the Maxwell field and the scalar field (Φ) responsible for inflation (∝ eαΦFµνF µν), here, Fµν is the electromagnetic field tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' This scenario can generate magnetic fields with a present-day strength up to nG with the coherence scale of a few Mpc depending on the parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' A similar mechanism to generate the magnetic fields during inflation is based on the superstring cosmology [277, 278].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The Lagrangian, simi- lar to considered by [276] with α = −1, naturally arises from the effective action in low-energy string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, inflation is driven by the kinetic part of the dilaton scalar field— Φ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Whereas in the article [276], it is driven by the false vac- uum scalar field potential— which is too steep for producing the slow-roll inflation [272, 277].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the article [279], authors argue that the back reaction of generated magnetic fields via inflation can spoil the inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Considering the backreaction, the authors put an upper bound on the present-day strength of magnetic fields to 10−32 G on the Mpc scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' This strength seems too small for galactic dynamos to amplify to explain the observed magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the recent article [280], it is shown that this issue can be circumvented for some parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The authors find that magnetic fields with a present-day strength of ∼ 10−13 G with a scale of Mpc can be generated while keeping the backreactions under control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Magnetic fields can also arise during electroweak [281] and quantum-chromo-dynamics [282] phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Other mechanisms include cosmic strings [283, 284], primor- dial plasma vorticity [285], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In this chapter, we obtain the upper bounds on present-day strength of PMFs for various values of spectral index in the light of EDGESc observation and excess radio background observed by the ARCADE 2 & LWA 1 observation [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, we obtain the bounds on PMFs in both the presence and absence of heating effects due to first stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' cRecently, the EDGES signal has been questioned in many articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We discuss this point in chapter (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 65 PMFs & Excess Radio Background Chapter 4 21 cm Line Astronomy and Constraining New Physics 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 Existing bounds on primordial magnetic fields The present-day strength, spectral index and coherence scale of PMFs depends on their generation mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, the constraints on PMFs can give a hint of the early Universe physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Recently in the Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [286], authors show that PMFs can be used as a remedy to resolve the Hubble tension between different observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The present-day amplitude of PMFs is constrained from the BBN, formation of structures and temperature anisotropies & polarization of CMB [259, 287, 288].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Authors of the Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [50], put an upper constraint to ∼ 10−10 G on 1 Mpc scale by considering Tgas ≲ TCMB (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' T21 ≲ 0) so that, PMFs can not erase the T21 absorptional signal in the redshift range 15 ≲ z ≲ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Planck 2015 results put individual upper constraints of the O(nG) for different cosmological scenarios on 1 Mpc scale [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The authors of the Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [289], in the context of EDGES observation, put an upper and lower constraint on the PMFs to be 6 × 10−3 nG and 5 × 10−4 nG respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Also, the lower bound on the present-day strength of PMFs found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [290–292].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Further, in the Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [21], authors put a lower bound on the strength of intergalactic magnetic fields of the order of 3 × 10−16 G using Fermi observations of TeV blazars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Authors of the reference [293], report upper bound of 2 × 109 G at the end of BBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Presence of PMFs can modify the present-day relic abundance of He4 and other light elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, magnetic fields can be constrained by observations of light element abundances [259, 294– 297].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The authors of the Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [298], put an upper bound of 47 pG for scale-invariant PMFs by comparing CMB anisotropies, reported by the Wilkinson Microwave Anisotropy Probe (WMAP) and Planck, with calculated CMB anisotropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3 Evolution of PMFs after recombination The generation of the magnetic fields in the early Universe for the various cosmo- logical scenarios has been studied in the earlier literature (for example see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Chapter 4 PMFs & Excess Radio Background 66 21 cm Line Astronomy and Constraining New Physics [272, 282, 290, 299, 300]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' It is to be noted that decaying magnetic fields has been studied in several literatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In these works, the authors consider the decay of the PMFs by ambipolar diffusion and turbulent decay [26, 50, 254, 259, 301].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Ambipo- lar diffusion of magnetic fields is important in neutral medium as it is inversely proportional to free-electron fraction (xe) and xe ∼ 10−4 after redshift z ≲ 100 [160, 254, 259].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The presence of PMFs can induce the Lorentz force in the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The force exerts only on free electrons and ions leaving the neutral components unaffected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' This can result in creating a velocity difference between charged and neutral components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The velocity difference can enhance the collision frequency in the gas, resulting in a dissipation of magnetic energy into the gas— known as the ambipolar diffusion of magnetic fields [302].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' After the recombination (z ∼ 1100), the radiative viscosity of fluid dramatically decreases, and velocity perturbations are no longer damped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, the tangled magnetic fields having length scale smaller than the magnetic Jeans length can dissipate via another mode— tur- bulent decay [254, 259, 303].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Magnetic heating of the gas due to the turbulent decay decreases with redshift but later when ionization fraction decreases, heat- ing increases due to ambipolar diffusion [254, 259].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We further discuss about the ambipolar and turbulent decay in section (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Decaying PMFs can inject mag- netic energy into the thermal energy of the IGM and heat the gas above 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7 K at z = 17, and even it can erase the EDGES absorption signal [50, 254, 259].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Still, the EDGES absorption signal can be explained by considering the possible early excess of radio radiation [304].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4 Background excess radio radiation The Absolute Radiometer for Cosmology, Astrophysics and Diffuse Emission (AR- CADE 2) collaborationd, a double-nulled balloon-borne instrument with seven ra- diometers, measured the absolute sky temperature in a frequency range of 3 − dhttps://asd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='gov/archive/arcade/ 67 PMFs & Excess Radio Background Chapter 4 21 cm Line Astronomy and Constraining New Physics 90 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The observation reported excess radio radiation in a frequency range of 3 − 10 GHz [27], T(ν) = T0 + Tr (ν/ν0)β , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1) here, ARCADE 2 observation fitted the parameters as: T0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='731 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='004 K, β = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6, Tr = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='0 K and ν0 = 310 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' By combining ARCADE 2 with the Low-frequency data [305–308] and Far Infrared Absolute Spectrophotometer (FIRAS) data [309], the parameters can be fitted as: T0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='725 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='001 K, β = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='599 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='036, Tr = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 K and ν0 = 310 MHz in a frequency range of 22 MHz−10 GHz [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' This is measured at present-day (z = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The radiation temperature maps with redshift as: ∝ (1 + z), we can multiply T(ν) with (1 + z) for past [304, 310–314], T(z) = T0 (1 + z) � 1 + Tr T0 � 78 310 �β × � ν 78 MHz �β � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2) This radiation is several times larger than the observed radio counts due to the known Galactic and extragalactic radio processes and sources, such as star-forming galaxies, AGN-driven sources— quasars and radio galaxies, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The presence of early excess radiation can not be completely ruled out at the time of cosmic dawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For example, in the redshift range z ≈ 30 to 16, accretion onto the first intermediate-mass black holes can produce a radio radiation [315].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Accreting supermassive black holes [316] or supernovae [317] can also produce radio background due to synchrotron emission at the time of cosmic down by accelerated electrons in the presence of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The enhancement in the background radiation is also supported by the first station of the Long Wavelength Array (LWA 1)e in frequency range 40 − 80 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The excess observed by LWA 1 can also be fitted by the same model given by equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' After the inclusion of LWA ehttps://leo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='unm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='edu/ lwa/ Chapter 4 PMFs & Excess Radio Background 68 21 cm Line Astronomy and Constraining New Physics 1 data with ARCADE 2 [27] and Low-frequency data [305–308], the parameters change as: T0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='722 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='022 K, β = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='05 and Tr = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6 K at ν0 = 310 MHz [30, 318].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For the observation of 21-cm signal, we can write: ν = 1420.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4/(1 + z) MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2), the factor of (ν/78 MHz)β can be defined as a fraction of excess radio background, Ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Depending on the origin, Ar can have different values— we discuss about this more in next sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, in the final analysis, we vary the value of excess radiation fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 Excess radiation during the cosmic dawn In this work, we use the EDGES signal in the presence of excess radio radiation to constrain the strength of PMFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Some of the processes which we have discussed responsible for the excess radio background can occur at earlier redshift (z ∼ 17) [315–317].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Also, one of the interesting proposals in the Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [304] is to argue that such a possibility can exist at the time of cosmic dawn, and it can help to explain the EDGES signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, authors show that the EDGES absorption signal can be explained by having only 10 percent of the observed radio background by ARCADE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [319, 320], the authors claim that thermal emission from the axion quark nugget dark matter model can explain the EDGES signal, and it can also contribute a fraction of the radiation excess observed by ARCADE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' At present, there exist several theoretical models to explain this excess at the time of cosmic dawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The stimulated emission from Bose stars can give a large contribution to the radio background and explain the EDGES and ARCADE 2 observations [321].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The radio emission from accreting Pop III black holes can produce the EDGES like signal by increasing background radiation temperature [322].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In other scenarios, the EDGES anomaly can be explained by axion-photon conversion in the presence of intergalactic magnetic fields [323] or by radiative decays of standard model neutrino induced by magnetic fields [324].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Radio excess can also be explained by the cusp region of superconducting cosmic strings [325].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [326], authors 69 PMFs & Excess Radio Background Chapter 4 21 cm Line Astronomy and Constraining New Physics consider radiative decays of relic neutrino and show that it can potentially explain the ARCADE 2 excess together with the EDGES signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Depending on the origin, the excess fraction of radio radiation can have a different value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We discuss the constraints on excess radiation later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Considering the above possibilities of having early excess radiation, we believe that it is important to analyze constraints on the primordial magnetic field in the presence of such radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 Phenomenological model for excess radiation As discussed in subsection (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1), the possibility of an excess radio radiation back- ground over the CMBR can not be denied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For the excess radio background, we consider the phenomenological model following the Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [310–314].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, Authors consider a uniform redshift-independent synchrotron-like radiation, motivated by the ARCADE 2 and LWA 1 observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' This model can explain the EDGES anomaly in addition to enhancement of cosmic down power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Accord- ingly, from equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2) and following the Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [310–314], TR = T0 (1 + z) � 1 + Ar � νobs 78 MHz �β � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3) where, T0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='725 K is the present day CMB temperature and β = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6 is the spectral index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, Ar is the amplitude defined relative to the CMB at reference frequency of 78 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For the 21 cm signal νobs is 1420.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4/(1 + z) MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Authors of the Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [310], put a limit on the excess radiation background to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9 < Ar < 418 at reference frequency of 78 MHz by considering the effect of an uniform radiation excess on the 21 cm signal from the cosmic dawn, dark ages and reionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Authors consider a synchrotron-like spectrum with spectral index −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The case with Ar ∼ 418 corresponds to the LWA 1 limit on Ar at the reference frequency of 78 MHz [30, 310].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The stringent constraint on excess radiation comes from the Low-Frequency Array (LOFAR) to Ar < 182 (95 percent CL) and Ar < 259 (99 percent CL) for a spectral index of −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6 [313].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Chapter 4 PMFs & Excess Radio Background 70 21 cm Line Astronomy and Constraining New Physics 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5 Impact on the thermal and ionization history due to primordial magnetic fields In the presence of decaying magnetic fields, the gas temperature can increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Tgas can even increase above the background radiation and can erase the 21 cm absorption signal reported by EDGES [50, 254, 259, 303].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, present-day PMFs strength can be constrained by the EDGES observation in the presence of excess radiation reported by ARCADE 2 and LWA 1 [5, 27, 30, 304, 310, 327].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the presence of turbulent decay and ambipolar diffusion, the thermal evolution of the gas with the redshift can be written as [254, 259, 302, 303, 328], dTgas dz = 2 Tgas 1 + z + Γc (1 + z) H (Tgas − TCMB) − 2 3 ntot(1 + z) H (Γturb + Γambi) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4) Here, fHe = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='079 and TCMB = T0 (1 + z) is the cosmic microwave background (CMB) temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' At early times, Tgas remains in equilibrium with CMB tem- perature due to Compton scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' However, the gas temperature will not be strongly affected by the comparatively small amount of energy in the non-thermal radio radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, Tgas and Tα can be assumed independent of the excess radiation [304].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The change in the free electron fraction (xe) with redshift is given by equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9) with E = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Heating rate per unit volume due to the ambipolar diffusion (Γambi) and turbulence decay (Γturb) is given by [254, 259], Γambi = (1 − xe) γ xe (MH Nb)2 |(∇ × B) × B|2 16 π2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5) Γturb = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5 m [ln(1 + ti/td)]m [ln(1 + ti/td) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5 ln{(1 + zi)/(1 + z)}]m+1H EB , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6) here, m = 2(nB + 3)/(nB + 5), zi = 1088 is the redshift when heating starts due the magnetic fields (recombination epoch), γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9 × 1014 (Tgas/K)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='375cm3/g/s is 71 PMFs & Excess Radio Background Chapter 4 21 cm Line Astronomy and Constraining New Physics the coupling coefficient, MH is the mass of hydrogen atom and Nb is the number density of baryons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' td = 1/ � kd VA(kd, z) � is the decay time for the turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For matter dominated era, ti = 2/ � 3 H(zi) � and VA(kd, z) = B(kd, z)/ � 4 π ρb(z) �1/2 is the Alfv´en wave velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' B(kd, z) is the magnetic field strength smoothed over the scale of kd at redshift z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' kd is constrained by the damping wavenumber of Alfv´en wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' PMFs with wavenumber (k) larger than kd, are strongly damped by the radiative-viscosity [259, 303, 329–332].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Moreover, EB = B2/(8π) is the magnetic field energy density, dEB dz = 4 EB 1 + z + 1 H (1 + z) ( Γturb + Γambi ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7) Here, we assume that PMFs are isotropic and homogeneous Gaussian random magnetic field, whose power spectrum is given by the following equation [50, 259, 262, 333] ⟨ ˜Bi(k) ˜B∗ j(q)⟩ = (2π)3 2 δ3 D(k − q) � δij − kikj k2 � PB(k) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='8) here, PB(k) is the magnetic power spectrum, k = |k| is the comoving wave number and δD is the Dirac delta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, we consider a power-law spectrum of the magnetic fields in the Fourier space for k < kd [50], PB(k) = (2π)2 Γ � (nB + 3)/2 � B2 0 � k Mpc−1 �nB Mpc3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9) Here, nB is the spectral index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In particular, nB = 2 for white noise [265], nB = 4 for the Batchelor spectrum [334] and nB = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9 for nearly scale invariant spec- trum [259].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As discussed above, magnetic fields are strongly damped by the large radiative-viscosity for wavenumber larger than kd before recombination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' There- fore, we consider a sharp cut-off for power spectrum of PMFs: PB(k) = 0 for k ≥ kd [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Following the Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [50], we take the time evolution of the Alfv´en wave damping scale: kd(z) = kd,i f(z) and f(zi) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, kd,i is the damping Chapter 4 PMFs & Excess Radio Background 72 21 cm Line Astronomy and Constraining New Physics wavenumber at recombination epoch, kd,i = 2π Mpc−1 � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='32 × 10−3 � B0 nG �2 � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='02 Ωbh2 � �Ωmh2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='15 �1/2 �− 1 nB+5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='10) Here, to smooth the magnetic field amplitude over the inverse length scale of kd,i , we choose the Gaussian window function in Fourier space (k) as [49, 50, 335], B2 kd,i = � ∞ 0 d3k (2π)3 e −k2� 2π kd,i �2 PB(k) = B2 0 � kd,i 2π Mpc−1 �nB+3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='11) The magnetic field strength smoothed over the scale of 1 Mpc, B2 1 Mpc = � (dk/2π)3 exp[−(k/Mpc−1)2] PB(k) = B2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Lorentz force and the magnetic energy density can be calculated as [50], |(∇ × B) × B|2 = � k1,k2 k2 1 PB(k1) PB(k2) f 2nB+8(z) (1 + z)10 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='12) here � k1,k2[· · · ] = � � d3k1/(2π)3 × d3k2/(2π)3 [· · · ], and EB = 1 8π � d3k (2π)3 PB(k) f nB+3(z) (1 + z)4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='13) We can get the redshift evolution of the function f(z), by substituting equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='13) in equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6 Impact on the thermal and ionization history due to background radiation Heating of IGM due to background radio radiation during cosmic dawn era has been discussed in chapter (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' After inclusion of heating due to excess radio radi- 73 PMFs & Excess Radio Background Chapter 4 21 cm Line Astronomy and Constraining New Physics ation and x-ray, the equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4) will modify, dTgas dz = dTgas dz ����� [eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4)] + dTgas dz ����� x−ray − ΓR (1 + z) (1 + fHe + xe) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='14) where, dTgas/dz �� [eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4)] stands for the gas temperature evolution represented in equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' To include the x-ray heating of the gas, we consider the tanh parameterization [51, 74, 75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the presence of x-ray radiation, the ionization fraction evolution will also change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For the present case, we consider the fiducial model, for x-ray heating and ionization fraction evolution, motivated by Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The heating effects of both the VDKZ18 (the last term in equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='14— discussed in chapter 2) and x-ray are shown in plots (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7 & 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7 Result and discussion We consider the following values for the cosmological parameters: Ωm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='31, Ωb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='048, h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='68, σ8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='82 and ns = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='97 [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' To study the gas temperature evolution with redshift in the presence of primordial magnetic field dissipation, we solve the coupled equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9 with E = 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' To get the Lorentz force term in equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5), we solve the equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Similarly, to get the magnetic field energy density in equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6), we solve the equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' To get the evolution of the f(z) with redshift, df(z)/dz, we substitute equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='13) in equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7) with initial condition f(zi) = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' To obtain upper constraint on PMFs strength, we solve the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='19) with equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9 with E = 0) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7) for T21 ≃ −300 mK or −500 mK by varying B0, nB and Ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For infinite Lyα coupling TS ≃ Tgas, therefore, TS solely depends on the gas temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' While, for finite Lyα coupling, TS depends on both the gas and background radiation temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In figures (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2) & (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3), we plot the gas temperature vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' redshift for different values of present-day strengths of PMFs (B0) and excess radio background Chapter 4 PMFs & Excess Radio Background 74 21 cm Line Astronomy and Constraining New Physics 100 101 102 10 20 30 40 Tgas (K) z No heating VDKZ18, Ar = 0 VDKZ18, Ar = 100 VDKZ18, Ar = 418 x-ray VDKZ18 + x-ray, Ar = 0 VDKZ18 + x-ray, Ar = 100 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1: The gas temperature evolution with redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The solid blue lines represent the case when there is no x-ray, VDKZ18 or magnetic heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' VDKZ18 corresponds to the heat transfer from the background radiation to gas mediated by Lyα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The shaded region represents the EDGES observation redshift range, 15 ≤ z ≤ 20 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In this figure, we consider only VDKZ18 and x-ray heating with excess radiation (Ar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 75 PMFs & Excess Radio Background Chapter 4 21 cm Line Astronomy and Constraining New Physics 100 101 102 10 20 30 40 50 Tgas (K) z No heating VDKZ18, Ar = 0, B0 = 0 nG B0=1×10-1 nG VDKZ18, Ar = 0, B0=1×10-1 nG VDKZ18, Ar=100, B0=1×10-1 nG VDKZ18+x-ray, Ar=100, B0=1×10-1 nG B0=3×10-1 nG VDKZ18, Ar=100, B0=3×10-1 nG x-ray, B0=3×10-1 nG VDKZ18+x-ray, Ar=100, B0=3×10-1 nG Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2: The caption is same as in figure (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1), except here, we include different combination of VDKZ18, x-ray and magnetic heating, and spectral index is fixed to −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='99 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 100 101 102 103 104 101 102 103 Tgas (K) z No heating B0 = 3×10-1 nG, nB = -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='99 B0 = 3×10-1 nG, nB = -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='50 B0 = 3×10-1 nG, nB = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='99 B0 = 1×10-1 nG, nB = -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='50 B0 = 1×10-1 nG, nB = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='99 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3: The caption is same as in figure (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1), except here, we vary the spectral index and plot magnetic heating of the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Chapter 4 PMFs & Excess Radio Background 76 21 cm Line Astronomy and Constraining New Physics fraction (Ar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The solid blue lines represent the case when there is no heating of the IGM gas, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' no x-ray, VDKZ18 or magnetic heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The pink shaded band in the figure shows the EDGES redshift range, 15 ≤ z ≤ 20, for the 21 cm absorption signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In plot (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1), we consider only VDKZ18 and x-ray heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The orange dashed line describes the heating due to VDKZ18 only while keeping Ar = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Next, we increase the value of Ar from 0 to 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' This case is described by the dashed-green line in plot (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1), which shows a significant rise in the gas temperature due to the excess radiation fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Further, if one increases the Ar to its LWA 1 limit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Ar = 418, the gas temperature does not change significantly from Ar = 100 case, as shown by the solid magenta curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' It happens because ΓR ∝ (TR/TS − 1) ∼ TR/TS, equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As we increase Ar, TR/TS increases slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For example, at z = 17, TR/TS is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5 for Ar = 0, 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4 for Ar = 100 and 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9 for Ar = 418.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, we can see that, even increasing Ar to ∼ 4 times (100 to 418), TR/TS increases by only 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='8 percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, increasing further Ar will not affect gas temperature significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' To analyse the role of x-ray heating, we have first considered the heating due to x-ray only, depicted by the red dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The inclusion of VDKZ18 for Ar = 0 further increases the gas temperature slightly, as shown by the black dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In this case of inclusion of x-ray heating, if we increase the value of Ar to 100, there is a significant increase in the gas temperature as shown by the solid green line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We find the contribution due to x-ray heating dominates for redshift values z ≲ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In plot (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1), we compare the contribution of VDKZ18 and x-ray heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In plot (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2), we compare the contributions of VDKZ18, x-ray and magnetic heating while keeping the spectral-index, nB = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='99 for a nearly scale-invariant magnetic field spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' While in figure (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3), we vary the magnetic spectral index (nB) and plot the magnetic heating of the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In plot (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2), we have included the effect of primordial magnetic fields on the IGM gas evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The solid blue line represents the case when there is no heating, and the dashed-black curve shows the case of VDKZ18 with no magnetic fields 77 PMFs & Excess Radio Background Chapter 4 21 cm Line Astronomy and Constraining New Physics and x-ray for Ar = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The double dot-dashed green curve represents the case when there is only the magnetic heating with a magnetic field strength of B0 = 1×10−1nG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Next, we include the case of VDKZ18 for Ar = 0 in the pure magnetic heating scenario, as shown by the red dashed curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Now, if we increase Ar from 0 to 100, the gas temperature rises significantly in the shaded region as shown by the dash-dotted red curve in figure (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Now the further addition of x-ray heating is shown by the cyan plot, which shows significant heating in the shaded region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Next, for more analysis, we increase the magnetic field strength from B0 = 1×10−1 nG to B0 = 3×10−1 nG and study cases with VDKZ18 and x-ray as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The magenta dashed line depicts the case with only magnetic heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The green dashed line shows the case of VDKZ18 with Ar = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The orange curve shows the case with magnetic and x-ray heating only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, as expected, the gas temperature decreases after the inclusion of the x-ray effect with the magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' It happens because the ionization fraction increases by x-ray radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Ambipolar diffusion evolves as Γambi ∝ (1−xe)/xe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' therefore, as ionization fraction increases, ambipolar diffusion of the magnetic field decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Thus, the heating due to magnetic fields also decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, including the x-ray contribution with the magnetic field decreases the magnetic field diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Hence, the gas temperature decreases (this effect also occurs for B0 = 1 × 10−1 nG, but it is not visible in the plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The black dot-dashed line includes all the three effects: magnetic and x-ray heating together with VDKZ18 for Ar = 100 and B0 = 3 × 10−1 nG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, the addition of the VDKZ18 heating for Ar = 100 increases the gas temperature above the solid orange line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' It is also lower than the magenta dashed line because of the inclusion of the x-ray contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' At the smaller redshift, x-ray heating dominates over all other heating mechanisms, and all lines merge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In figure (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3), we plot the magnetic heating of the gas for the different spectral index (nB) and B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The solid lines, except the blue one, represent the magnetic heating for B0 = 3 × 10−1 nG, while double dot-dashed lines are for B0 = 1 × 10−1 nG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Increasing the spectral index, the magnetic heating due to ambipolar Chapter 4 PMFs & Excess Radio Background 78 21 cm Line Astronomy and Constraining New Physics diffusion and turbulent decay increases as Γambi ∝ � 1/Γ[(nB + 3)/2] �2 and Γturb ∝ 1/Γ[(nB + 3)/2] (by ignoring the logarithmic and integral dependencies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For example, if one changes nB from its value −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='99 to −1 then 1/Γ[(nB+3)/2] changes from 5 × 10−3 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, by increasing nB from −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='99 to −1, magnetic heating enhances significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' To get T21 (equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='19) around −500 mK or −300 mK at z = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2, one needs to ensure that even by increasing nB, that the factor xHI (1 − TR/TS) remains same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Thus from equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9) when we increase nB, we have to decrease B0 so that the magnetic heating contribution to the gas remains the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, by increasing nB, the upper bound on B0 will become more stringent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, we also include the collisional ionization of the gas in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9), as this term is important only when gas temperature is ≳ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='58 × 105 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Otherwise this term is exponentially suppressed as ∝ exp[−(13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6 eV)/Tgas] [259, 336, 337].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In plot (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3), the gas temperature rises by increasing B0, as more magnetic energy is getting injected into thermal energy of the gas via Γambi ∝ E2 B and Γturb ∝ EB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' However, for redshift z ≲ 100, the gas temperature starts decreasing as the cooling effect due to expansion of the Universe become dominant, as can be seen in equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4) & (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7) (it also depends on the strength and spectral index of the magnetic field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Since, with the expansion of the Universe, magnetic energy density (EB) also dilutes, the contributions from Γambi and Γturb decreases as can be seen from equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In figure (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4), we plot the spin (dashed lines) and gas (solid lines) tempera- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For Ar = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' TR = TCMB, we get Tgas ≃ TS as seen by the overlapping dashed and solid blue lines in the shaded region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' xα and xc are ∝ 1/TR as can be seen from equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='8) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, the coupling between the gas and spin temperature decreases by increasing Ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As discussed before, increasing the value of Ar above ∼ 100, the spin temperature increases, but the increment in gas temperature becomes insignificant, and the TR/TS ratio increases slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' There- fore, as xα and xc decreases, the difference between the gas and spin temperature increases, as shown in the plot (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Increasing the values of Ar from 100 (green 79 PMFs & Excess Radio Background Chapter 4 21 cm Line Astronomy and Constraining New Physics 100 101 102 10 20 30 40 B0 = 3×10-1 nG, nB = -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='99 T (K) z VDKZ18, Ar = 0 VDKZ18, Ar = 100 VDKZ18, Ar = 418 VDKZ18+x-ray, Ar = 100 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4: This figure shows the gas (solid lines) and spin (dashed lines) temper- ature evolution, The shaded region corresponds to the redshift 15 ≤ z ≤ 20 — the redshift range for EDGES reported signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 1000 800 600 400 200 0 10 15 20 25 30 B0 = 3×10-1 nG, nB = -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='99 T21 (mK) z VDKZ18, Ar = 0 VDKZ18, Ar = 100 VDKZ18, Ar = 418 VDKZ18+x-ray, Ar = 100 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5: This figure shows the 21 cm differential brightness temperature with redshift for same cases in plot (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Chapter 4 PMFs & Excess Radio Background 80 21 cm Line Astronomy and Constraining New Physics 101 102 Ar 10 15 10 14 10 13 10 12 10 11 10 10 10 9 10 8 B0 (Gauss) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='99 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='99 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='99 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='00 nB 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='99 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='00 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6: In this figure, we study upper bounds on present-day magnetic field strength (B0) with excess radiation fraction (Ar) for different values of the spectral index, nB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The green-yellow and red-grey colour schemes represent the cases when T21|z=17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 ≃ −500 mK and −300 mK, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For T21|z=17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 ≃ −300 mK case the value of nB written with blue coloured text , while for −500 mK case it is written with black coloured text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, we consider TS ≃ Tgas and do not take into account the x-ray and VDKZ18 effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' lines) to 418 (black lines), the difference between gas and spin temperatures in- creases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Figure (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5), shows the plots for 21 cm differential brightness temperature vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' redshift, for all the cases discussed in plot (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As we increase the Ar from 0 to 100 the |T21| increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' By increasing Ar from 100 to 418, values of T21 does not change significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Further, including x-ray heating and magnetic heating (for B0 = 3 × 10−1 nG and nB = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='99) the gas temperature rises and |T21| decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In figures (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7), we plot the maximally allowed values of B0 versus radiation excess (Ar) for different spectral indexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The colour-bars represent the 81 PMFs & Excess Radio Background Chapter 4 21 cm Line Astronomy and Constraining New Physics 101 102 Ar 10 15 10 14 10 13 10 12 10 11 10 10 10 9 10 8 B0 (Gauss) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='99 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='99 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='99 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='00 nB 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='99 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='00 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7: The caption is same as in figure (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6), except here, we consider the effects of VDKZ18 and x-ray heating on the gas due to first stars after z ≲ 35 and consider finite Lyα coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Chapter 4 PMFs & Excess Radio Background 82 21 cm Line Astronomy and Constraining New Physics variation of the magnetic field spectral index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the plots, the spectral index varies from its nearly scale-invariant value (-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='99) to -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, we consider both the EDGES best fit and upper constraint on the 21 cm absorption signal for constraining B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The green-yellow colour scheme represents the case with T21|z=17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 ≃ −500 mK, while the red-grey colour scheme represents the case with T21|z=17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 ≃ −300 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Numerical values of nB for the different colour bands are written with different colour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For T21|z=17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 ≃ −300 mK case the value of nB written with blue coloured text , while for T21|z=17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 ≃ −500 mK case it is written with black coloured text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The colour-bars are common for both the plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In figure (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6), we consider infinite Lyα coupling (xα ≫ xc, 1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' TS ≃ Tgas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, we do not consider the x-ray and VDKZ18 effects on the gas and thus the 21 cm signal T21 ∝ (1 − TR/Tgas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As we increase Ar, the amplitude of |T21| increases, and we get more window to increase the gas temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In this plot, we consider heating only due to the decaying magnetohydrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, we can increase B0 as we increase Ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As discussed earlier, by decreasing nB, the amplitude of the magnetic field power spectrum also decreases, resulting in less magnetic energy dissipation into the gas kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Thus by reducing values of nB from -1 to -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='99, we get more window to increase B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Next, when one increases T21 from -500 mK to -300 mK, the allowed value of B0 also increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' This is shown by the red-grey colour scheme in figures (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In figure (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7), we consider the effects of VDKZ18 and x-ray on IGM gas evolution due to first stars after z ≲ 35 and consider finite Lyα coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As discussed earlier, Tgas ̸= TS for Ar > 0 and the difference between gas and spin temperature increases as Ar increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Thus, in the presence of first star’s effects, the upper bound on the present-day strength of PMFs modifies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Following the Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [51, 74, 75], we consider WF coupling coefficient, xα = 2Aα(z) × (T0/TR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, Aα(z) = Aα(1 + tanh[(zα0 − z)/∆zα]), the step height Aα = 100, pivot redshift zα0 = 17 and duration ∆zα = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The collisional coupling coefficient, xc = T10/TR×(NH kHH 10 )/A10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' After the inclusion of x-ray and VDKZ18 heating effects, the gas temperature remains > 10 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, 83 PMFs & Excess Radio Background Chapter 4 21 cm Line Astronomy and Constraining New Physics we can take kHH 10 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 ×10−11 (Tgas/K)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='357 exp(−32 K/Tgas) cm3/sec for 10 K < Tgas < 103 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As illustrated in plot (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4) & (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5), increasing excess radiation fraction Ar above ∼ 100, the TR/TS remains nearly constant and this also mean that T21 remain unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Consequently one can not increase the value of B0 and one gets nearly flat profile for B0 for Ar ≳ 100 in figure (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In figures (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='8) & (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9), we plot the maximally allowed values of B0 vs nB for various values of Ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The colour-bars represent the variation in Ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the plots, Ar varies from 5 to LWA 1 limit ∼ 418.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We consider both the EDGES best fit and upper constraint on 21 cm absorption signal for constraining B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The green-yellow scheme represent the case with T21|z=17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 ≃ −500 mK, while the red-grey colour scheme represent the case T21|z=17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 ≃ −300 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Numerical values of Ar for the different colour bands are written in different colours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For T21|z=17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 ≃ −300 mK case the value of Ar written with blue coloured text , while for T21|z=17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 ≃ −500 mK case it is written with black coloured text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The spectral index ranges from -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='99 to -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The red dashed line represents the Planck 2015 upper constraint on the present-day magnetic field strength with spectral index in both plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' This constraint has been taken from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In plot (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='8), we consider TS ≃ Tgas and we do not take into account the x- ray and VDKZ18 effects on IGM gas evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The zoomed inset in the figure shows the contour plot when T21|z=17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 ≃ −300 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, considering T21|z=17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 ≃ −300 mK, for nB < −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='98 the Ar ≳ 200 is excluded similarly for nB < −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='96 the Ar ≳ 280 is excluded by Planck 2015 upper constraint on B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Likewise, for T21|z=17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 ≃ −500 mK, for nB < −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='97 the Ar ≳ 280 is excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For spectral index 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9 and excess radiation fraction 418, we get the upper constraint on B0 to be ∼ 1 nG and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3 nG by requiring T21|z=17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 ≃ −500 mK (EDGES best fit constraint) and −300 mK (EDGES upper constraint), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' While for nB = −1, these bound change to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 × 10−3 nG and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6 × 10−3 nG for T21|z=17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 ≃ −500 mK and −300 mK, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In plot (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9), we include both the VDKZ18 and x- ray effect and consider finite Lyα coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As discusses earlier, for Ar ≳ 100, Chapter 4 PMFs & Excess Radio Background 84 21 cm Line Astronomy and Constraining New Physics 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='0 nB 10 15 10 14 10 13 10 12 10 11 10 10 10 9 10 8 B0 (Gauss) 418 418 280 280 200 120 60 10 5 5 10 60 120 200 280 418 Ar 5 10 60 120 200 280 418 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='8: In this figure, we study upper bounds on the present-day magnetic field strength (B0) with spectral index (nB) for different values of excess radiation fraction (Ar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The green-yellow and red-grey colour schemes represent the cases when T21|z=17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 ≃ −500 mK and −300 mK, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For T21|z=17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 ≃ −300 mK case the value of nB written with blue coloured text , while for −500 mK case it is written with black coloured text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The red dashed line depicts the Planck 2015 upper constraint on the present-day magnetic field strength [49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, we consider TS ≃ Tgas and do not take into account the x-ray and VDKZ18 effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 85 PMFs & Excess Radio Background Chapter 4 21 cm Line Astronomy and Constraining New Physics 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='0 nB 10 15 10 14 10 13 10 12 10 11 10 10 10 9 10 8 B0 (Gauss) 418 60 418 60 10 5 5 10 60 120 200 280 418 Ar 5 10 60 120 200 280 418 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9: The caption is same as in figure (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='8), except here, we consider the heating effects of VDKZ18 and x-ray on IGM gas due to first stars after z ≲ 35 and consider finite Lyα coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The colour-bars are common for both plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Chapter 4 PMFs & Excess Radio Background 86 21 cm Line Astronomy and Constraining New Physics TR/TS ratio remain nearly constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Thus, in the plot (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9), we can see that for Ar ≳ 100, the upper bound on B0 is not changing significantly— the plots are merged for Ar ≳ 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' These plots have been shown by the zoomed inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The right upper zoomed inset is shown for T21 ≃ −300 mK, while left lower zoomed inset is shown for green-yellow contour plots when T21 ≃ −500 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Hence, further increasing Ar > 100 will not change significantly the upper bound on B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As illustrated in figure (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4), TS > Tgas for Ar > 0, and T21 ∝ (1−TR/TS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, to get T21 ≃ −300 mK or −500 mK, we need to lower B0 compared to previous scenario— figure (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Hence, we get the more stringent upper bound on present- day magnetic field strength in figure (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For spectral index -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9 and excess radiation fraction 418, we get the upper constraint on B0 to be ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7×10−1 nG and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 × 10−1 nG by requiring T21|z=17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 ≃ −300 mK and −500 mK, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For nB = −1, we get B0 ≲ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9×10−5 nG and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7×10−5 nG by requiring EDGES upper and best fit constraint on 21 cm differential brightness temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Decreasing the values of Ar, the upper constraint on B0 becomes more stringent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For example, when Ar = 5, we get upper bound on present day magnetic field strength to be ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4 × 10−1 nG for spectral index -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='99, and for spectral index nB = −1 we get B0 ≲ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='8 × 10−6 nG by requiring EDGES best fit constraint on T21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The upper bounds are also well below the Planck 2015 constraint [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='8 Conclusions In the present work, we study the upper constraint on the strength of the pri- mordial magnetic fields for different spectral index using the bound of EDGES observation on T21, in the presence of uniform redshift-independent synchrotron like radiation reported by ARCADE 2 and LWA 1 [27, 30, 304, 310].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We have considered excess radiation fraction up to the LWA 1 limit (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Ar ∼ 418) at the reference frequency of 78 MHz [30, 310].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' To get the upper constraint on B0, we have used both the EDGES upper and best-fit constraints on T21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We have 87 PMFs & Excess Radio Background Chapter 4 21 cm Line Astronomy and Constraining New Physics considered two scenarios: First, infinite Lyα coupling (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' xα ≫ xc, 1) without the effects of x-ray and VDKZ18 on IGM gas evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In another scenario, we consider the finite Lyα coupling with x-ray and VDKZ18 effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The following summarises our results for T21 = −500 mK: In the first scenario, for Ar = 418, we get B0 ≲ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7 nG for spectral index -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='99, while for nB = −1 we get B0 ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 × 10−3 nG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' When Ar = 5, upper constraint on present-day magnetic field strength varies from B0 ≲ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9×10−1 nG to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='8×10−5 nG by varying nB from -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='99 to -1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the second scenario, the upper bounds on B0 will modify [34, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For Ar = 418, we get the upper constraint on magnetic field to be B0(nB = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='99) ≲ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9 × 10−1 nG and B0(nB = −1) ≲ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7 × 10−5 nG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' While for Ar = 5, we get upper bound on present day magnetic field strength to be ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4 × 10−1 nG for spectral index -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='99, and for spectral index -1 we get B0 ≲ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='8 × 10−6 nG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We would like to note that these upper bounds on B0 that we have reported here are also consistent with the Planck observations [49, 338].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Chapter 4 PMFs & Excess Radio Background 88 “Who sees the future?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Let us have free scope for all directions of research” Ludwig Eduard Boltzmann, “Lectures on Gas Theory” translated by Stephen G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Brush 5 Primordial Magnetic Fields and Baryon-Dark matter Interaction In the previous chapter (4), we have analysed the upper bound on present-day strength of PMFs in the light of EDGES observation and excess radio background reported by ARCADE 2 and LWA 1 observations [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As discussed earlier in chapter (1), to explain EDGES observation one requires that either the background radio radiation should be grater than ∼104 K in the absence of any non-standard mechanism for the evolution of the gas temperature or the gas temperature should be less than 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 K for the standard evolution of CMB temperature at the centre of the “U” profile for the best fitting amplitude [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The first possibility has been investigated by authors of the Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [339–342].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the second scenario, IGM gas can be cooled by emitting the photons between the Ly-limit to Ly-γ wavelengths [343, 344].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' There are very few mechanisms to cool the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Since the dark matter 21 cm Line Astronomy and Constraining New Physics is colder than the gas, effective cooling of the gas can be obtained by elastic scattering between the dark matter and baryon particles [73, 345, 346].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' A new kind of interaction between dark matter and baryons was proposed by the authors of reference [345, 347] to explain the EDGES absorptional signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The authors consider a non-standard “Coulomb-like” interaction: σ = ˆσ v−4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' v is the relative velocity between the dark matter and baryons and ˆσ is the strength of baryon-dark matter interaction cross-section [73, 345–348].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, the interaction between dark matter and baryons does not depend on whether the baryons are free or bound within atoms [345].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The cooling of the gas, by transferring energy to the dark matter, is tightly constrained because of constraints on the dark matter mass and cross-section by cosmological and astrophysical observations [73, 345, 349, 350].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the present chapter, we reanalyse the constraints on PMFs in the presence of baryon-dark matter interaction proposed by the authors of reference [345].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the presence of baryon-dark matter interaction the bounds on magnetic field, baryon dark matter cross-section strength (ˆσ) and dark matter mass (MDM) can strongly influence each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' This requires to rework the bounds on ˆσ , MDM and B0 which can explain the observed absorption signal by EDGES collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The upper limit on the magnetic field strength can modify in presence of baryon-dark matter interaction cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the presence of a strong magnetic field, a large baryon-dark matter interaction cross-section is required to balance magnetic heating of gas to explain the EDGES signal as compared to a weak magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Subsequently, the strong magnetic-fields can even erase the 21 cm signal— this gives an upper bound on the strength of magnetic-fields, dark matter mass and baryon-dark matter cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In order to explain the EDGES absorption signal, the gas temperature needs to be cooler than the ΛCDM prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' During the Cosmic dawn era, the Universe was at its coldest phase, and the relative velocity between the dark matter and baryon was very small, O(10−6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Also, the temperature of the dark matter was colder than the baryon temperature during this period, so an interaction of the Chapter 5 PMFs & Baryon-Dark matter Interaction 90 21 cm Line Astronomy and Constraining New Physics baryon with dark matter can cool the gas temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Since the relative velocity is small, scattering cross section of the type σ = ˆσ v−4 can enhance the interaction rate and cool the gas sufficiently to explain EDGES absorption dip [345, 346, 348].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In this chapter, we consider magnetic heating of the gas and dark matter via ambipolar and turbulent decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, we take cosmological parameters Ωb, Ωm, and h as Ωb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='04859, Ωm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='315 and h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='68 [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 Baryon-dark matter interaction in presence of magnetic fields In this section, we discuss the effects of magnetic fields on the gas temperature in the presence of baryon-dark matter interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The gas temperature evolves as discussed in the chapter (4), except here, the cooling rate (dQgas/dt) will add due to the energy transfer from gas to dark matter [254, 347], dTgas dz = 2 Tgas 1 + z + ΓC (1 + z)H (Tgas − TCMB) − 2 3 ntot(1 + z) H (Γturb + Γambi) + 2 3 (1 + z) H dQgas dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1) The cooling rate (dQgas/dt) depends on the temperature difference and relative velocity between dark matter and baryons, dQgas dt = 2 Mb ρDM ˆσ e−r2/2 √ 2 π (Mb + MDM)2 u3 th � Tgas − TDM � − µ ρDM ρM v D(v) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2) here, Mb ≈ MH is the baryon mass and can be taken as mass of hydrogen atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' ρDM and ρM are the dark matter and total matter energy density, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Moreover, r = v/uth, v is the relative motion between baryons and dark matter while u2 th = Tgas/Mb + TDM/MDM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, TDM is the dark matter temperature and µ = Mb MDM/(Mb + MDM) is the reduced mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The first term in equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2), arises due to the temperature difference between dark matter and gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As 91 PMFs & Baryon-Dark matter Interaction Chapter 5 21 cm Line Astronomy and Constraining New Physics TDM < Tgas, the first term is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' It implies that the energy of gas is being transferred to dark matter with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The second term in equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2), comes due to the friction between two fluids caused by velocity difference— drag term, and it is given by µ (ρDM/ρM) v D(v), D(v) ≡ ρM ˆσ Mb + MDM 1 v2 F(r) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3) here, r = v/uth and the function F(r) is defined as, F(r) ≡ erf � r √ 2 � − � 2 π r e−r2/2 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4) here, erf() is the Gauss error function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' When the relative velocity between dark matter and baryons is zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' r = 0, one gets F(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In this case there will not be any drag heating of gas and dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As r → ∞, F(r) → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For any value of r ≥ 0, one finds that F(r) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, the last term in equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2) always remains negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' It implies that the energy of gas always increases due to the drag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2), one can check that the heating gets maximize due to drag as MDM → Mb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The dark matter temperature evolution can be written as, dTDM dz = 2 TDM (1 + z) + 2 3 (1 + z) H dQDM dt , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5) here, first term represents the cooling of the dark matter due to expansion of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Heat transfer rate for dark matter (dQDM/dt) can be obtained by interchanging b ↔ DM and Tgas ↔ TDM in equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As drag term (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3) remains symmetric under the transformation b ↔ DM, it heats the dark matter also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We can also check that total energy density of the system is conserved [347], NDM dQDM dt + Nb dQgas dt − ρDM ρb ρM v D(v) = 0 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6) here, NDM and Nb are number density of dark matter and baryons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As the relative motion between dark matter and baryons is damped due to friction between both Chapter 5 PMFs & Baryon-Dark matter Interaction 92 21 cm Line Astronomy and Constraining New Physics fluids and expansion of the Universe, one can write the evolution of relative motion as, dv dz = v 1 + z + D(v) (1 + z) H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7) Temperature evolutions of the gas and dark matter require free electron fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' It is given by equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9) with E = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As it has been confirmed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [50], that cooling due to effects like Lyα emission, Bremsstrahlung and recombination does not have that much effects on the dynamics of the gas and dark matter, therefore, we have not considered these effects in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 Results and Discussion Solving coupled equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9 with E = 0 , 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7) with initial conditions Tgas(1010) ≃ TCMB(1010), TDM(1010) ∼ 0 K, xe(1010) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='057 and B(z) = B0 (1+z)2|z=1010 is the initial magnetic field strength, we get the tempera- ture evolution of the dark matter and gas for different dark matter masses, strength of baryon-dark matter interaction cross-sections and magnetic field’s strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Figures (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3) show the evolution of the gas and dark matter tem- perature with redshift (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The solid blue line in all these figures correspond to gas temperature when both the magnetic field and baryon-dark matter interaction are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In this case, gas temperature falls as Tgas ∝ (1 + z)2 after z ∼ 200 and reaches 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='8 K at z = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In figure (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1), temperature evolution of the gas and dark matter is given for different strength of PMFs at constant ˆσ = 10−41 cm2 and MDM = 10−1 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For both the cases B0 = 10−5 G and 10−6 G, gas temperature falls down due to Hubble expansion and baryon-dark matter interaction till z ∼ 30 and ∼ 20, respectively, then temperature rises due to magnetic heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We note that, TDM also increases due to the energy transfer from gas to dark matter depending on ˆσ and MDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 93 PMFs & Baryon-Dark matter Interaction Chapter 5 21 cm Line Astronomy and Constraining New Physics 10 100 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 1 10 100 1000 Redshift ( z ) Temperature ( in Kelvin ) σ\uf111=10-41 cm2, md =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 GeV T d , B 0 = 10 - 6 G T gas , B 0 = 10 - 6 G T d , B 0 = 10 - 5 G T gas , B 0 = 10 - 5 G T gas , σ\uf111 = 0 , B 0 = 0 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1: This figure shows the temperature evolutions of baryon and dark matter in the presence of PMFs and baryon-dark matter interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Blue line corresponds to temperature evolution of gas in the absence of both magnetic heat- ing and baryon-dark matter interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The red (green) solid lines represents the variation of the gas temperature and the dotted red (green) line shows the vari- ation of the dark matter temperature in presence of PMFs and the baryon-dark matter interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In this plot we vary the strength of PMFs, and keep ˆσ & dark matter mass constant to 10−41 cm2 & 10−1 GeV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In all figures, notation for the mass of dark matter is written with md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' While in the text, it is written as MDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Chapter 5 PMFs & Baryon-Dark matter Interaction 94 21 cm Line Astronomy and Constraining New Physics 10 100 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 1 10 100 1000 Redshift ( z ) Temperature ( in Kelvin ) B0=10-6 G, md =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 GeV T d , σ\uf111 = 10 - 42 cm 2 T gas , σ\uf111 = 10 - 42 cm 2 T d , σ\uf111 = 10 - 41 cm 2 T gas , σ\uf111 = 10 - 41 cm 2 T gas , σ\uf111 = 0 , B 0 = 0 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2: The caption is same as in figure (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1), except here, we only vary the strength of baryon-dark matter cross-section, and keep B0 & dark matter mass constant to 10−6 G & 10−1 GeV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 10 100 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 1 10 100 1000 Redshift ( z ) Temperature ( in Kelvin ) B0=10-6 G, σ\uf111=10-41 cm2 T d , m d = 1 GeV T gas , m d = 1 GeV T d , m d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 GeV T gas , m d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 GeV T gas , σ\uf111 = 0 , B 0 = 0 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3: The caption is same as in figure (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1), except here, we only vary the dark matter mass, and keep B0 & ˆσ constant to 10−6 G & 10−41 cm2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 95 PMFs & Baryon-Dark matter Interaction Chapter 5 21 cm Line Astronomy and Constraining New Physics Larger the strength of magnetic fields, earlier the heating begins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For example, heating for the case with B0 = 10−5 G starts earlier compared to the case with B0 = 10−6 G in figure (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Although TDM at z ∼ 1010 is taken to be zero, it increases due to the energy transfer from baryons to dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' By increasing B0, magnetic-heating of the gas rises, subsequently, the value of TDM also rises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' It can be seen in figure (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1), temperature of dark matter for B0 = 10−5 G is larger compared to B0 = 10−6 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Figure (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2) shows the temperature evolution of gas and dark matter for dif- ferent strength of baryon-dark matter interaction cross-section when B0 = 10−6 G and MDM = 10−1 GeV are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Larger the ˆσ, more heat transfers from gas to dark matter and cools the gas efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For the green lines ˆσ = 10−42 cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As we increase ˆσ to 10−41 cm2, the gas temperature decreases— shown by red solid line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' It decreases because the energy transfer from gas to dark matter becomes more efficient by increasing interaction between dark matter and baryons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' It results in more heating of dark matter— shown by red dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For B0 = 10−6 G and ˆσ = 10−41 cm2, temperature evolution for different dark matter mass is shown in Figure (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As we increase the dark matter mass from 10−1 GeV to 1 GeV, temperature of both the dark matter and gas increases, and it becomes more efficient for large dark matter mass [347].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' This drag heating is important when mass of dark matter is around ∼ 1 GeV [347].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' When MDM approaches to 1 GeV, in addition to magnetic heating of the gas, the heating due to drag term also becomes effective .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, the gas temperature for MDM = 1 GeV is higher than MDM = 10−1 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 Correlation between dark matter mass and baryon- dark matter cross section In this subsection, we analyse the effect of B0, MDM and ˆσ on gas and dark matter temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4), we study constraints on MDM and ˆσ for T21 ≃ −500 mK Chapter 5 PMFs & Baryon-Dark matter Interaction 96 21 cm Line Astronomy and Constraining New Physics 10-15 10-14 10-13 10-6 10-5 10-4 10-3 10-2 10-1 100 Planck 2015 excluded region CMB-S4 excluded region σ̂ (GeV -2) md (GeV) B0 =3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='48×10-6 G B0 =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='67×10-6 G B0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='0 ×10-6 G B0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='0 ×10-9 G Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4: The figure shows the minimal cross-section required to get T21 ≃ −500 mK (solid lines) and T21 ≃ −300 mK (dashed line) at z = 17 as a function of mass for different strengths of PMFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, we assume no x-ray heating of gas, and spin temperature is completely coupled to gas temperature, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Tgas ≃ TS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The solid (dashed) magenta, black, blue and red line correspond to B0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='48×10−6 G, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='67×10−6 G, 10−6 G and 10−9 G respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The CMB-S4 (forecast) and Planck 2015 constraints on ˆσ and MDM with 95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' have been taken from the Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [51, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The green and gold regions are excluded by Planck 2015 and CMB-S4 forecast respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (1 GeV−2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='89 × 10−28 cm2) 97 PMFs & Baryon-Dark matter Interaction Chapter 5 21 cm Line Astronomy and Constraining New Physics (Tgas ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='26 K) and −300 mK (Tgas ≃ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, we have taken xα ≫ 1 to plot T21 profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Thus, from equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7) one can get TS ≈ Tgas as xc is already ≪ 1 at required redshift due to the small number density of hydrogen, free electrons and protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Subsequently, one can calculate T21 from equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4), we consider cases B0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='48 × 10−6 G, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='67 × 10−6 G, 10−6 G and 10−9 G and solve equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9 with E = 0) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5 & 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7) for Tgas ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='26 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 K at z = 17 to get ˆσ vs MDM plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The solid and dashed lines represent the cases when T21 ≃ −500 mK and −300 mK, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The gold and green regions respectively show the CMB-S4 (forecast) and Planck 2015 upper constraint on ˆσ − MDM with 95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [51, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The magenta, black, blue and red lines corresponds to B0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='48 × 10−6 G, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='67 × 10−6 G, 10−6 G and 10−9 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As we increase the magnetic field strength from 10−9 G to ∼ 10−6 G, larger value of ˆσ is required for MDM ∈ {10−6, 1} GeV to maintain T21 ≃ −500 or −300 mK at z=17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' To get EDGES upper limit on T21 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' −300 mK), required ˆσ is smaller compared to the case when T21 = −500 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' This is because we need to transfer less energy from gas to the dark matter to obtain EDGES upper limit on T21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We get the upper limit on PMFs strength to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='67 × 10−6 G by CMB- S4 (forecast) constraint on ˆσ − MDM and maintaining T21 ≃ −300 mK at z=17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For B0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='67 × 10−6 G, MDM ≳ 10−2 GeV is excluded by CMB-S4 forecast for T21 ≃ −300 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' By Planck 2015 constraint on ˆσ − MDM, the allowed maximum strength of PMFs is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='48×10−6 G by requiring EDGES upper constraint on T21 at z=17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For B0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='48 × 10−6 G, mass of dark matter ≳ 1 × 10−2 GeV is excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Similarly, for the B0 = 10−6 G, MDM ≳ 8 × 10−1 GeV is excluded by CMB-S4 forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' When the dark matter mass approaches mass of hydrogen, the drag term in equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3) also starts to contribute significantly in heating of the gas in addition to the magnetic heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, higher mass of dark matter is excluded for higher magnetic field as shown by figure (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As discussed in [347], when MDM ∼ 1 GeV, the drag term heat up both the gas and dark matter in such a way that we can not obtain Tgas = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='26 K at z = 17 as required for the EDGES Chapter 5 PMFs & Baryon-Dark matter Interaction 98 21 cm Line Astronomy and Constraining New Physics 15 16 17 18 19 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 Redshift ( z ) T 21 in Kelvin md =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 GeV, σ\uf111=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='22×10-15 GeV-2 Standard EDGES B 0 =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='65 ×10 - 6 G B 0 =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='35 ×10 - 6 G B 0 =10 - 6 G Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5: 21 cm differential brightness temperature (assuming infinite Lyα cou- pling) vs redshift when their is no x-ray heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The dotted black (orange) colour represents standard ΛCDM (EDGES) predictions for the global T21 signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Green, red and blue solid curves correspond to B0 = 1×10−6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='35×10−6 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='65×10−6 G respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, MDM = 10−1 GeV and ˆσ = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='22 × 10−15 GeV−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' There is a independent bound on the primordial magnetic field from CMB of the order of ≲ nG [287, 288].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' This constraint, in our analysis, restricts value of ˆσ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, we note that in our analysis further decreasing value of B0 below 10−9 G, does not change our result in significant way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 Effect of primordial magnetic fields on the global 21 cm signal We have discussed above that, with increase in the strength of the magnetic field the temperature of the gas increases for a fix MDM and ˆσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In figures (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5) & (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6), we plot 21 cm differential brightness temperature with redshift for different magnetic field strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' These figures are obtained by keeping MDM = 10−1 GeV 99 PMFs & Baryon-Dark matter Interaction Chapter 5 21 cm Line Astronomy and Constraining New Physics 500 400 300 200 100 0 10 15 20 25 30 T21 (mK) z B0= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='0 ×10-6 G B0= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='0 ×10-6 G B0= 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='0 ×10-7 G B0= 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='0 ×10-7 G Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6: T21 plot with redshift when x-ray heating and finite Lyα coupling are considered [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Black, blue, green and red solid curves correspond to B0 = 2 × 10−6, 1 × 10−6, 8 × 10−7 and 6 × 10−7 G respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The magenta dashed line is corresponds to the EDGES upper bound on T21 : −300 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The values of MDM and ˆσ are same as considered in figure (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Chapter 5 PMFs & Baryon-Dark matter Interaction 100 21 cm Line Astronomy and Constraining New Physics and ˆσ = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='22×10−15 GeV−2 constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In figure (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5), to plot T21 we assume infinite Lyα coupling (xα → ∞ ⇒ TS ≃ Tgas) and do not include the x-ray heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For B0 = 1 × 10−6 G, the 21 cm line absorption signal reported by EDGES (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' −500 mK) can be explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In figure (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6), we include the x-ray heating and consider finite Lyα coupling (xα) [51, 74–76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As we decrease B0 from 2 × 10−6 G, the minimum value of T21 profile decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For the case when B0 = 1 × 10−6 G (blue solid line), minimum of T21 profile is well below the EDGES upper limit on T21 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' −300 mK— magenta dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In figure (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5), when there is infinite Lyα coupling, T21 = −300 mK corresponds to B0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='35 × 10−6 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Thus, we need to lower B0 values when the finite Lyα coupling is considered to get desired value of T21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As shown in figures (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6), brightness temperature is suppressed by the increase of the strength of the magnetic field and it can even erase the standard 21 cm signal when the magnetic field strength increases above ∼ 1 × 10−6 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' This sets the upper limit on the strength of the magnetic field for MDM = 10−1 GeV and ˆσ = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='22 × 10−15 GeV−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3 Conclusions Magnetic fields in [50, 259] have shown to heat the gas during the cosmic dawn era by the ambipolar diffusion and the turbulence decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Since, it could erase the observed 21 cm absorption signal, one can calculate the upper bound on the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' One of the promising mechanisms to explain the absorption signal of the 21 cm line is to have interaction between the dark matter and baryons [73, 345].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In this work, we have shown that in the presence of such an interaction the upper bound on the strength of magnetic fields can significantly be altered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The magnetic-energy converted to the thermal energy and it heats both the gas and dark matter when ˆσ is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' This is an extra heating effect of dark matter in addition to the drag heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The drag term heats the dark matter and baryons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' but in the lower range of dark matter mass (≪ 1 GeV) it becomes small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' To 101 PMFs & Baryon-Dark matter Interaction Chapter 5 21 cm Line Astronomy and Constraining New Physics explain the observed anomaly in the 21 cm signal by the EDGES, a large baryon- dark matter scattering cross-section is required to balance the magnetic heating effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' An earlier saturation occurs in baryon-dark matter cross-section in the presence of the strong magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We have also explored the millicharged dark matter scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In this case, we are not able to reproduce the EDGES signal by considering the upper bound on ˆσ − MDM by Planck 2015 and CMB-S4 (forecast) [51, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Recently, the similar results about “millicharged” and “Coulomb-like” dark matters also have been obtained in reference [351].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' They find that 100% millicharged dark matter scenario can not reproduce the EDGES result for any parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The inclusion of PMFs will further increase the gas temperature and reduce the amplitude of 21-cm absorptional signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, it will further worsen the situation for millicharged dark matter scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Considering upper bound on ˆσ − MDM by Planck 2015 [52] and EDGES upper constraint on T21 (−300 mK) at z = 17 [5], we found upper bound on the magnetic field strength: B0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='48 × 10−6 G, while considering CMB-S4 forecast constraint [51] we get B0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='67 × 10−6 G for the dark matter mass ≲ 10−2 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Chapter 5 PMFs & Baryon-Dark matter Interaction 102 “I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.” Isaac Newton, “Memoirs of Newton” (1855), Vol II By David Brewster 6 Summary and Future outlook 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 Summary The 21 cm signal is shown to be a prestigious probe in the cosmological laboratory to provide robust bounds on the physics of the early and late time Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The signal can give a good insight into the period when the galaxies and first stars were formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the thesis, I have analysed bounds on the present-day strength of primordial magnetic fields, sterile neutrino lifetime & mixing angle with active neutrinos, and primordial black hole dark matter fraction using the global 21 cm signal during the cosmic dawn era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The 21 cm line corresponds to the wavelength of hyperfine transition between 1S singlet and triplet states of the neutral hydrogen atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The corresponding frequency for the 21 cm line is 1420.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For a transition at redshift z, the frequency can be mapped for a present-day observed frequency as 1420.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4/(1 + z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 21 cm Line Astronomy and Constraining New Physics In the ΛCDM framework of cosmology, the evolution of the gas temperature and ionization fraction are well-established during the cosmic dawn era [153].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The addition of any exotic source of energy can significantly impact the ionization and thermal history of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The change in the gas temperature can significantly modify the absorption feature in the global 21 cm signal during cosmic dawn [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' This can provide constraints on the properties of such exotic sources of energy injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The EDGES collaboration has reported the 21 cm differential brightness tem- perature: T21 = −500+200 −500 mK with 99 percent confidence limit centred at 78 MHz or redshift z = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' By considering TS = Tgas, the observed brightness temper- ature translates to gas temperature as Tgas(z = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='26+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='94 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='58 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the ΛCDM framework, the gas temperature at redshift z = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 remains around 7 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' This corresponds to differential brightness temperature T21(z = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2) ≃ −220 mK— equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='19) for TS ≃ Tgas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' To resolve the tension between the theoretical prediction based on ΛCDM model and EDGES observation, one requires to in- crease the ratio of TR/TS in equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='19) over theoretical predictions in redshift range 15 ≤ z ≤ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' This can be achieved either by increasing the background radiation or decrease the gas temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Both possibilities have been studied by several authors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' for example, see the Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [25–27, 289, 304, 315–317, 319– 321, 325, 326, 345, 346, 352–354].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' However, such mechanisms to increase the background radio radiation or cooling the gas are debatable issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' One of such mechanisms to cool gas is baryon dark matter interaction [345].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' This approach has been questioned by several authors [51, 73, 340, 349, 355–358].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, it is to be noted that the authors do not consider heating of the gas by decaying or annihi- lating dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Injection of electrons and photons by decaying or annihilating dark matter into IGM can heat the gas more than cooling of the gas [156, 157].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Moreover, the EDGES measurement has been also questioned in several articles [6– 8, 77, 78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Recently, SARAS 3 observation reported that the EDGES observation is not of an astrophysical origin and it is rejected with the 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3 percent confidence Chapter 6 Summary and Future outlook 104 21 cm Line Astronomy and Constraining New Physics level [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [77], the authors have questioned the fitting parameters for the foreground emission and data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' There is a possibility that the absorption fea- ture in the EDGES observation can be a ground screen artifact [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The absorption amplitude may modify depending on modelling of foreground [8, 78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [359], the authors perform the Bayesian comparison of fitting models for EDGES data and argue that the highest evidence models favour an amplitude of |T21| < 209 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the light of these controversies, it is require to verify the EDGES result by other observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The future updated version of the hydrogen Epoch of Reionization Array (HERA)a, Thirty Meter Telescope (TMT)b, JWST, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=', will be able to probe the cosmic dawn era more precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The following summarizes the results reported in the thesis: 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 Bounds on dark matter candidates About 85 per cent of the total matter content in the Universe is dominated by dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the last decades, many dark matter models have been proposed to explain various astrophysical observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' However, the microscopic nature of dark matter is still unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' During my doctoral research, I have considered sterile neutrinos and primordial black holes as dark matter candidates and constrain their properties using the absorption feature in 21 cm differential brightness temperature during the cosmic dawn era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As discussed earlier, here, we have taken 21 cm differential brightness temperature such that it does not change from its standard theoretical value (∼ −220 mK) by more than a factor of 1/4 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' −150 mK) or 1/2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' −100 mK) at redshift 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Sterile Neutrino Dark Matter In the warm dark matter models, one of the theoretically well-motivated candidates is KeV mass sterile neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We have constrained the sterile neutrino dark matter ahttp://reionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='org/ bhttp://tmt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='org/ 105 Summary and Future outlook Chapter 6 21 cm Line Astronomy and Constraining New Physics lifetime and mixing angle with active neutrino as a function of sterile neutrino mass [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, we have considered the two scenarios to get the bounds: First, IGM evolution without the heat transfer from the background radiation to gas mediated by Lyα photons (VDKZ18 effect).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Next, we have considered additional VDKZ18 heating effects on the IGM gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The following summarises our results for T21 = −150 mK : In the first scenario, the lower bound on the sterile neutrino lifetime varies from 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3 × 1027 sec to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4 × 1025 sec by varying sterile neutrino mass from 2 KeV to 50 KeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' While the upper bound on the mixing angle varies from 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='8 × 10−9 to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 × 10−14 by varying sterile neutrino mass from 2 KeV to 50 KeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the second scenario, the lower bound on the sterile neutrino lifetime varies from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5 × 1028 sec to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7 × 1026 sec by varying sterile neutrino mass from 2 KeV to 50 KeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' While the upper bound on the mixing angle varies from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='8 × 10−9 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='42 × 10−14 by varying sterile neutrino mass from 2 KeV to 50 KeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Primordial Black Hole Dark Matter Spinning primordial black holes can substantially affect the ionization and thermal history of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Subsequently, it can modify the 21 cm absorption signal during cosmic dawn era by injecting energy due to Hawking evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We study the upper projected bounds on the fraction of dark matter in the form of PBHs as a function of mass and spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Our projected constraints are stringent compared to DSNB, INTEGRAL observation of the 511 KeV line, IGRB, Planck, Leo T and COMPTEL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the near future, AMEGO collaboration will be able to probe some parameter space in our considered mass range of PBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the present work, we have considered the monochromatic mass distribution of PBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The allowed parameter space can also be explored for different PBHs mass distributions such as log-normal, power-law, critical collapse, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [251].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We find the most robust lower projected constraint on the mass of PBHs, which is allowed to constitute the entire dark matter, to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5 × 1017 g, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9 × 1017 g, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9 × 1017 g and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7 × 1017 g Chapter 6 Summary and Future outlook 106 21 cm Line Astronomy and Constraining New Physics for PBH spins 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9999, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The lower bound on MPBH for ΩPBH = ΩDM, for extremal spinning PBHs is nearly four times larger than non-spinning ones [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 Primordial Magnetic Fields Observations suggest that the magnetic fields are ubiquitous in the Universe— from the length scale of planets and stars to the cluster of galaxies [17–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The origin and evolution of PMFs are one of the outstanding problems of modern cosmology (Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [23, 24] and references cited therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Decaying PMFs can inject magnetic energy into the thermal energy of the IGM and heat the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As discussed earlier, the EDGES collaboration reported an absorption profile for the global 21 cm signal with an amplitude of −500+200 −500 mK in the redshift range 15 − 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' To explain the EDGES anomaly, one requires to enhance the background radio radiation above the CMB radiation or lower the gas temperature below 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 K at redshift ∼ 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We have explored the upper bounds on the present-day strength of the PMFs in both scenarios by considering different models [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the Presence of Excess Radio Radiation As discussed, one requires to enhance the background radiation above the CMBR to explain the EDGES anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For excess radiation fraction to be LWA 1 limit, we have reported upper bounds on the present-day PMFs strength, B0 on the scale of 1 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The following summarises our results for T21 = −500 mK (EDGES best fit result): We have reported B0 ≲ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7 nG for spectral index nB = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='99 for excess radiation fraction to be LWA 1 limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' While for nB = −1, the upper bound gets more stringent: B0 ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 × 10−3 nG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We also discuss the effects of first stars on IGM gas evolution and the allowed value of B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' By decreasing excess radiation fraction below the LWA 1 limit, we get a more stringent bound on B0 [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' 107 Summary and Future outlook Chapter 6 21 cm Line Astronomy and Constraining New Physics In the Presence of Baryon-Dark Matter Interaction One of the alternatives to explain the EDGES anomaly is by cooling the gas below 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Since the dark matter is colder than the gas, adequate cooling of the gas can be obtained by introducing the baryon-dark matter interaction beyond the ΛCDM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The introduction of baryon-dark matter interaction relaxes the upper bound on B0 by transferring energy of the gas to the dark matter using drag between gas and dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Considering upper bound on ˆσ−md by Planck 2015 and EDGES upper constraint on T21 (−300 mK) at z = 17, we found upper bound on the present-day strength of PMFs: B0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='48×10−6 G, while considering CMB-S4 forecast constraint we get B0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='67 × 10−6 G for the dark matter mass ≲ 10−2 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' We have also discussed the bounds on ˆσ − md by considering Planck 2018 upper bound on B0 ∼ 10−9 G for EDGES best fit and upper bound on T21 [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Chapter 6 Summary and Future outlook 108 A A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1 Spin temperature of hydrogen In the presence of collisions, rate of change in the population of singlet state [2], dnS dt = −nS P C ST + nT P C TS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1) In the steady state, the transition coefficients from equation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1): nT/nS = P C ST/P C TS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' In the presence of collisions, the spin temperature will be kinetic tem- perature of gas only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, from equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5), P C ST = 3 exp � − TTS Tgas � × P C TS ≃ 3 � 1 − TTS Tgas � × P C TS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2) As discussed in the section (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2), Tgas, Tα ≫ TTS : exp [−TTS/Tgas] ≃ 1 − TTS/Tgas .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Similarly, for the Lyα radiation, Tgas and P C TS will be replaced by Tα and P α TS , 21 cm Line Astronomy and Constraining New Physics respectively, in equation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2), P α ST ≃ 3 � 1 − TTS Tα � × P α TS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3) In the hydrogen atom, there can be spontaneous and induced emissions by back- ground radiation also, P R TS = A10 + B10 IR ν , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4) here, B10 IR ν is the induced emission due to background radiation and IR ν is the specific intensity for 21 cm transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, A10 and B10 are Einstein coefficients and their relation is given by A10 = 2 ν2 TS TTS B10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For the background radiation, in the Rayleigh-Jeans limit from equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='11): IR ν = 2 ν2 TS TR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Therefore, from equation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4), P R TS = � 1 + TR TTS � A10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5) The induced transition from singlet to triplet due to background radiation [2], P R ST = B01 IR ν = 3 B10 IR ν = 3 A10 TR TTS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6) Using equations (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6), P R ST P R TS ≃ 3 � 1 − TTS TR � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7) In the detailed balance between the population of 1S singlet and triplet states (dnS/dt = 0), by solving the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='6) with the use of equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='7), we get, � 1 − TTS TS � = � 1 − TTS TR � + xα � 1 − TTS Tα � + xc � 1 − TTS Tgas � 1 + xα + xc , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='8) Chapter A Appendix 110 21 cm Line Astronomy and Constraining New Physics here, xα = P α TS/P R TS and xc = P C TS/P R TS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Solving the equation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='8), we get [2, 3], T −1 S = T −1 R + xα T −1 α + xc T −1 gas 1 + xα + xc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='9) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2 Emergent brightness temperature Solving differential equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='13) with initial conditions: when, l = 0 → τν = 0 and Iν = Iν0 (figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='4), Iν = Sν (1 − e−τν) + Iν0 e−τν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='10) Here, using equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='11), Iν = 2 ν2 T ′ R is the final/emergent specific intensity of light— of frequency ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Sν = 2 ν2 Texc is the specific intensity due to the medium having an excitation temperature, Texc, at a frequency of ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Iν0 = 2 ν2 TR is the initial specific intensity of the light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As a result, we find the final/emergent brightness temperature as [2, 3], T ′ R = Texc (1 − e−τν) + TR e−τν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='11) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='3 Optical depth of hydrogen medium The radiative transfer equation in the presence of emission and absorption of a light with travelled distance dl in the medium, dIν dl = TTS 4 π φ(ν) [ nT A10 + nT B10 Iν − nS B01 Iν ] , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='12) here, TTS = 2 π νTS, and φ(ν) represents line profile of the light beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' TTS/(4 π) represents the energy of light beam per unit solid angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The first term in the bracket is due to the spontaneous emission from the triplet to the singlet state, and it is proportional to the population density of the triplet state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The second 111 Appendix Chapter A 21 cm Line Astronomy and Constraining New Physics and third terms in the bracket are due to the stimulated/induced emission and absorption, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Comparing equations (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='12) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='12), we get, αν = TTS 4 π φ(ν) [ nS B01 − nT B10 ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='13) To get the optical depth of hydrogen medium, we can integrate equation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='13) over dl (equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='14), τν = 3 A10 32 π ν2 TS × TTS TS × nHI � φ(ν)dl , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='14) here, we have used the relations: A10 = 2 ν2 TS TTS B10 and B01 = 3 B10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' As the neutral hydrogen number density: nHI = nS + nT, the singlet state population density can be approximated by nS ≃ nHI/4 — from equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The ratio nT/nS , has given by equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' By solving the integral in equation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='14) for a line profile φ(ν) = 1/∆ν with the Doppler shift in the frequency due to the moving medium with a proper velocity v along the line of sight in the comoving coordinate (∆r = (1 + z) ∆l );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' we find the optical depth for hydrogen medium as [3], τν = 3 nHI 32 π ν3 TS × TTS TS × A10 H × �H/(1 + z) ∂v/∂r � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='15) Here, ∂v/∂r is the proper velocity gradient along the line of sight, and it can be taken as H/(1+z) for high redshift or in the absence of peculiar velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, nHI can be written as xHI nH, and xHI is the neutral hydrogen fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' The hydrogen number density can be expressed in the form of dimensionless baryon energy den- sity: nH ≃ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='5 × 10−6 (1 + δb) Ωb h2 (1 + z)3 cm−3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, δb = (ρb − ¯ρb)/¯ρb is the baryon density contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' ρb and ¯ρb are total and average baryon energy density, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' For the matter dominated era, we can take H = H0 √Ωm (1 + z)3/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Here, H0 and Ωm are present-day values of Hubble parameter and dimensionless matter energy density parameter, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' After some manipulation, we get Chapter A Appendix 112 21 cm Line Astronomy and Constraining New Physics the final expression for optical depth of hydrogen medium for 21 cm line [3, 68–71], τν ≃ 27 xHI (1 + δb) (1 + z) �mK TS � � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='15 Ωm h2 1 + z 10 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='physletb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content='136955.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [10] John Michell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' On the 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' [102] Mark R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Lovell, Violeta Gonzalez-Perez, Sownak Bose, Alexey Boyarsky, Shaun Cole, Carlos S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Frenk, and Oleg Ruchayskiy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' Addressing the too big to fail problem with baryon physics and sterile neutrino dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfyALn/content/2301.02655v1.pdf'} +page_content=' MNRAS, 468(3):2836–2849, 2017.' metadata={'source': 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+++ b/EtAzT4oBgHgl3EQfiv1a/content/tmp_files/2301.01504v1.pdf.txt @@ -0,0 +1,871 @@ +Generative models for scalar field theories: how to deal +with poor scaling? +Javad Komijani𝑎,∗ and Marina K. Marinkovic𝑎 +𝑎Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland +E-mail: jkomijani@phys.ethz.ch +Generative models, such as the method of normalizing flows, have been suggested as alternatives to +the standard algorithms for generating lattice gauge field configurations. Studies with the method +of normalizing flows demonstrate the proof of principle for simple models in two dimensions. +However, further studies indicate that the training cost can be, in general, very high for large +lattices. The poor scaling traits of current models indicate that moderate-size networks cannot +efficiently handle the inherently multi-scale aspects of the problem, especially around critical +points. We explore current models with limited acceptance rates for large lattices and examine +new architectures inspired by effective field theories to improve scaling traits. We also discuss +alternative ways of handling poor acceptance rates for large lattices. +The 39th International Symposium on Lattice Field Theory, +8th-13th August, 2022, +Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany +∗Speaker +© Copyright owned by the author(s) under the terms of the Creative Commons +Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). +https://pos.sissa.it/ +arXiv:2301.01504v1 [hep-lat] 4 Jan 2023 + +Generative models for scalar field theories: how to deal with poor scaling? +Javad Komijani +1. +Introduction +The method of trivializing maps was formulated by Lüscher [1] to improve the efficiency of +Markov Chain Monte Carlo (MCMC) simulations of lattice QCD by mapping the theory to another +one that is easier to simulate, ideally to a theory in which the degrees of freedom are decoupled. +Lüscher discussed how to construct such a map systematically by integrating certain flow equations +in field space and pointed out that, once such a map is constructed, the theory “can be simulated +simply by generating uniformly distributed random gauge fields” [1]. Although the last remark +seemed “likely to remain an academic one” [1], it took less than one decade that a similar idea, +which is called the method of normalizing flows (NF), flourished with many applications such as +image generation; for review, see Refs. [2, 3]. The method of normalizing flows is implemented +using deep neural networks rather than integrating certain flow equations. Deep neural networks +can approximate a huge class of functions and, as a result, provide a way to tackle complicated +problems without a need to model them first, in this case, constructing some flow equations and +integrating them. This, however, does not mean that one cannot use theoretical developments to +construct more suitable neural network architectures for NF. +Li and Wang [4] used a flow-based method for sampling from a dual version of two-dimensional +Ising model that resembles a scalar field theory. Albergo et.al. [5, 6] extended the study by applying +NF to scalar field theories with quartic potential in two-dimensional lattices up to 142 sites and +discussed in detail different aspects of the algorithm such as effects on the autocorrelation time. Del +Debbio et.al. [7] explored the scalability of the method by investigating lattices up 202 sites using +different architectures. Their study indicates that, in general, the method’s efficiency deteriorates +as the lattice size increases (for a fixed architecture). For a review of applications of generative +models on lattice field theory, we refer the reader to Ref. [8]. In this manuscript, we expand the +study of scalar field theories with quartic potential in two dimensions by introducing a novel flow +model inspired by effective field theories, discuss the scalability of the model, and present a way to +deal with the low acceptance rate at large volumes. +2. +Background and review of widely used architectures for NF +Let us start with a quick comment about the method of inverse transform sampling (ITS). This +method can be used to draw samples from the probability density function (PDF) of a continuous +random variable, 𝑓𝑌 (𝑦), by sampling from a simpler one, 𝑓𝑋 (𝑥), and transforming the samples +using +𝑦 = 𝐹−1 +𝑌 ◦ 𝐹𝑋 (𝑥) , +(1) +in which 𝐹𝑋 and 𝐹𝑌 stand for the cumulative distribution functions of 𝑥 (the prior) and 𝑦 (the +target) variables. The method of NF can be considered a generalization of the ITS method to +higher dimensional distributions. With the method of NF, we deal with a series of invertible and +differentiable transformations that are typically implemented by deep neural networks. The series +of transformations map the prior variable/distribution to a new one that we simply refer to as the +transformed variable/distribution. Training a NF-based model is then nothing but optimizing the +parameters of the model such that the transformed distribution resembles the target distribution. To +2 + +Generative models for scalar field theories: how to deal with poor scaling? +Javad Komijani +prior +transform +action +switch +/ +Metropolis +delay +gradient descent +TRAIN +GENERATE +ACCEPT/REJECT +ξ(x) +r[ξ] +φ(x) +q[φ] +p[φ] +log q/p +on/off +Figure 1: Block diagram for the method of normalizing flows. 𝜉(𝑥) and 𝜙(𝑥) are the prior and transformed +fields at position 𝑥, and and 𝑟[𝜉] and 𝑞[𝜙] are corresponding probability densities. The “GENERATE“ block +illustrates the integration of NF and MCMC by including an accept/reject step. +this end, one can minimize the relative entropy of the transformed and target distributions using the +Kullback-Leibler (KL) divergence +𝐷KL(𝑞||𝑝) ≡ +∫ +𝑑𝜙 𝑞[𝜙] log 𝑞[𝜙] +𝑝[𝜙] +≥ 0. +(2) +Here, 𝜙 denotes the transformed variable; 𝑝[𝜙] is the target PDF; and 𝑞[𝜙], which can be written +in terms of the prior PDF and the Jacobian of transformation, is the transformed PDF. The equality +in KL divergence holds only if 𝑝[𝜙] = 𝑞[𝜙]. The “TRAIN” block in Fig. 1 depicts the described +training procedure. Here, 𝜉(𝑥) and 𝜙(𝑥) are the prior and transformed variables (fields) at position +𝑥, and 𝑟[𝜉] and 𝑞[𝜙] are corresponding probability densities. +For the prior, we use a set of +independent normal distributions. The target PDF is +𝑝[𝜙] = 1 +𝑍 𝑒−𝑆[𝜙], +where 𝑆 is the action of the theory and the normalization factor 𝑍 is typically not known, indicating +that the lower bound in (2) is not known. +Once the model is perfectly trained, one can use it to draw samples from the target distribution. +In practice, however, it is unlikely to find a perfectly trained model, especially when the degrees of +freedom increase. To correct the samples, one can integrate the method of NF with MCMC. For +example, Ref. [5] introduced an accept/reject step as used in the Metropolis-Hastings algorithm to +ensure exactness. The “GENERATE” block in Fig. 1 illustrates such an integration, in which the +accept/reject step is formulated using the logarithm of the ratio of transformed and target densities, +log(𝑞[𝜙]/𝑝[𝜙]), of consecutive proposed fields as input. +The method of normalizing flows requires invertible transformations, putting some restrictions +on NF architectures. Coupling flows are one of the most widely used architectures; see Refs. [2, 3] +for reviews of different types of flows. +With coupling flows, one divides the field degrees of +freedom into two partitions, which can be labeled as 𝑎 (active) and 𝑓 (frozen/fixed) partitions. A +checkerboard-like mask is convenient for such partitioning. Each coupling-flow layer transforms +the active partition by a function parametrized with the frozen partition of the data: +𝑥𝑎 → 𝑇(𝑥𝑎; Θ(𝑥 𝑓 )) . +3 + +Generative models for scalar field theories: how to deal with poor scaling? +Javad Komijani +It is convenient to employ element-wise operations for 𝑇, e.g., element-wise linear (affine) and +spline transformations. With such transformations, the Jacobian matrix is triangular, making it easy +to calculate its determinant. Contrary to 𝑇, the form of Θ can be extremely complicated, which is +usually implemented by deep neural networks. +There are two widely used neural networks to model Θ: linear (dense) networks and convo- +lutional networks. The former is great for small-size lattices, but the number of parameters grows +fast as the size of the lattice grows. The latter takes advantage of the translational symmetry of the +underlying theory and in general needs much fewer parameters. However, the latter requires many +layers of neural networks to propagate the correlation throughout the data. +3. +Designing new architectures for normalizing flows +3.1 Effective action and power spectral density +Inspired by symmetries of the action and effective theories of scalar fields, our primary goal in +this section is to construct a novel flow layer that can propagate the correlation in data in an efficient +way. To this end, we start with an effective description of a real, scalar field theory. The action of +such a field in 𝑑 spacetime dimensions is +𝑆[𝜙] = +∫ +𝑑𝑑𝑥 +� +𝜁 +2 𝜕𝜇𝜙(𝑥)𝜕𝜇𝜙(𝑥) + 𝑚2 +2 𝜙2(𝑥) + +∞ +∑︁ +𝑛=3 +𝑔𝑛𝜙𝑛(𝑥) +� +. +(3) +The corresponding quantum effective action reads +Γ[𝜙] = 1 +2 +∫ +𝑑𝑑𝑘 +(2𝜋)𝑑 +� +𝜁𝑘2 + 𝑚2 + Π(𝑘2) +� +| ˜𝜙(𝑘)|2 + · · · , +(4) +where ˜𝜙(𝑘) is the scalar field in Fourier space. The quantum effective action has the following +property: the tree-level Feynman diagrams that it generates give the complete scattering amplitude +of the original theory [9]. Note that +� +𝜁𝑘2 + 𝑚2 + Π(𝑘2) +� +is the inverse of the two-point correlator +and, employing the engineering terminology, it is proportional to the inverse of power spectral +density (PSD) generalized to 𝑑 dimensions. +As manifested in (4), an element-wise operation on ˜𝜙(𝑘) can map the PSD to the one of interest. +Depending on the properties of PSD, one can restrict the map even further. For example, the Lorentz +invariance of PSD implies that the element-wise operation depends only on 𝑘2. We now examine a +couple of examples for further restrictions. Figure 2 shows the inverse of PSD of a 𝜙4 scalar theory +with double-well potential in one and two dimensions obtained from MCMC simulations plotted +against ˆ𝑘2 = � +𝑖 4 sin2(𝑘𝑖/2). The figure indicates that the inverse of PSD can be modeled using +a positive, monotonically increasing function of ˆ𝑘2. Here, we model the inverse of PSD with a +rational quadratic spline (RQS) [10–12] as a function of ˆ𝑘2, and we scale ˜𝜙(𝑘) accordingly. +In the case of a two-dimensional problem, PSD blows up at 𝑘2 = 0 in the broken phase. +This special point can be handled using the mean field theory: the mean-field potential turns to a +double-well potential at the broken phase. Therefore, at 𝑘2 = 0, instead of scaling ˜𝜙(0), we feed it +to a separate RQS, which can change the distribution of ˜𝜙(0) to a multi-modal distribution. +We use the term PSD flow to denote the described transformation. Note that a PSD flow can +change the correlation in data at the largest and shortest scales. In the next part, we investigate an +architecture with one PSD-flow layer followed by four coupling-flow layers. +4 + +Generative models for scalar field theories: how to deal with poor scaling? +Javad Komijani +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +ˆk2 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +1/PSD +0.000 +0.025 +0.050 +0.075 +0.00 +0.05 +0.10 +3.6 +3.7 +3.8 +3.9 +4.0 +3.6 +3.8 +4.0 +0 +1 +2 +3 +4 +5 +6 +7 +8 +ˆk2 +0 +1 +2 +3 +4 +5 +6 +7 +8 +1/PSD +Figure 2: The inverse of PSD of quartic scaler field theories with double-well potential (from MCMC +simulations). Left: One-dimensional lattice with size 𝐿 = 1024 and parameters 𝜁 = 1, 𝑚2 = −1.6, 𝑔4 = 0.1 +(with lattice spacing set to 0.125). Right: Two-dimensional lattice with size 𝐿2 = 322 and parameters +𝜁 = 0.7, 𝑚2 = −2.8, and 𝑔4 = 0.5. The dashed lines are guide for the eye. +3.2 A new architecture +In this part, we explain how we use a PSD-flow layer to construct a new NF architecture for a +real, scalar field theory in two dimensions, and we investigate the scalability of the new architecture. +The architecture that we investigated contains three blocks. A PSD-flow layer, followed by two +blocks of affine coupling flows, each block has two layers alternating the active and frozen partitions. +(In total, there are four coupling-flow layers.) For the Θ function in the affine coupling flows, we +use convolutional neural networks. Each of the three blocks has its own activation: symmetric +RQS, tanh, symmetric RQS, respectively. Unlike the tanh activation, the symmetric RQS splines +that we use have free parameters. We use symmetric RQS activations because they respect the 𝑍2 +symmetry of 𝜙4 scalar theories. In total, there are about 3.4 K parameters in the model. We use +this NF model for 𝜙4 scalar fields in two dimensions with several values of 𝐿: 8, 12, 16, · · · , 64. +We train the model for the 82 lattice with 10 K epochs. For the 122 lattice, instead of training from +scratch, we rely on transfer learning: we start from the model trained for the 82 lattice and train it +for 5 K epochs. Then, we use the model trained for the 122 lattice as the starting point for the 162 +lattice and analogously for larger lattices. +In order to compare our results with the literature, we fix the parameters of the action in (3) +as follows: 𝜁 = 𝜅, 𝑚2 = −4𝜅, 𝑔4 = 1/2, and 𝑔𝑛 = 0 for 𝑛 ≠ 4. Varying 𝜅 from 0.5 to 0.8, we can +compare our results with Ref. [7][Fig. 4]. The left panel of Fig. 3 shows the acceptance rate plotted +against 𝜅 for several values of lattice size. For 𝐿 = 8 lattices, the acceptance rate of the trained +models has a mild dependence on 𝜅. As the lattice size increases, the acceptance rate decreases. +Similar to Ref. [7], we observe that as 𝜅 approaches its critical value (𝜅𝑐 ≈ 0.67), the acceptance +rate deteriorates faster as the lattice size increases. +The middle and right panels of Fig. 3 show the acceptance rate plotted against 𝐿 and 𝐿2, +respectively. +The acceptance rate drops exponentially fast as 𝐿 increases, but the asymptotic +dependence cannot be reliably extracted from the graphs. To investigate this behavior, we examine +the acceptance rate and its dependence on the distribution of log 𝑞[𝜙]/𝑝[𝜙] by introducing a toy +5 + +Generative models for scalar field theories: how to deal with poor scaling? +Javad Komijani +0.5 +0.6 +0.7 +0.8 +κ +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +acceptance rate +L +8 +12 +16 +20 +26 +32 +46 +64 +20 +40 +60 +L +10−3 +10−2 +10−1 +100 +acceptance rate +κ +0.5 +0.54 +0.58 +0.6 +0.62 +0.64 +0.65 +0.66 +0.67 +0.68 +0.69 +0.7 +0.72 +0.76 +0.8 +0 +1000 +2000 +3000 +4000 +L2 +10−4 +10−3 +10−2 +10−1 +100 +acceptance rate +κ +0.5 +0.54 +0.58 +0.6 +0.62 +0.64 +0.65 +0.66 +0.67 +0.68 +0.69 +0.7 +0.72 +0.76 +0.8 +Figure 3: Acceptance rate plotted against 𝜅, 𝐿, and 𝐿2. The dashed lines are guide for the eye. +Figure 4: Histograms of snapshots of 𝜙(𝑥) from the prior and outputs of three blocks of transformations. +The lower panels show corresponding 2-point correlation functions. +model in the next part. +Before concluding this part, it is in order to examine the effect of employing a PSD-flow layer. +To this end, we present results from the model with parameters 𝜅 = 0.6 and 𝐿 = 32. Figure 4 shows +histograms of snapshots of 𝜙(𝑥) from the prior (upper left panel) and the outputs of three blocks of +transformations (2nd, 3rd, and 4th columns, respectively). The lower panels show corresponding 2- +point correlation functions. We observe that the PSD-flow block (the second column) can introduce +a correlation to the data that roughly remains unchanged in the next blocks. +4. +Variance in log(𝑞/𝑝), acceptance rate, and poor scaling at large volumes +The distribution of log(𝑞[𝜙]/𝑝[𝜙]) determines the acceptance rate of the model. For the +architecture investigated here, we observe that the variance of log(𝑞/𝑝) roughly scales with the +volume of the lattice in most cases. This rough behavior can be heuristically explained as follows. +One can divide a large lattice into multiple blocks. +If the blocks are large enough, the field +6 + +Hist of on +F'D +field +0.8 - +0.6 +f'n +0.3 - +0.6 +0.3 +0.4 +0.2 - +0.4 - +0.2 +0.2 +0.1 - +0.2 - +0.2 - +0.1. +0.0 +0.0 +0.0 +0.0 +5 +2.5 +2.5 +2.5 +2.5 +-2 +2 +1. +0 +0.0 +0.0 +0 +-2 +0 +2 +Cor pnpn+m [2-dim colormesh) +30 +30 +30 +30 +30 +20 - +20 - +20 - +20 - +20 - +10 - +10 - +10 - +10 - +10 - ++ D +0 - +-D +0 - +0 +20 +0 +20 +0 +20 +0 +20 +0 +20 +Corr onontm +10° ++pD +10-3 +10-1, +10-1- +0-1 +17-1 +: +104 +0-3 +. +: +03 +10~3. +. +10~3. +0 +20 +0 +20 +0 +20 +0 +20 +0 +20Hist of on +F'D +field +0.8 - +0.6 +f'n +0.3 - +0.6 +0.3 +0.4 +0.2 - +0.4 - +0.2 +0.2 +0.1 - +0.2 - +0.2 - +0.1. +0.0 +0.0 +0.0 +0.0 +5 +2.5 +2.5 +2.5 +2.5 +-2 +2 +1. +0 +0.0 +0.0 +0 +-2 +0 +2 +Cor pnpn+m [2-dim colormesh) +30 +30 +30 +30 +30 +20 - +20 - +20 - +20 - +20 - +10 - +10 - +10 - +10 - +10 - ++ D +0 - +-D +0 - +0 +20 +0 +20 +0 +20 +0 +20 +0 +20 +Corr onontm +10° ++pD +10-3 +10-1, +10-1- +0-1 +17-1 +: +104 +0-3 +. +: +03 +10~3. +. +10~3. +0 +20 +0 +20 +0 +20 +0 +20 +0 +20Generative models for scalar field theories: how to deal with poor scaling? +Javad Komijani +fluctuations in one block can be considered independent of other blocks. Then, the variance of +log(𝑞/𝑝) is proportional to the number of blocks and, in turn, to the volume of the lattice. +It is easy to compose models that yield a large acceptance rate for a small lattice. As the +lattice volume increases, the given model reaches a poor scaling area. One can improve the model’s +performance by changing the hyperparameters, adding more layers, or using more complicated +architectures. +As an alternative approach, we introduce and use a method that we call block +updating. To this end, we first introduce a toy model, investigate it, and explain the block-updating +approach. +4.1 Toy model +Let 𝑥𝑛 be a sequence of independent and identically distributed (iid) random variables with +normal distribution N (0, 𝜎2). We define a new random sequence based on the Metropolis-Hastings +accept/reject step as +𝑦𝑛 = ℎ(𝑥𝑛, 𝑦𝑛−1) = +� +𝑥𝑛 +with probability 𝑒−Relu(𝑥𝑛−𝑦𝑛−1) +𝑦𝑛−1 +otherwise +(5) +for 𝑛 > 0 and 𝑦0 = 𝑥0. The conditional probability distribution, for 𝑛 > 0, is +𝑓𝑌𝑛 |𝑋𝑛,𝑌𝑛−1(𝑦𝑛|𝑥𝑛, 𝑦𝑛−1) = 𝛿(𝑦𝑛 − 𝑥𝑛)𝑒−Relu(𝑥𝑛−𝑦𝑛−1) + 𝛿(𝑦𝑛 − 𝑦𝑛−1) +� +1 − 𝑒−Relu(𝑥𝑛−𝑦𝑛−1)� +. +(6) +We are interested to calculate the (static) distribution of the 𝑦𝑛 sequence for large values of 𝑛. From +𝑓𝑌𝑛(𝑦𝑛) = +∫ +𝑑𝑥𝑛 𝑑𝑦𝑛−1 𝑓𝑋𝑛(𝑥𝑛) 𝑓𝑌𝑛−1(𝑦𝑛−1) 𝑓𝑌𝑛 |𝑋𝑛,𝑌𝑛−1(𝑦𝑛|𝑥𝑛, 𝑦𝑛−1) , +(7) +we conclude that 𝑌𝑛 ∼ N (−𝜎2, 𝜎2) for large 𝑛. The acceptance rate is then +∫ +𝑑𝑥𝑛𝑑𝑦𝑛−1 𝑓𝑋𝑛(𝑥𝑛) 𝑓𝑌𝑛−1(𝑦𝑛−1)𝑒−Relu(𝑥𝑛−𝑦𝑛−1) = erfc(𝜎/2) . +(8) +The left panel in Fig. 5 illustrates erfc(𝜎/2) and also the simulation values of acceptance rate +plotted against 𝜎. For later use, let us calculate the asymptotic form of the acceptance rate. From +the asymptotic behavior of the complementary error function, we conclude that for large 𝜎, +− log(accept rate) = 1 +4𝜎2 + O(log(𝜎)) . +(9) +We now study the autocorrelation in the 𝑦𝑛 sequence defined as 𝑅[𝑛]/𝑅[0], with +𝑅[𝑛] = E +� +𝑦𝑘 + 𝜎2� � +𝑦𝑛+𝑘 + 𝜎2� +(10) +for 𝑘 large enough. The autocorrelation function can be calculated asymptotically for large 𝑛; the +expression is lengthy, and we do not reproduce it here. The middle panel in Fig. 5 shows the +autocorrelation in 𝑦𝑛 for several values of 𝜎. The decay of the autocorrelation function is sub- +exponential, in agreement with the corresponding asymptotic expression shown by dashed lines. +For a fraction of points, rough estimates of uncertainties in determining the autocorrelation are +shown with error bars. +7 + +Generative models for scalar field theories: how to deal with poor scaling? +Javad Komijani +0 +1 +2 +3 +4 +σ +0.0 +0.2 +0.4 +0.6 +0.8 +accept rate +erfc(σ/2) +0 +1 +2 +3 +4 +σ +10−2 +10−1 +100 +accept rate +erfc(σ/2) +0 +100 +200 +n +10−5 +10−3 +10−1 +R[n]/R[0] +σ +2 +1 +0.5 +0.25 +0 +25 +50 +75 +100 +[log(1 + n)]2 +10−5 +10−3 +10−1 +R[n]/R[0] +σ +2 +1 +0.5 +0.25 +0 +250 +500 +750 +1000 +n × nblocks +10−5 +10−3 +10−1 +R[n]/R[0] +nblocks +1 +4 +16 +64 +0 +25 +50 +75 +100 +[log(1 + n × nblocks)]2 +10−5 +10−3 +10−1 +R[n]/R[0] +nblocks +1 +4 +16 +64 +Figure 5: Acceptance rate (left panel) and autocorrelation function (middle and right panels) for the toy +model introduced in Sec. 4.1. +The dashed lines show the asymptotic expression of the autocorrelation +function. In the right panel, the block-updating procedure is used for several numbers of blocks and 𝜎 = 2. +We aim to modify the model to decrease the autocorrelation in the 𝑦𝑛 sequence. We implement +a method that we call block updating. +First, we assume that 𝑥𝑛 is obtained by adding 𝑛blocks +iid normal variables with mean 0 and variance 𝜎2/𝑛blocks as 𝑥𝑛 = �𝑛blocks +𝑏=1 +𝑥 {𝑏} +𝑛 +. Similarly, we +decompose 𝑦𝑛 as 𝑦𝑛 = �𝑛blocks +𝑏=1 +𝑦{𝑏} +𝑛 +. Then, instead of proposing independent values of 𝑥𝑛 at each +step, we divide each step to 𝑛blocks substeps. At each substep, we draw a new value for one block of +𝑥𝑛, i.e., for 𝑥 {𝑏} +𝑛 +, and propose it to update 𝑦{𝑏} +𝑛 +: +𝑦{𝑏} +𝑛 += ℎ +� +𝑥 {𝑏} +𝑛 +, 𝑦{𝑏} +𝑛−1 +� +. +(11) +Because the blocks are independent, the problem can be reduced to having 𝑛blocks independent copies +of the original problem with reduced variance 𝜎2/𝑛blocks. As the number of blocks increases, +the reduced variance decreases and consequently the acceptance rate increases (at the price of +splitting each step into 𝑛blocks substeps or having 𝑛blocks copies). +From equation (9) one may +conclude that we do not gain any advantages because the probability of getting a completely fresh +vector (𝑦{1} +𝑛 , · · · , 𝑦{𝑛blocks} +𝑛 +) compared to (𝑦{1} +𝑛−1, · · · , 𝑦{𝑛blocks} +𝑛−1 +), in which all blocks are replaced with +proposed ones, does not change asymptotically because +𝑛blocks log erfc +� +𝜎 +2√𝑛blocks +� += 𝜎2 + O(log(𝜎)) +(12) +when 𝜎2/𝑛blocks is large enough. However, the block-updating procedure has a significant effect on +the autocorrelation in 𝑦𝑛. +There are two competing aspects in the block-updating procedure. On the one hand, as 𝑛blocks +increases, the outputs of consecutive substeps get more correlated because we update only a block +of the data at each substep. On the other hand, the acceptance rate increases for each substep, which +in general reduces the autocorrelation in the output. The effects of these two competing aspects can +be seen in the right panel of Fig. 5, which illustrates the autocorrelation in 𝑦𝑛 with 𝜎 = 2 for several +blocks: 1, 4, 16, 64. In this panel, to take into account the cost of block updating, i.e., splitting +each step into 𝑛blocks substeps, the argument of the autocorrelation function (the horizontal axis) is +inflated by the number of blocks. We observe that the decay of the autocorrelation function speeds +up as the number of blocks increases from 1, indicating that the effects of the second aspect are +dominant. But, after a certain point, the effects of the first aspect dominate and autocorrelation time +increases. We leave detailed discussions on the integrated autocorrelation time to future work. +8 + +Generative models for scalar field theories: how to deal with poor scaling? +Javad Komijani +4.2 Variance of log(𝑞/𝑝), block size, and acceptance rate +There are similarities and differences between the toy model introduced in the previous part +and the main problem investigated in this manuscript. Assuming the distribution of log(𝑞/𝑝) is +normal, one could identify the sequence of proposed values of log(𝑞/𝑝) with 𝑥𝑛 in the toy model +and the sequence of accepted values of log(𝑞/𝑝) with 𝑦𝑛. Then, one could apply the results of +the previous section to study log(𝑞/𝑝) and, to some extent, other quantities. There are three main +differences. Firstly, the distribution of log(𝑞/𝑝) is not necessarily normal. Secondly, all quantities +do not necessarily suffer from the same autocorrelation in the sequence of accepted values of +log(𝑞/𝑝). Finally, the effects of applying a block updating procedure cannot be reduced to having +𝑛blocks independent copies of a similar problem. +We first examine the relation between acceptance rate and volume. As mentioned above, for the +architecture studied here, we observe that the variance of log(𝑞/𝑝) roughly scales with the volume +of the lattice in most cases. Based on this observation and assuming the distribution of log(𝑞/𝑝) is +normal, one can employ the asymptotic relation in (9) and argue that as 𝑉 → ∞, +− log(acceptance rate) ∝ 𝑉 + O(log(𝑉)) . +(13) +In practice, however, the above assumptions are not completely correct, and by comparing the +middle and right panels of Fig. 3, one may conclude that dependence of the logarithm of the +acceptance rate on the volume is milder than what equation (13) suggests for large volumes. Even +in some cases, the dependence looks more consistent with scaling by +√ +𝑉 rather than 𝑉, but this +might be because the volume is not large enough to use the asymptotic relation. Moreover, note +that these observations may change once one varies the settings, e.g., by using a different model or +increasing the number of epochs. +Similar to the toy model, we can use the block-updating procedure to improve acceptance rate +and integrated autocorrelation time. To this end, instead of proposing completely independent +configurations at each step, we split the lattice into several blocks, and at each substep, we update +only the prior fields on the corresponding block. +Figure 6 shows the effect of block-updating +procedure applied on the largest lattice, 𝐿2 = 642, for three values of 𝜅 close to the critical point +of theory. The circle, cross, and square points show the acceptance rate for 1, 4, and 16 blocks, +respectively. As expected, the acceptance rate improves as we increase the number of blocks. +Our primary investigation shows that the block-updating procedure introduced here also im- +proves the autocorrelation in various quantities. We leave this discussion to another work. +5. +Summary and outlook +In this manuscript, we reviewed coupling flows as one of the widely use building blocks to +construct NF architectures. Inspired by effective field theories, we presented a new transformation +called PSD flow. With a new architecture that employs a PSD-flow layer and (in total) 4 coupling- +flow layers, we investigated lattices up to 642 sites. Although the new architecture allows us to +increase the lattice size, the model’s acceptance rate deteriorates at large volumes in a fashion +similar to what was observed in Ref. [7]. +To investigate the behavior of the acceptance rate as a function of the volume of the lattice, +we introduced a toy model and discussed how one could handle the poor acceptance rate and long +9 + +Generative models for scalar field theories: how to deal with poor scaling? +Javad Komijani +20 +40 +60 +L +10−3 +10−2 +10−1 +100 +acceptance rate +κ +0.66 +0.67 +0.68 +Figure 6: Effects of block updating on acceptance rate for three values of 𝜅 close to the critical point. The +circle, cross, and square points show the acceptance rate for 1, 4, and 16 blocks applied on the largest lattice. +integrated autocorrelation time of the toy model by block updating. +Based on the similarities +between the toy model and the problem at hand, we proposed that a block-updating procedure can +be employed to handle the poor scaling of the acceptance rates for large lattices. +We are extending our studies to other theories, e.g., gauge theories, and applying the PSD flow +to these theories. We are also exploring variants of the PSD flow. Moreover, we are investigating +the effects of the block-updating procedure on various quantities related to the scalar field theory. +References +[1] M. Luscher, Commun. Math. Phys. 293, 899 (2010), arXiv:0907.5491 [hep-lat] . +[2] I. Kobyzev, S. J. Prince, and M. A. Brubaker, IEEE Transactions on Pattern Analysis and +Machine Intelligence 43, 3964 (2021). +[3] G. Papamakarios et al., Journal of Machine Learning Research 22, 1 (2021). +[4] S.-H. Li and L. Wang, Physical Review Letters 121 (2018). +[5] M. S. Albergo, G. Kanwar, and P. E. Shanahan, Phys. Rev. D 100, 034515 (2019) . +[6] M. S. Albergo et al. (2021), arXiv:2101.08176 [hep-lat] . +[7] L. Del Debbio, J. M. Rossney, and M. Wilson, Phys. Rev. D 104, 094507 (2021) . +[8] D. Boyda et al., in 2022 Snowmass Summer Study (2022) arXiv:2202.05838 [hep-lat] . +[9] M. Srednicki, Quantum field theory (Cambridge University Press, 2007). +[10] J. A. Gregory and R. Delbourgo, IMA Journal of Numerical Analysis 2, 123 (1982) . +[11] R. Delbourgo and J. A. Gregory, IMA Journal of Numerical Analysis 3, 141 (1983) . +[12] C. Durkan, A. Bekasov, I. Murray, and G. Papamakarios, (2019), arXiv:1906.04032 . +10 + diff --git a/EtAzT4oBgHgl3EQfiv1a/content/tmp_files/load_file.txt b/EtAzT4oBgHgl3EQfiv1a/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3350d7b3dfc579ca169c985677560fce4471248d --- /dev/null +++ b/EtAzT4oBgHgl3EQfiv1a/content/tmp_files/load_file.txt @@ -0,0 +1,466 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf,len=465 +page_content='Generative models for scalar field theories: how to deal with poor scaling?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Javad Komijani𝑎,∗ and Marina K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Marinkovic𝑎 𝑎Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland E-mail: jkomijani@phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='ch Generative models, such as the method of normalizing flows, have been suggested as alternatives to the standard algorithms for generating lattice gauge field configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Studies with the method of normalizing flows demonstrate the proof of principle for simple models in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' However, further studies indicate that the training cost can be, in general, very high for large lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' The poor scaling traits of current models indicate that moderate-size networks cannot efficiently handle the inherently multi-scale aspects of the problem, especially around critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' We explore current models with limited acceptance rates for large lattices and examine new architectures inspired by effective field theories to improve scaling traits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' We also discuss alternative ways of handling poor acceptance rates for large lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' The 39th International Symposium on Lattice Field Theory, 8th-13th August, 2022, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany ∗Speaker © Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='0 International License (CC BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' https://pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='it/ arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='01504v1 [hep-lat] 4 Jan 2023 Generative models for scalar field theories: how to deal with poor scaling?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Javad Komijani 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Introduction The method of trivializing maps was formulated by Lüscher [1] to improve the efficiency of Markov Chain Monte Carlo (MCMC) simulations of lattice QCD by mapping the theory to another one that is easier to simulate, ideally to a theory in which the degrees of freedom are decoupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Lüscher discussed how to construct such a map systematically by integrating certain flow equations in field space and pointed out that, once such a map is constructed, the theory “can be simulated simply by generating uniformly distributed random gauge fields” [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Although the last remark seemed “likely to remain an academic one” [1], it took less than one decade that a similar idea, which is called the method of normalizing flows (NF), flourished with many applications such as image generation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' for review, see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' The method of normalizing flows is implemented using deep neural networks rather than integrating certain flow equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Deep neural networks can approximate a huge class of functions and, as a result, provide a way to tackle complicated problems without a need to model them first, in this case, constructing some flow equations and integrating them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' This, however, does not mean that one cannot use theoretical developments to construct more suitable neural network architectures for NF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Li and Wang [4] used a flow-based method for sampling from a dual version of two-dimensional Ising model that resembles a scalar field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Albergo et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' [5, 6] extended the study by applying NF to scalar field theories with quartic potential in two-dimensional lattices up to 142 sites and discussed in detail different aspects of the algorithm such as effects on the autocorrelation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Del Debbio et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' [7] explored the scalability of the method by investigating lattices up 202 sites using different architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Their study indicates that, in general, the method’s efficiency deteriorates as the lattice size increases (for a fixed architecture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' For a review of applications of generative models on lattice field theory, we refer the reader to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' In this manuscript, we expand the study of scalar field theories with quartic potential in two dimensions by introducing a novel flow model inspired by effective field theories, discuss the scalability of the model, and present a way to deal with the low acceptance rate at large volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Background and review of widely used architectures for NF Let us start with a quick comment about the method of inverse transform sampling (ITS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' This method can be used to draw samples from the probability density function (PDF) of a continuous random variable, 𝑓𝑌 (𝑦), by sampling from a simpler one, 𝑓𝑋 (𝑥), and transforming the samples using 𝑦 = 𝐹−1 𝑌 ◦ 𝐹𝑋 (𝑥) , (1) in which 𝐹𝑋 and 𝐹𝑌 stand for the cumulative distribution functions of 𝑥 (the prior) and 𝑦 (the target) variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' The method of NF can be considered a generalization of the ITS method to higher dimensional distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' With the method of NF, we deal with a series of invertible and differentiable transformations that are typically implemented by deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' The series of transformations map the prior variable/distribution to a new one that we simply refer to as the transformed variable/distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Training a NF-based model is then nothing but optimizing the parameters of the model such that the transformed distribution resembles the target distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' To 2 Generative models for scalar field theories: how to deal with poor scaling?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Javad Komijani prior transform action switch / Metropolis delay gradient descent TRAIN GENERATE ACCEPT/REJECT ξ(x) r[ξ] φ(x) q[φ] p[φ] log q/p on/off Figure 1: Block diagram for the method of normalizing flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' 𝜉(𝑥) and 𝜙(𝑥) are the prior and transformed fields at position 𝑥, and and 𝑟[𝜉] and 𝑞[𝜙] are corresponding probability densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' The “GENERATE“ block illustrates the integration of NF and MCMC by including an accept/reject step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' this end, one can minimize the relative entropy of the transformed and target distributions using the Kullback-Leibler (KL) divergence 𝐷KL(𝑞||𝑝) ≡ ∫ 𝑑𝜙 𝑞[𝜙] log 𝑞[𝜙] 𝑝[𝜙] ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' (2) Here, 𝜙 denotes the transformed variable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' 𝑝[𝜙] is the target PDF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' and 𝑞[𝜙], which can be written in terms of the prior PDF and the Jacobian of transformation, is the transformed PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' The equality in KL divergence holds only if 𝑝[𝜙] = 𝑞[𝜙].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' The “TRAIN” block in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' 1 depicts the described training procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Here, 𝜉(𝑥) and 𝜙(𝑥) are the prior and transformed variables (fields) at position 𝑥, and 𝑟[𝜉] and 𝑞[𝜙] are corresponding probability densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' For the prior, we use a set of independent normal distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' The target PDF is 𝑝[𝜙] = 1 𝑍 𝑒−𝑆[𝜙], where 𝑆 is the action of the theory and the normalization factor 𝑍 is typically not known, indicating that the lower bound in (2) is not known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Once the model is perfectly trained, one can use it to draw samples from the target distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' In practice, however, it is unlikely to find a perfectly trained model, especially when the degrees of freedom increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' To correct the samples, one can integrate the method of NF with MCMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' For example, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' [5] introduced an accept/reject step as used in the Metropolis-Hastings algorithm to ensure exactness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' The “GENERATE” block in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' 1 illustrates such an integration, in which the accept/reject step is formulated using the logarithm of the ratio of transformed and target densities, log(𝑞[𝜙]/𝑝[𝜙]), of consecutive proposed fields as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' The method of normalizing flows requires invertible transformations, putting some restrictions on NF architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Coupling flows are one of the most widely used architectures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' [2, 3] for reviews of different types of flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' With coupling flows, one divides the field degrees of freedom into two partitions, which can be labeled as 𝑎 (active) and 𝑓 (frozen/fixed) partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' A checkerboard-like mask is convenient for such partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Each coupling-flow layer transforms the active partition by a function parametrized with the frozen partition of the data: 𝑥𝑎 → 𝑇(𝑥𝑎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Θ(𝑥 𝑓 )) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' 3 Generative models for scalar field theories: how to deal with poor scaling?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Javad Komijani It is convenient to employ element-wise operations for 𝑇, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=', element-wise linear (affine) and spline transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' With such transformations, the Jacobian matrix is triangular, making it easy to calculate its determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Contrary to 𝑇, the form of Θ can be extremely complicated, which is usually implemented by deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' There are two widely used neural networks to model Θ: linear (dense) networks and convo- lutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' The former is great for small-size lattices, but the number of parameters grows fast as the size of the lattice grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' The latter takes advantage of the translational symmetry of the underlying theory and in general needs much fewer parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' However, the latter requires many layers of neural networks to propagate the correlation throughout the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Designing new architectures for normalizing flows 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='1 Effective action and power spectral density Inspired by symmetries of the action and effective theories of scalar fields, our primary goal in this section is to construct a novel flow layer that can propagate the correlation in data in an efficient way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' To this end, we start with an effective description of a real, scalar field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' The action of such a field in 𝑑 spacetime dimensions is 𝑆[𝜙] = ∫ 𝑑𝑑𝑥 � 𝜁 2 𝜕𝜇𝜙(𝑥)𝜕𝜇𝜙(𝑥) + 𝑚2 2 𝜙2(𝑥) + ∞ ∑︁ 𝑛=3 𝑔𝑛𝜙𝑛(𝑥) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' (3) The corresponding quantum effective action reads Γ[𝜙] = 1 2 ∫ 𝑑𝑑𝑘 (2𝜋)𝑑 � 𝜁𝑘2 + 𝑚2 + Π(𝑘2) � | ˜𝜙(𝑘)|2 + · · · , (4) where ˜𝜙(𝑘) is the scalar field in Fourier space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' The quantum effective action has the following property: the tree-level Feynman diagrams that it generates give the complete scattering amplitude of the original theory [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Note that � 𝜁𝑘2 + 𝑚2 + Π(𝑘2) � is the inverse of the two-point correlator and, employing the engineering terminology, it is proportional to the inverse of power spectral density (PSD) generalized to 𝑑 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' As manifested in (4), an element-wise operation on ˜𝜙(𝑘) can map the PSD to the one of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Depending on the properties of PSD, one can restrict the map even further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' For example, the Lorentz invariance of PSD implies that the element-wise operation depends only on 𝑘2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' We now examine a couple of examples for further restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Figure 2 shows the inverse of PSD of a 𝜙4 scalar theory with double-well potential in one and two dimensions obtained from MCMC simulations plotted against ˆ𝑘2 = � 𝑖 4 sin2(𝑘𝑖/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' The figure indicates that the inverse of PSD can be modeled using a positive, monotonically increasing function of ˆ𝑘2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Here, we model the inverse of PSD with a rational quadratic spline (RQS) [10–12] as a function of ˆ𝑘2, and we scale ˜𝜙(𝑘) accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' In the case of a two-dimensional problem, PSD blows up at 𝑘2 = 0 in the broken phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' This special point can be handled using the mean field theory: the mean-field potential turns to a double-well potential at the broken phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Therefore, at 𝑘2 = 0, instead of scaling ˜𝜙(0), we feed it to a separate RQS, which can change the distribution of ˜𝜙(0) to a multi-modal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' We use the term PSD flow to denote the described transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Note that a PSD flow can change the correlation in data at the largest and shortest scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' In the next part, we investigate an architecture with one PSD-flow layer followed by four coupling-flow layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' 4 Generative models for scalar field theories: how to deal with poor scaling?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Javad Komijani 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='0 ˆk2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='5 1/PSD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='0 0 1 2 3 4 5 6 7 8 ˆk2 0 1 2 3 4 5 6 7 8 1/PSD Figure 2: The inverse of PSD of quartic scaler field theories with double-well potential (from MCMC simulations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Left: One-dimensional lattice with size 𝐿 = 1024 and parameters 𝜁 = 1, 𝑚2 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='6, 𝑔4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='1 (with lattice spacing set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='125).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Right: Two-dimensional lattice with size 𝐿2 = 322 and parameters 𝜁 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='7, 𝑚2 = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='8, and 𝑔4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' The dashed lines are guide for the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='2 A new architecture In this part, we explain how we use a PSD-flow layer to construct a new NF architecture for a real, scalar field theory in two dimensions, and we investigate the scalability of the new architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' The architecture that we investigated contains three blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' A PSD-flow layer, followed by two blocks of affine coupling flows, each block has two layers alternating the active and frozen partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' (In total, there are four coupling-flow layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=') For the Θ function in the affine coupling flows, we use convolutional neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Each of the three blocks has its own activation: symmetric RQS, tanh, symmetric RQS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Unlike the tanh activation, the symmetric RQS splines that we use have free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' We use symmetric RQS activations because they respect the 𝑍2 symmetry of 𝜙4 scalar theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' In total, there are about 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='4 K parameters in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' We use this NF model for 𝜙4 scalar fields in two dimensions with several values of 𝐿: 8, 12, 16, · · · , 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' We train the model for the 82 lattice with 10 K epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' For the 122 lattice, instead of training from scratch, we rely on transfer learning: we start from the model trained for the 82 lattice and train it for 5 K epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Then, we use the model trained for the 122 lattice as the starting point for the 162 lattice and analogously for larger lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' In order to compare our results with the literature, we fix the parameters of the action in (3) as follows: 𝜁 = 𝜅, 𝑚2 = −4𝜅, 𝑔4 = 1/2, and 𝑔𝑛 = 0 for 𝑛 ≠ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Varying 𝜅 from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='5 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='8, we can compare our results with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' [7][Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' The left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' 3 shows the acceptance rate plotted against 𝜅 for several values of lattice size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' For 𝐿 = 8 lattices, the acceptance rate of the trained models has a mild dependence on 𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' As the lattice size increases, the acceptance rate decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Similar to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' [7], we observe that as 𝜅 approaches its critical value (𝜅𝑐 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='67), the acceptance rate deteriorates faster as the lattice size increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' The middle and right panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' 3 show the acceptance rate plotted against 𝐿 and 𝐿2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' The acceptance rate drops exponentially fast as 𝐿 increases, but the asymptotic dependence cannot be reliably extracted from the graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' To investigate this behavior, we examine the acceptance rate and its dependence on the distribution of log 𝑞[𝜙]/𝑝[𝜙] by introducing a toy 5 Generative models for scalar field theories: how to deal with poor scaling?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Javad Komijani 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='8 κ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='0 acceptance rate L 8 12 16 20 26 32 46 64 20 40 60 L 10−3 10−2 10−1 100 acceptance rate κ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='8 0 1000 2000 3000 4000 L2 10−4 10−3 10−2 10−1 100 acceptance rate κ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='8 Figure 3: Acceptance rate plotted against 𝜅, 𝐿, and 𝐿2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' The dashed lines are guide for the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Figure 4: Histograms of snapshots of 𝜙(𝑥) from the prior and outputs of three blocks of transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' The lower panels show corresponding 2-point correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' model in the next part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Before concluding this part, it is in order to examine the effect of employing a PSD-flow layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' To this end, we present results from the model with parameters 𝜅 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='6 and 𝐿 = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Figure 4 shows histograms of snapshots of 𝜙(𝑥) from the prior (upper left panel) and the outputs of three blocks of transformations (2nd, 3rd, and 4th columns, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' The lower panels show corresponding 2- point correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' We observe that the PSD-flow block (the second column) can introduce a correlation to the data that roughly remains unchanged in the next blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Variance in log(𝑞/𝑝), acceptance rate, and poor scaling at large volumes The distribution of log(𝑞[𝜙]/𝑝[𝜙]) determines the acceptance rate of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' For the architecture investigated here, we observe that the variance of log(𝑞/𝑝) roughly scales with the volume of the lattice in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' This rough behavior can be heuristically explained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' One can divide a large lattice into multiple blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=" If the blocks are large enough, the field 6 Hist of on F'D field 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='8 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content="6 f'n 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='3 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='3 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='5 2 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='0 0 2 0 2 Cor pnpn+m [2-dim colormesh) 30 30 30 30 30 20 - 20 - 20 - 20 - 20 - 10 - 10 - 10 - 10 - 10 - + D 0 - D 0 - 0 20 0 20 0 20 0 20 0 20 Corr onontm 10° +pD 10-3 10-1, 10-1- 0-1 17-1 : 104 0-3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' : 03 10~3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' 10~3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=" 0 20 0 20 0 20 0 20 0 20Hist of on F'D field 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='8 - 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='5 2 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='0 0 2 0 2 Cor pnpn+m [2-dim colormesh) 30 30 30 30 30 20 - 20 - 20 - 20 - 20 - 10 - 10 - 10 - 10 - 10 - + D 0 - D 0 - 0 20 0 20 0 20 0 20 0 20 Corr onontm 10° +pD 10-3 10-1, 10-1- 0-1 17-1 : 104 0-3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' : 03 10~3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' 10~3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' 0 20 0 20 0 20 0 20 0 20Generative models for scalar field theories: how to deal with poor scaling?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Javad Komijani fluctuations in one block can be considered independent of other blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Then, the variance of log(𝑞/𝑝) is proportional to the number of blocks and, in turn, to the volume of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' It is easy to compose models that yield a large acceptance rate for a small lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' As the lattice volume increases, the given model reaches a poor scaling area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' One can improve the model’s performance by changing the hyperparameters, adding more layers, or using more complicated architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' As an alternative approach, we introduce and use a method that we call block updating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' To this end, we first introduce a toy model, investigate it, and explain the block-updating approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='1 Toy model Let 𝑥𝑛 be a sequence of independent and identically distributed (iid) random variables with normal distribution N (0, 𝜎2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' We define a new random sequence based on the Metropolis-Hastings accept/reject step as 𝑦𝑛 = ℎ(𝑥𝑛, 𝑦𝑛−1) = � 𝑥𝑛 with probability 𝑒−Relu(𝑥𝑛−𝑦𝑛−1) 𝑦𝑛−1 otherwise (5) for 𝑛 > 0 and 𝑦0 = 𝑥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' The conditional probability distribution, for 𝑛 > 0, is 𝑓𝑌𝑛 |𝑋𝑛,𝑌𝑛−1(𝑦𝑛|𝑥𝑛, 𝑦𝑛−1) = 𝛿(𝑦𝑛 − 𝑥𝑛)𝑒−Relu(𝑥𝑛−𝑦𝑛−1) + 𝛿(𝑦𝑛 − 𝑦𝑛−1) � 1 − 𝑒−Relu(𝑥𝑛−𝑦𝑛−1)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' (6) We are interested to calculate the (static) distribution of the 𝑦𝑛 sequence for large values of 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' From 𝑓𝑌𝑛(𝑦𝑛) = ∫ 𝑑𝑥𝑛 𝑑𝑦𝑛−1 𝑓𝑋𝑛(𝑥𝑛) 𝑓𝑌𝑛−1(𝑦𝑛−1) 𝑓𝑌𝑛 |𝑋𝑛,𝑌𝑛−1(𝑦𝑛|𝑥𝑛, 𝑦𝑛−1) , (7) we conclude that 𝑌𝑛 ∼ N (−𝜎2, 𝜎2) for large 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' The acceptance rate is then ∫ 𝑑𝑥𝑛𝑑𝑦𝑛−1 𝑓𝑋𝑛(𝑥𝑛) 𝑓𝑌𝑛−1(𝑦𝑛−1)𝑒−Relu(𝑥𝑛−𝑦𝑛−1) = erfc(𝜎/2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' (8) The left panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' 5 illustrates erfc(𝜎/2) and also the simulation values of acceptance rate plotted against 𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' For later use, let us calculate the asymptotic form of the acceptance rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' From the asymptotic behavior of the complementary error function, we conclude that for large 𝜎, − log(accept rate) = 1 4𝜎2 + O(log(𝜎)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' (9) We now study the autocorrelation in the 𝑦𝑛 sequence defined as 𝑅[𝑛]/𝑅[0], with 𝑅[𝑛] = E � 𝑦𝑘 + 𝜎2� � 𝑦𝑛+𝑘 + 𝜎2� (10) for 𝑘 large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' The autocorrelation function can be calculated asymptotically for large 𝑛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' the expression is lengthy, and we do not reproduce it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' The middle panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' 5 shows the autocorrelation in 𝑦𝑛 for several values of 𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' The decay of the autocorrelation function is sub- exponential, in agreement with the corresponding asymptotic expression shown by dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' For a fraction of points, rough estimates of uncertainties in determining the autocorrelation are shown with error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' 7 Generative models for scalar field theories: how to deal with poor scaling?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Javad Komijani 0 1 2 3 4 σ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='8 accept rate erfc(σ/2) 0 1 2 3 4 σ 10−2 10−1 100 accept rate erfc(σ/2) 0 100 200 n 10−5 10−3 10−1 R[n]/R[0] σ 2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='25 0 25 50 75 100 [log(1 + n)]2 10−5 10−3 10−1 R[n]/R[0] σ 2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='25 0 250 500 750 1000 n × nblocks 10−5 10−3 10−1 R[n]/R[0] nblocks 1 4 16 64 0 25 50 75 100 [log(1 + n × nblocks)]2 10−5 10−3 10−1 R[n]/R[0] nblocks 1 4 16 64 Figure 5: Acceptance rate (left panel) and autocorrelation function (middle and right panels) for the toy model introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' The dashed lines show the asymptotic expression of the autocorrelation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' In the right panel, the block-updating procedure is used for several numbers of blocks and 𝜎 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' We aim to modify the model to decrease the autocorrelation in the 𝑦𝑛 sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' We implement a method that we call block updating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' First, we assume that 𝑥𝑛 is obtained by adding 𝑛blocks iid normal variables with mean 0 and variance 𝜎2/𝑛blocks as 𝑥𝑛 = �𝑛blocks 𝑏=1 𝑥 {𝑏} 𝑛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Similarly, we decompose 𝑦𝑛 as 𝑦𝑛 = �𝑛blocks 𝑏=1 𝑦{𝑏} 𝑛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Then, instead of proposing independent values of 𝑥𝑛 at each step, we divide each step to 𝑛blocks substeps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' At each substep, we draw a new value for one block of 𝑥𝑛, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=', for 𝑥 {𝑏} 𝑛 , and propose it to update 𝑦{𝑏} 𝑛 : 𝑦{𝑏} 𝑛 = ℎ � 𝑥 {𝑏} 𝑛 , 𝑦{𝑏} 𝑛−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' (11) Because the blocks are independent, the problem can be reduced to having 𝑛blocks independent copies of the original problem with reduced variance 𝜎2/𝑛blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' As the number of blocks increases, the reduced variance decreases and consequently the acceptance rate increases (at the price of splitting each step into 𝑛blocks substeps or having 𝑛blocks copies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' From equation (9) one may conclude that we do not gain any advantages because the probability of getting a completely fresh vector (𝑦{1} 𝑛 , · · · , 𝑦{𝑛blocks} 𝑛 ) compared to (𝑦{1} 𝑛−1, · · · , 𝑦{𝑛blocks} 𝑛−1 ), in which all blocks are replaced with proposed ones, does not change asymptotically because 𝑛blocks log erfc � 𝜎 2√𝑛blocks � = 𝜎2 + O(log(𝜎)) (12) when 𝜎2/𝑛blocks is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' However, the block-updating procedure has a significant effect on the autocorrelation in 𝑦𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' There are two competing aspects in the block-updating procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' On the one hand, as 𝑛blocks increases, the outputs of consecutive substeps get more correlated because we update only a block of the data at each substep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' On the other hand, the acceptance rate increases for each substep, which in general reduces the autocorrelation in the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' The effects of these two competing aspects can be seen in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' 5, which illustrates the autocorrelation in 𝑦𝑛 with 𝜎 = 2 for several blocks: 1, 4, 16, 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' In this panel, to take into account the cost of block updating, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=', splitting each step into 𝑛blocks substeps, the argument of the autocorrelation function (the horizontal axis) is inflated by the number of blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' We observe that the decay of the autocorrelation function speeds up as the number of blocks increases from 1, indicating that the effects of the second aspect are dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' But, after a certain point, the effects of the first aspect dominate and autocorrelation time increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' We leave detailed discussions on the integrated autocorrelation time to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' 8 Generative models for scalar field theories: how to deal with poor scaling?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Javad Komijani 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='2 Variance of log(𝑞/𝑝), block size, and acceptance rate There are similarities and differences between the toy model introduced in the previous part and the main problem investigated in this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Assuming the distribution of log(𝑞/𝑝) is normal, one could identify the sequence of proposed values of log(𝑞/𝑝) with 𝑥𝑛 in the toy model and the sequence of accepted values of log(𝑞/𝑝) with 𝑦𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Then, one could apply the results of the previous section to study log(𝑞/𝑝) and, to some extent, other quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' There are three main differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Firstly, the distribution of log(𝑞/𝑝) is not necessarily normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Secondly, all quantities do not necessarily suffer from the same autocorrelation in the sequence of accepted values of log(𝑞/𝑝).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Finally, the effects of applying a block updating procedure cannot be reduced to having 𝑛blocks independent copies of a similar problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' We first examine the relation between acceptance rate and volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' As mentioned above, for the architecture studied here, we observe that the variance of log(𝑞/𝑝) roughly scales with the volume of the lattice in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Based on this observation and assuming the distribution of log(𝑞/𝑝) is normal, one can employ the asymptotic relation in (9) and argue that as 𝑉 → ∞, − log(acceptance rate) ∝ 𝑉 + O(log(𝑉)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' (13) In practice, however, the above assumptions are not completely correct, and by comparing the middle and right panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' 3, one may conclude that dependence of the logarithm of the acceptance rate on the volume is milder than what equation (13) suggests for large volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Even in some cases, the dependence looks more consistent with scaling by √ 𝑉 rather than 𝑉, but this might be because the volume is not large enough to use the asymptotic relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Moreover, note that these observations may change once one varies the settings, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=', by using a different model or increasing the number of epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Similar to the toy model, we can use the block-updating procedure to improve acceptance rate and integrated autocorrelation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' To this end, instead of proposing completely independent configurations at each step, we split the lattice into several blocks, and at each substep, we update only the prior fields on the corresponding block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Figure 6 shows the effect of block-updating procedure applied on the largest lattice, 𝐿2 = 642, for three values of 𝜅 close to the critical point of theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' The circle, cross, and square points show the acceptance rate for 1, 4, and 16 blocks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' As expected, the acceptance rate improves as we increase the number of blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Our primary investigation shows that the block-updating procedure introduced here also im- proves the autocorrelation in various quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' We leave this discussion to another work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Summary and outlook In this manuscript, we reviewed coupling flows as one of the widely use building blocks to construct NF architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Inspired by effective field theories, we presented a new transformation called PSD flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' With a new architecture that employs a PSD-flow layer and (in total) 4 coupling- flow layers, we investigated lattices up to 642 sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Although the new architecture allows us to increase the lattice size, the model’s acceptance rate deteriorates at large volumes in a fashion similar to what was observed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' To investigate the behavior of the acceptance rate as a function of the volume of the lattice, we introduced a toy model and discussed how one could handle the poor acceptance rate and long 9 Generative models for scalar field theories: how to deal with poor scaling?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Javad Komijani 20 40 60 L 10−3 10−2 10−1 100 acceptance rate κ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='68 Figure 6: Effects of block updating on acceptance rate for three values of 𝜅 close to the critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' The circle, cross, and square points show the acceptance rate for 1, 4, and 16 blocks applied on the largest lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' integrated autocorrelation time of the toy model by block updating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' Based on the similarities between the toy model and the problem at hand, we proposed that a block-updating procedure can be employed to handle the poor scaling of the acceptance rates for large lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content=' We are extending our studies to other theories, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfiv1a/content/2301.01504v1.pdf'} +page_content='g.' metadata={'source': 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a/EtFKT4oBgHgl3EQfaC59/content/tmp_files/2301.11806v1.pdf.txt b/EtFKT4oBgHgl3EQfaC59/content/tmp_files/2301.11806v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5630de97e6a8ad9dda08f406c6c45fef01a7f80b --- /dev/null +++ b/EtFKT4oBgHgl3EQfaC59/content/tmp_files/2301.11806v1.pdf.txt @@ -0,0 +1,882 @@ +PCV: A POINT CLOUD-BASED NETWORK VERIFIER +Arup Kumar Sarker +Department of Computer Science +University of Virginia +Charlottesville, VA 22903 +djy8hg@virginia.edu +Farzana Yasmin Ahmad +Department of Computer Science +University of Virginia +Charlottesville, VA 22903 +fa7sa@virginia.edu +Matthew B. Dwyer +Department of Computer Science +University of Virginia +Charlottesville, VA 22903 +matthewbdwyer@virginia.edu +January 30, 2023 +ABSTRACT +3D vision with real-time LiDAR-based point cloud data became a vital part of autonomous system +research, especially perception and prediction modules use it for object classification, segmentation, +and detection. Despite their success, point cloud-based network models are vulnerable to multiple +adversarial attacks, where certain factor of changes in the validation set causes significant performance +drop in well-trained networks. Most of the existing verifiers work perfectly on 2D convolution. Due +to complex architecture, dimension of hyper-parameter, and 3D convolution, no verifiers can perform +the basic layer-wise verification. It is difficult to conclude the robustness of a 3D vision model without +performing the verification. Because there will be always corner cases and adversarial input that can +compromise the model’s effectiveness. +In this project, we describe a point cloud-based network verifier that successfully deals state of the art +3D classifier PointNet and verifies the robustness by generating adversarial inputs. We have used +extracted properties from the trained PointNet and changed certain factors for perturbation input. +We calculate the impact on model accuracy versus property factor and can test PointNet networks’ +robustness against a small collection of perturbing input states resulting from adversarial attacks +like the suggested hybrid reverse signed attack. The experimental results reveal that the resilience +property of PointNet is affected by our hybrid reverse signed perturbation strategy. +1 +Introduction +The point cloud is an important type of geometric data +structure percept from the LiDAR in the autonomous sys- +tem. In 1, the perception module process the data and +detect 3D object. The planning tasks get the prediction +results along with localization and send the driving pol- +icy to control so that control can send it to actuators. To +implement weight sharing and other kernel optimizations +in perception, typical convolutional architectures require +extremely regular input data formats, such as picture grids +and 3D voxels [24]. Because point clouds or meshes aren’t +in a standard format, most researchers convert them to 3D +voxel grids or collections of images (e.g., views) before +feeding them to a deep net architecture, which produces +enormous data that obscures natural invariances. As a +result, we concentrate on an alternative input representa- +tion for 3D geometry — point clouds – and call the deep +nets PointNet[3]. Point clouds are easy to understand be- +cause they are basic and unified structures that avoid the +combinatorial irregularities and complexities of meshes. +There could be adversarial assaults by placing dynamic +noise on the input due to the widespread use of PointNet[3] +and PointNet++[2] in perception modules of autonomous +vehicles and robotics. Because tiny changes in the input +could cause the network’s accuracy and robustness prop- +erties to be violated at different layers (2). Because of +these concerns about employing models in safety-critical +applications due to their opacity, formally measuring the +robustness of a trained PointNet is critical. +The majority of existing approaches focus on verifying the +safety and robustness properties of feedforward neural net- +works (FNN) with the Rectified Linear Unit activation func- +tion (ReLU). There are several different approaches such +as Mixed Integer Linear Programming (MILP) [5, 12, 13], +Satisfiability (SAT), and Satisfiability Modulo Theory +(SMT) techniques [7, 10], Optimization [6, 9], Geometric +Reachability [20, 23] etc. Most of these works focus on +2D convolutional Neural Networks. All of these existing +approaches use L0 distance between two images. Their +optimization-based approach computes a tight bound on +the number of pixels that may be changed in an image +arXiv:2301.11806v1 [cs.CV] 27 Jan 2023 + +PCV: A point cloud-based network verifier +Perception +Localization +Downstream +Adaptation +Planning +Control +Figure 1: High-Level flow of an Autonomous System +without affecting the classification result of the network. +These approaches do not fit with point cloud-based data +distribution. Overlapping points in the foreground bound- +ing box, will create adversarial examples to improve the +robustness of the network. +In this project, we implement a verifier for point cloud- +based network model. Proposed method does not pro- +vide the robustness in terms of number of points that are +allowed to be changed (L0 distance), attacks by distur- +bances, bounded with arbitrary linear constraints. These +approaches are applied to CNN-based network verification. +Even if we applied this to a point cloud-based network +with a variety of measurements, it will be a novel approach. +Rather, we add disturbance bounded with signed gradi- +ents and clipping into foreground bounding box points. In +addition, we add the reachability properties of a Region +Pooling Layer for each validation set. It is a set-based +analysis method by detecting the correctness and robust- +ness properties. The representation can be used as a set +of distorted points by an adversarial attack into the input +domain. Target is to construct the reachable set of outputs +from an adversarial attack that are used to reason about +the overall robustness of the network. When a PoinNet- +based network violates the robustness property, let’s say +for detecting overlapping objects (e.g. pedestrian riding +bi-cycle), an exact reachability scheme will construct a set +of concrete adversarial examples. The contribution of this +project is as follows. +• PCV, a framework for point cloud network model +based on efficient reachability analysis. +• Verification of correctness and robustness prop- +erties with the set of reachable objects that are +considered adversarial objects. +• Implementation of PCV with reachability algo- +rithm based on the over-approximate method +• Release the generated adversarial dataset for the +future benchmark. +2 +Background +2.1 +Point Clouds +Point clouds refer to a set of points in space and these +points represent the 3D shape of the object. Cartesian +coordinates(X, Y, Z) are used to define the point position. +Along with these coordinates point clouds data might in- +clude other information related to objects such as height, +width, and length. An example of point clouds data from +the ModelNet dataset is shown in figure 4 Point clouds +are usually generated from a 3D laser scanner and Li- +DAR (light detection and ranging) technology. There are +several point clouds based dataset such as ModelNet[22], +ShapeNet[1], ScanNet[4], ApolloCar3D[18], PartNet[14] +etc.. Point cloud data is used in construction, highway +planning, engineering, developing a self-driving car, aug- +mented virtual reality, and housekeeping robots. +However, its irregular format of data is highly inconvenient +to work with typical convolutional architecture as it re- +quires a regular format of the input. To overcome this issue, +researchers transform these point clouds or meshes into +image grids or 3D voxels which are regular formats. Image +grids or multi-view-based methods turn these unstructured +point clouds data into 2D images, while volumetric-based +method converts point clouds into 3D volumetric repre- +sentations. Then the researchers can apply existing 2D or +3D convolutional networks which might cost the loss of +information. On the other hand, point-based methods such +as PointNet[3], PointNet++[2] use direct point cloud data +without any voxelization or projection. These methods do +not cause any explicit information loss. PointNet can learn +pointwise features and use the max pooling layer to gather +global features. PointNet++ is a hierarchical network to +detect fine geometric structures from the neighborhood of +each point. However, a point-based method like PointNet +can not detect overlapping objects which is a violation of +the robustness property. For example, similar to Figure 3, +we have a point clouds data of a biker riding a bike. This +PointNet-based network might not detect this point clouds +data as human and bike separately. Then this example data +can be considered as an adversarial example. Similarly, +Figure 5 shows that distortion of points in the point clouds +can be a way of adversarial attack. +2 + +PCV: A point cloud-based network verifier +Nx3 +Nx3 +Nx64 +Nx1024 +Nx64 +N x1088 +shared +shared +Nx128 +shared +NxM +mlp (512,256,128) +mlp(128,M) +Output Scores +Point Features +shared +Input +Transform +mlp (64,64) +Global Features +Output Scores +mlp (64,128,1024) +Max +Pool +1024 +mlp (512,256,k) +Feature +Transform +T-Net +Matrix +Multiply +3x3 +Transform +Input Points +T-Net +Matrix +Multiply +64x64 +Transform +Classification Network +Segmentation Network +Figure 2: Architecture of Pointnet [3] +Figure 3: Point clouds data of two overlapping objects which might be detected as a single object by PointNet +Figure 4: An example of Point Clouds data from ModelNet dataset: bed +3 + +0.2 +Z +0 +0.2 +0.5 +0 +0.5 +- +0.4 +0.2 +0PCV: A point cloud-based network verifier +Figure 5: This is a bird’s eye view of LiDAR data. Objects inside the green boxes are cars. Here points are distorted. +This opens a door for the adversarial attack +. +2.2 +Target model: PointNet +PointNet[3] directly takes point clouds as input and classi- +fies the entire input. Also, it can do per-point segmentation +or labeling. The basic architecture includes points with +three coordinates(x, y, z). In figure 2 we can see two net- +works: Classification and Segmentation networks. The +classification network takes n points as input. There are +transformations: input and feature transformation which +are applied to these points respectively. This task is done +by a mini network which is called T-net. An affine transfor- +mation matrix is determined by this T-net and then directly +applied to the coordinates of the points. Then there is +an aggregation of point features using the Max Pooling +operation in order to make the model invariant to input +permutation. In the output, we get classification scores for +k classes. The segmentation network is an extension of +the classification network, concatenating global and local +features of the points to get per-point scores. PointNet is +an ideal model to consider as the target model because of +its simplicity and primitive design. With this, we can set +a standard for verifying a model which has a common 3D +vision task. +We face several difficulties while working with this model. +We can not find any pertained model with onnx format. +The original model is implemented in TensorFlow by the +authors of the paper [3]. However, we have looked for +the pytorch implementation of PointNet although there are +some API conflicts for pytorch. We have to communicate +with the authors to get the pytorch source. However, we +have to spend a significant amount of time modifying the +architecture to get it in the onnx format. The actual ar- +chitecture is complicated. That is why we simply remove +the segmentation network from our implementation of the +verifier. +3 +Problem Definition +The robustness of a machine learning model defines how +well a model is performing to classify or cluster the object. +As a significant part of the training, data will come from +multiple sources and we do not have full control over that. +Therefore, it is very important to have robustness metrics +for constant monitoring of the model, so that we can get a +flag when it fails to result in a certain level of confidence. +We are proposing a robustness framework consisting of the +noise calibration module, target model, and robustness ver- +ifier. The final goal is to produce an adversarial set along +with feature-wise impact analysis that is causing lower +confidence in classification for the target model used. +Let’s define the robustness properties with respect to a +model f : Rm → Rk. Let X denote the input space +and Y indicate the input and output spaces, respectively, +and let D be a random distribution over X from which +input items are taken. Because we’re dealing with a multi- +label instance, it can be labeled with many values where +Y = {−1, +1}k. The learner is given a labeled sample +S = +� +(x1, y1), (x2, y2), ..., (xm, ym) +� +∈ (X ×Y)m with +x1, ..., xm drawn according to D, and yi = f(xi) for all +i ∈ [1, m], where f : X → Y is the target labeling +function. If we add a noise C with X, then +���X − C +�� ≤ T +� +r1 ← f(x) +r2 ← f(x − c) +� +class(r1) = class(r2) +� +Here T is the tripping point, after that the model will fail +to predict and class(r1) ̸= class(r2) +4 + +PCV: A point cloud-based network verifier +Figure 6: Block Diagram of PCV +We consider the reachability of a PointNet P that con- +sists of a series of layers L that include fully connected +layers, max-pooling layers, average pooling layers, region +pooling layers, and ReLU activation layers. Mathemati- +cally, we define a PointNet with p layers as P = Li, i = +1, 2, 3, ..., p. The reachability of the PointNet P is defined +based on the concept of reachable sets corresponding to a +linear set I. +RL1 +∆= y1|y1 = L1(x), x ∈ I +RL2 +∆= y2|y2 = L2(y1), y1 ∈ RL1 +RL3 +∆= y3|y3 = L3(y2), y2 ∈ RL2 +· · · +RP = RLp +∆= yp|yp = Lp(yp−1), yp−1 ∈ RLp−1 +The definition shows that the reachable set of the +PointNet P can be constructed layer-by-layer. The core +computation is constructing the reachable set of each layer +Li defined by a specific operation. +So, for reachable set RL1,2,...,p, the final output yp should +hold the true value of postcondition i.e. the detected object +with higher precision. +4 +PCV Overview +In Figure 6, we have shown a basic block diagram of PCV. +It will output the verification state of each layer whether +the property is satisfied or not. At the same time, it will +have an adversarial set generator. There will be multiple +modules. Property Generator will generate the possible +properties for the input layer of the model. As our target +is to verify each layer, we will have a property translator +from the second layer and onwards. Property Translator +plays a vital role in collecting the state of each output of +the previous layer from the network segregator and loops +back to the property generator for the new property of the +downstream layers. Layer Verifier (LVi), gets properties +for each layer to verify the output for the reachability set. +It will run the algorithm of exact and over-approximate +to ensure the correctness and robustness properties. Our +primary plan is to take the onnx and PTH versions of Point- +Net and make a compatible verifier. There are two main +components of PCV: Noise Model and Robustness Verifier. +4.1 +Noise Model +The noise Model is the first basic component of PCV. How +well a model can handle noisy data, define how robust +the model is. The hybrid perturbation method adds noise +into three stages. In the first stage, for PCV noise genera- +tor, we can define the function that creates the adversarial +examples by perturbing the original inputs. The hybrid per- +turbation function takes three inputs, original clean input +(x), ‘epsilon’ is the element-wise perturbation amount (ϵ), +and data_grad is the gradient of the loss w.r.t the input +(∇xJ(θ, x, y)) and η is the noise function. After that same +ϵ factor of Gaussian noise will be added to the perturbed +input. Every real point will have neighbor noise points that +have an impact during the convolution operation. Finally, +a clipping operation will reset the value. The function then +creates perturbed input as +perturbed_input = original_input ++ epsilon ∗ sign(data_grad) += x + ϵ ∗ sign(∇xJ(θ, x, y)) += η(perturbed_input) += Q(perturbed_input) +5 + +L1 +Trained +Network +PointNet +Segregator +L2 +[SAT, +Layer Verifier +UNSAT, +(AT) +UNKNOWN) +Ln +Property +Adversarial Set +Input Set +Property Translator +Generator +GeneratorPCV: A point cloud-based network verifier +Figure 7: Model Performance with original validation data +Figure 8: Architecture of Robustness Verifier +In order to maintain the original range of the data, the +perturbed input is clipped Q() to range [0, 1]. This hybrid +method generates more robust perturbation than FGSM[8] +4.2 +Robustness Verifier +The overall goal is to create the intended misclassification +with the least amount of disruption to the input data. The +enemy only cares about the output classification is incor- +rect, not the new classification. The concept is simple: +instead of altering weights based on backpropagated gra- +dients to minimize loss, the attack alters the input data to +increase loss based on the same backpropagated gradients. +In other words, the attack maximizes the loss by using the +gradient of the loss in relation to the input data. +To clear the confusion, we state that all the perturbation +process is done in the verification dataset. All models are +trained with the original training set. There are multiple +stages of the verification process that are handled by dif- +ferent modules. After the training process, we saved a +trained model. The verification process starts by loading +the trained model and generating the Model Output from +the verification dataset. It has two parts. At first, we calcu- +late the model Verification Accuracy and save it in a data +structure. In parallel, we have to calculate the verification +accuracy with perturb input. To do that, we Calculate +Loss from the model output and Reset Gradients. Loss +Backward is an important step for resetting the model state. +In the next stage, hybrid perturbation is started. There +are three different parameters: trained model, epsilon, and +Data grad. In the beginning, we generate signs from the +data grid and change the gradient steps by multiplying +the input with epsilon. After that, add these two results +and regenerate the Perturb Input. In addition, we add an +epsilon factor of random noise to perturb input and exploit +the property for the model. Finally, a clipped function +is applied to perturb input and output of the final perturb +verification dataset. +With that perturb input, we calculate the model output and +generate Perturb Verification Accuracy. The robustness +6 + +Model Output +Calculate Loss +Reset Gradient +Loss Backward +Adversarial Set +Verification +Robustness +Accuracy +Comparator +Data Grad +Perturb Input +Generator +Random Noise +Perturb PC +Generator +Perturb +Perturb Model +Verification +Output +AccuracyPCV: A point cloud-based network verifier +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +Epsilon +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Accuracy +Accuracy vs Epsilon +Figure 9: Impact on accuracy due to increase of epsilon factors on PointNet +comparator will take two verification accuracy as input +and results from the impact of noise into deviation of the +accuracy. Based on the logic, if the verification results go +down to the tipping point, we append that perturb dataset +into Adversarial Set. We define the tipping point based +on the standard of accuracy for a model. If it is an n-class +classification problem, then below 50% is the safest place +to count on. Because there will be at least 2 classes for a +multi-class model. +4.3 +Verifier Algorithm +We have presented the core ideas for the robustness analy- +sis of PointNet. The adversarial set is constructed by every +input with a certain factor of perturbation for which the +accuracy of output drops. The full algorithm has two parts. +We present the Algorithm-1 for step by step verification +process of every single input dataset. It takes trained Point- +Net, epsilon, and validation dataset as input and returns +the threshold tipping and adversarial data set. Algorithm-2 +is the process of generating noisy data into three stages. +The robustness verification function uses it for generating +perturbed input data. HY BRIDP also takes three param- +eters as input: single point cloud frame, epsilon, and the +gradients generated from the original set data. The epsilon +factor is used for both steps to maximize loss and for gen- +erating Gaussian noise. These unique factors make the +perturbation process more robust. +Algorithm 1 Robustness Verification in PointNet +Input: PNet, ϵ, V Set +Output: T {...}, A{...} +for all ϵ do +for all V Set_datafromV Set do +data, target ← V Set_data +data.requires_grad ← TRUE +output = model(data) +i_pred ← output.max(1, keepdim = True) +if i_pred ! = target then +I_CC+ = f_pred.item().sum() +end if +loss = F.nll_loss(output, target) +model.zero_grad() +loss.backward() +data_grad = data.grad.data +p_data = HY BRID_P(data, ϵ, data_grad) +output = model(p_data) +f_pred ← output.max(1, keepdim = True) +if i_pred ! = f_pred then +A{} ← V Set_data +else +F_CC+ = f_pred.item().sum() +end if +i_acc = I_CC/len(V Set) +f_acc = F_CC/len(V Set) +if f_acc ≤ i_acc × 50% then +T {} ← f_acc +end if +end for +end for +return T {...}, A{...} +7 + +PCV: A point cloud-based network verifier +Algorithm 2 HYBRID_P() Perturbation process into input +data +Input: x, ϵ, data_grad +Output: perturb_x +s_dg, ← data_grad.sign() +perturb_x = x + ϵ × s_dg +perturb_x = η(perturb_x) +perturb_x = Q(perturb_x) +return perturb_x +5 +Evaluation +5.1 +Experimental Setup +We have used PointNet[3] as the target model to verify. +There are two parts: classification and segmentation. In +the current scope, the verification module works only for +the classification part. We trained PointNet with the Mod- +elNet dataset for 20 epochs with 10 batch sizes. It has 10 +classes of point cloud data. Training and validation data +sizes are 3991 and 908. After training with clean data, +we saved the model into ’onnx’ and ’pth’ format. Our +development environment configuration was Ubuntu 20.04 +with Intel Core i9, 64GB physical memory, and Titan XP +GPU with 12GB integrated memory. It took nearly 4 hours +to train the whole model. For executing the robustness +process with 7 different epsilon values, it took nearly 1hr +15 minutes. +5.2 +Experiment Results +We re-sampled the input into [64, 3, 1024], where 64 is +the input dimension with 1024 sampled point clouds that +have 3-dimensional(x, y, z) coordinates. The validation +function is the project’s most important outcome. Each +call to this test function runs the ModelNet test sets in +full and reports the final accuracy. This function, how- +ever, also accepts an epsilon input. Because the validation +function reports the accuracy of a model under attack from +an adversary with strength ϵ, this is the case. The func- +tion computes the gradient of the loss w.r.t. the input +data (data_grad), generate a perturbed image with "gra- +dient_sign_p" (perturbed_data), and then tests to see if +the perturbed example is adversarial for each sample in the +test set. The function saves and returns several successful +adversarial samples in addition to verifying the model’s +correctness. +The final step in the process is to actually run the assault. +For each epsilon value in the ‘epsilons’ input, we conduct +a whole test step. We also reserve the final accuracy and a +few successful adversarial samples for each epsilon, which +will be plotted in the following sections. As the epsilon +value grows, the printed accuracies drop. Also, keep in +mind that the epsilon = 0 condition represents the initial +test accuracy without any attacks. An increase of epsilon, +means an increase in step size to maximize the loss. That +means accuracy will be decreased. We will analyze the +pattern of decreasing accuracy. +In Figure-7, the validation results in accuracy for each +class with 908 validation inputs. Although we developed a +custom PointNet model, with only 10 epochs of training, +validation shows good results. That is our base accuracy +for verifying the robustness. We try to find the robustness +threshold T when the accuracy goes down less than 50% +for PointNet. +5.3 +Results Analysis +For the verification of robustness, we plot the accuracy ver- +sus epsilon. The target is to find the tipping point where the +model will fail to predict the desired classification. As pre- +viously stated, as epsilon increases, test accuracy should +decrease. This is because greater epsilons indicate that +we are taking a larger step in the direction of maximum +loss. Even though the epsilon numbers are linearly spaced, +the curve’s trend is not linear. For instance, in Figure-9, +while the accuracy at epsilon = 0.05 is just roughly 5% +lower than epsilon = 0, the accuracy at epsilon = 0.2 +is 20% lower than epsilon = 0.15. Also, the model’s +accuracy for a 10-class classifier between epsilon = 0.25 +and epsilon = 0.3 hits random accuracy. +The test accuracy drops as epsilon grows, making the per- +turbations more visible. In reality, an attacker must weigh +the tradeoff between accuracy degradation and perceptibil- +ity. At each epsilon value, we demonstrate some examples +of effective adversarial examples in Figure-10. The epsilon +value for each row of the figure is different. The first row +shows the epsilon = 0 samples, which are the clean pho- +tos that have not been altered. The initial classification → +adversarial classification is shown in the title of each im- +age. At epsilon = 0.15, the perturbations become visible, +and at epsilon = 0.3, they are fairly visible. Despite the +increased noise, humans are still capable of selecting the +correct class in all circumstances. From Figure-9, when +the value of ϵ = 0.20 the accuracy falls to 43.1%, which is +below 50%. Tipping pointT for that specific adversarial +input (ϵ = 0.20) +We have submitted our project files into the github reposi- +tory: https://github.com/arupcsedu/PCV +5.4 +Limitations and Future Work +Due to complicated network architectures, we had to +rewrite the PointNet and removed the segmentation part. +PCV only works for the classification module. On top of +PointNet, PointNet++ is currently state of art with the con- +cept of hierarchical convolution, and PCV is not designed +to verify that. It is the first verifier for 3D vision and we +plan to support n-dimensional convolution in the future. In +addition, any bounding box-based object detection mod- +els (e.g., PointRCNN[15]), has more than 3D dimension +input. Height(h), Width(w), Length(l), Orientation(θ) +are important dimension for real time 3D object detection. +8 + +PCV: A point cloud-based network verifier +Figure 10: Impact of noise on Point Clouds data +The KITTI dataset has that support. PCV only supports +(x, y, z) coordinates of data(e.g., ModelNet or ShapeNet). +But by design, PCV can be customized with a more com- +plex and robust dataset (e.g KITTI, Argoverse). In the +next phase, we plan to add verification of the segmentation +module with PointNet. There are other 3D object detection +models such as PointRCNN, PointNet++, VoxelNet[25] +where Voxelnet has voxel 3D data. PCV can be extended +for these models. +6 +Related Work +In our proposal, we mentioned using ImageStar[19], a +set-based framework for the verifier of CNN. The set- +based representation and reachability algorithm is built +in NNV. We could not use this verifier later due to some +challenges. One of the major complications is its imple- +mentation which is available in MatLab and there are no +full guidelines to use it given that it is a very new verifier. +It was way complicated to convert this MatLab implemen- +tation to PyTorch implementation. Also, there is no source +available for PointNet in MatLab. +Other than ImageStar we try different existing verifiers. +For this task we take the help of DNNV [16], an open- +source tool supporting 13 neural network verifiers with +extensive documentation. This DNNV framework makes +the life of the DNN verifier researchers and developers +easier by standardizing the inputs and output formats. For +property, DSL is used. In figure 12, the architecture of this +tool is explained. It takes the network in ONNX format as +input, along with a property file written in Domain Specific +Language DNNP and the name of the verifier. DNNV then +translates the network format and property to be applica- +ble for the target verifier and finally gives an output in a +standard format. To verify our PointNet model we use +two verifiers Neurify and Marabou. Neurify [21] gives a +tight output bound of a network for a given input range. +It is also applicable to a larger network. Marabou[11] is +an extension of Reluplex [10] algorithm. Marabou pro- +vides native support for fully connected and convolutional +DNNs with arbitrary piecewise-linear activation functions. +However, we fail to verify our Pointnet model in both ver- +ifiers. We got the error message “NotImplementedError: +Non-2D convolutions are not supported". This is because +DNNV currently only supports 2d convolutions and we +have identified that at least one of the convolutions in our +model is not 2d. This is because our point cloud input has +3-dimensional coordinates as x,y,z. None of the verifiers +will run because DNNV does not support the network. +We use another tool DNNF[17] which is based on falsi- +fication. A falsification is a complementary approach to +verification. It tries to find out the violations of the prop- +erties. It can sometimes give output more quickly than +verifiers when the property is false. DNNF can transform +the correctness problem to equivalent sets of adversarial +robustness problems using reduction11. We decide to try +this tool because DNNF supports a wide range of convo- +lutional layers. However, we also fail here to run DNNF +with the same error message as DNNV. +9 + +0.5 +-0.5 +0.4 +0.2 +0? +y +AA +0.5 +0.5 +0.5PCV: A point cloud-based network verifier +Reduction +Falsifiers +SAT/UNKNOWN +DNNF +ONNX +DNNF +Safety +Robustness +Figure 11: Architecture of DNNF [17] +Translate +Input +Property in +DNNF +Reduced Property +Simplify Network +Run Verifier +Translate +Output +SAT +UNSAT +UNKNOWN +Network in +ONNX +DNNV +Figure 12: Architecture of DNNV [16] +7 +Conclusions +The correctness and Robustness properties of a Machine +Learning model play a vital role to handle unknown and +corner cases in large input domains. PCV is the first Point- +Cloud Network verifier for the robustness of the network +and successfully verifies the model properties by gener- +ating adversarial sets. Our proposed hybrid perturbation +technique can exploit the properties of the model and com- +promise it. 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In Proceed- +ings of the IEEE conference on computer vision and +pattern recognition, pages 4490–4499, 2018. +11 + diff --git a/EtFKT4oBgHgl3EQfaC59/content/tmp_files/load_file.txt b/EtFKT4oBgHgl3EQfaC59/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..00b8297244292a35716f120440ac84bff9245eef --- /dev/null +++ b/EtFKT4oBgHgl3EQfaC59/content/tmp_files/load_file.txt @@ -0,0 +1,629 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf,len=628 +page_content='PCV: A POINT CLOUD-BASED NETWORK VERIFIER Arup Kumar Sarker Department of Computer Science University of Virginia Charlottesville, VA 22903 djy8hg@virginia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='edu Farzana Yasmin Ahmad Department of Computer Science University of Virginia Charlottesville, VA 22903 fa7sa@virginia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='edu Matthew B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Dwyer Department of Computer Science University of Virginia Charlottesville, VA 22903 matthewbdwyer@virginia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='edu January 30, 2023 ABSTRACT 3D vision with real-time LiDAR-based point cloud data became a vital part of autonomous system research, especially perception and prediction modules use it for object classification, segmentation, and detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Despite their success, point cloud-based network models are vulnerable to multiple adversarial attacks, where certain factor of changes in the validation set causes significant performance drop in well-trained networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Most of the existing verifiers work perfectly on 2D convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Due to complex architecture, dimension of hyper-parameter, and 3D convolution, no verifiers can perform the basic layer-wise verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' It is difficult to conclude the robustness of a 3D vision model without performing the verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Because there will be always corner cases and adversarial input that can compromise the model’s effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' In this project, we describe a point cloud-based network verifier that successfully deals state of the art 3D classifier PointNet and verifies the robustness by generating adversarial inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' We have used extracted properties from the trained PointNet and changed certain factors for perturbation input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' We calculate the impact on model accuracy versus property factor and can test PointNet networks’ robustness against a small collection of perturbing input states resulting from adversarial attacks like the suggested hybrid reverse signed attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' The experimental results reveal that the resilience property of PointNet is affected by our hybrid reverse signed perturbation strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' 1 Introduction The point cloud is an important type of geometric data structure percept from the LiDAR in the autonomous sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' In 1, the perception module process the data and detect 3D object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' The planning tasks get the prediction results along with localization and send the driving pol- icy to control so that control can send it to actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' To implement weight sharing and other kernel optimizations in perception, typical convolutional architectures require extremely regular input data formats, such as picture grids and 3D voxels [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Because point clouds or meshes aren’t in a standard format, most researchers convert them to 3D voxel grids or collections of images (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=', views) before feeding them to a deep net architecture, which produces enormous data that obscures natural invariances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' As a result, we concentrate on an alternative input representa- tion for 3D geometry — point clouds – and call the deep nets PointNet[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Point clouds are easy to understand be- cause they are basic and unified structures that avoid the combinatorial irregularities and complexities of meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' There could be adversarial assaults by placing dynamic noise on the input due to the widespread use of PointNet[3] and PointNet++[2] in perception modules of autonomous vehicles and robotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Because tiny changes in the input could cause the network’s accuracy and robustness prop- erties to be violated at different layers (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Because of these concerns about employing models in safety-critical applications due to their opacity, formally measuring the robustness of a trained PointNet is critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' The majority of existing approaches focus on verifying the safety and robustness properties of feedforward neural net- works (FNN) with the Rectified Linear Unit activation func- tion (ReLU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' There are several different approaches such as Mixed Integer Linear Programming (MILP) [5, 12, 13], Satisfiability (SAT), and Satisfiability Modulo Theory (SMT) techniques [7, 10], Optimization [6, 9], Geometric Reachability [20, 23] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Most of these works focus on 2D convolutional Neural Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' All of these existing approaches use L0 distance between two images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Their optimization-based approach computes a tight bound on the number of pixels that may be changed in an image arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='11806v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='CV] 27 Jan 2023 PCV: A point cloud-based network verifier Perception Localization Downstream Adaptation Planning Control Figure 1: High-Level flow of an Autonomous System without affecting the classification result of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' These approaches do not fit with point cloud-based data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Overlapping points in the foreground bound- ing box, will create adversarial examples to improve the robustness of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' In this project, we implement a verifier for point cloud- based network model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Proposed method does not pro- vide the robustness in terms of number of points that are allowed to be changed (L0 distance), attacks by distur- bances, bounded with arbitrary linear constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' These approaches are applied to CNN-based network verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Even if we applied this to a point cloud-based network with a variety of measurements, it will be a novel approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Rather, we add disturbance bounded with signed gradi- ents and clipping into foreground bounding box points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' In addition, we add the reachability properties of a Region Pooling Layer for each validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' It is a set-based analysis method by detecting the correctness and robust- ness properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' The representation can be used as a set of distorted points by an adversarial attack into the input domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Target is to construct the reachable set of outputs from an adversarial attack that are used to reason about the overall robustness of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' When a PoinNet- based network violates the robustness property, let’s say for detecting overlapping objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' pedestrian riding bi-cycle), an exact reachability scheme will construct a set of concrete adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' The contribution of this project is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' PCV, a framework for point cloud network model based on efficient reachability analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Verification of correctness and robustness prop- erties with the set of reachable objects that are considered adversarial objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Implementation of PCV with reachability algo- rithm based on the over-approximate method Release the generated adversarial dataset for the future benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' 2 Background 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='1 Point Clouds Point clouds refer to a set of points in space and these points represent the 3D shape of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Cartesian coordinates(X, Y, Z) are used to define the point position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Along with these coordinates point clouds data might in- clude other information related to objects such as height, width, and length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' An example of point clouds data from the ModelNet dataset is shown in figure 4 Point clouds are usually generated from a 3D laser scanner and Li- DAR (light detection and ranging) technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' There are several point clouds based dataset such as ModelNet[22], ShapeNet[1], ScanNet[4], ApolloCar3D[18], PartNet[14] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='. Point cloud data is used in construction, highway planning, engineering, developing a self-driving car, aug- mented virtual reality, and housekeeping robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' However, its irregular format of data is highly inconvenient to work with typical convolutional architecture as it re- quires a regular format of the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' To overcome this issue, researchers transform these point clouds or meshes into image grids or 3D voxels which are regular formats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Image grids or multi-view-based methods turn these unstructured point clouds data into 2D images, while volumetric-based method converts point clouds into 3D volumetric repre- sentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Then the researchers can apply existing 2D or 3D convolutional networks which might cost the loss of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' On the other hand, point-based methods such as PointNet[3], PointNet++[2] use direct point cloud data without any voxelization or projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' These methods do not cause any explicit information loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' PointNet can learn pointwise features and use the max pooling layer to gather global features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' PointNet++ is a hierarchical network to detect fine geometric structures from the neighborhood of each point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' However, a point-based method like PointNet can not detect overlapping objects which is a violation of the robustness property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' For example, similar to Figure 3, we have a point clouds data of a biker riding a bike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' This PointNet-based network might not detect this point clouds data as human and bike separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Then this example data can be considered as an adversarial example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Similarly, Figure 5 shows that distortion of points in the point clouds can be a way of adversarial attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' 2 PCV: A point cloud-based network verifier Nx3 Nx3 Nx64 Nx1024 Nx64 N x1088 shared shared Nx128 shared NxM mlp (512,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='256,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='128) mlp(128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='M) Output Scores Point Features shared Input Transform mlp (64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='64) Global Features Output Scores mlp (64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='1024) Max Pool 1024 mlp (512,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='256,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='k) Feature Transform T-Net Matrix Multiply 3x3 Transform Input Points T-Net Matrix Multiply 64x64 Transform Classification Network Segmentation Network Figure 2: Architecture of Pointnet [3] Figure 3: Point clouds data of two overlapping objects which might be detected as a single object by PointNet Figure 4: An example of Point Clouds data from ModelNet dataset: bed 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='2 Z 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='2 0PCV: A point cloud-based network verifier Figure 5: This is a bird’s eye view of LiDAR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Objects inside the green boxes are cars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Here points are distorted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' This opens a door for the adversarial attack .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='2 Target model: PointNet PointNet[3] directly takes point clouds as input and classi- fies the entire input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Also, it can do per-point segmentation or labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' The basic architecture includes points with three coordinates(x, y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' In figure 2 we can see two net- works: Classification and Segmentation networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' The classification network takes n points as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' There are transformations: input and feature transformation which are applied to these points respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' This task is done by a mini network which is called T-net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' An affine transfor- mation matrix is determined by this T-net and then directly applied to the coordinates of the points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Then there is an aggregation of point features using the Max Pooling operation in order to make the model invariant to input permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' In the output, we get classification scores for k classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' The segmentation network is an extension of the classification network, concatenating global and local features of the points to get per-point scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' PointNet is an ideal model to consider as the target model because of its simplicity and primitive design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' With this, we can set a standard for verifying a model which has a common 3D vision task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' We face several difficulties while working with this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' We can not find any pertained model with onnx format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' The original model is implemented in TensorFlow by the authors of the paper [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' However, we have looked for the pytorch implementation of PointNet although there are some API conflicts for pytorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' We have to communicate with the authors to get the pytorch source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' However, we have to spend a significant amount of time modifying the architecture to get it in the onnx format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' The actual ar- chitecture is complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' That is why we simply remove the segmentation network from our implementation of the verifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' 3 Problem Definition The robustness of a machine learning model defines how well a model is performing to classify or cluster the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' As a significant part of the training, data will come from multiple sources and we do not have full control over that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Therefore, it is very important to have robustness metrics for constant monitoring of the model, so that we can get a flag when it fails to result in a certain level of confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' We are proposing a robustness framework consisting of the noise calibration module, target model, and robustness ver- ifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' The final goal is to produce an adversarial set along with feature-wise impact analysis that is causing lower confidence in classification for the target model used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Let’s define the robustness properties with respect to a model f : Rm → Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Let X denote the input space and Y indicate the input and output spaces, respectively, and let D be a random distribution over X from which input items are taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Because we’re dealing with a multi- label instance, it can be labeled with many values where Y = {−1, +1}k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' The learner is given a labeled sample S = � (x1, y1), (x2, y2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=', (xm, ym) � ∈ (X ×Y)m with x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=', xm drawn according to D, and yi = f(xi) for all i ∈ [1, m], where f : X → Y is the target labeling function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' If we add a noise C with X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' then ���X − C �� ≤ T � r1 ← f(x) r2 ← f(x − c) � class(r1) = class(r2) � Here T is the tripping point,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' after that the model will fail to predict and class(r1) ̸= class(r2) 4 PCV: A point cloud-based network verifier Figure 6: Block Diagram of PCV We consider the reachability of a PointNet P that con- sists of a series of layers L that include fully connected layers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' max-pooling layers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' average pooling layers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' region pooling layers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' and ReLU activation layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Mathemati- cally, we define a PointNet with p layers as P = Li, i = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=', p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' The reachability of the PointNet P is defined based on the concept of reachable sets corresponding to a linear set I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' RL1 ∆= y1|y1 = L1(x), x ∈ I RL2 ∆= y2|y2 = L2(y1), y1 ∈ RL1 RL3 ∆= y3|y3 = L3(y2), y2 ∈ RL2 · · RP = RLp ∆= yp|yp = Lp(yp−1), yp−1 ∈ RLp−1 The definition shows that the reachable set of the PointNet P can be constructed layer-by-layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' The core computation is constructing the reachable set of each layer Li defined by a specific operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' So, for reachable set RL1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=',p, the final output yp should hold the true value of postcondition i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' the detected object with higher precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' 4 PCV Overview In Figure 6, we have shown a basic block diagram of PCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' It will output the verification state of each layer whether the property is satisfied or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' At the same time, it will have an adversarial set generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' There will be multiple modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Property Generator will generate the possible properties for the input layer of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' As our target is to verify each layer, we will have a property translator from the second layer and onwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Property Translator plays a vital role in collecting the state of each output of the previous layer from the network segregator and loops back to the property generator for the new property of the downstream layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Layer Verifier (LVi), gets properties for each layer to verify the output for the reachability set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' It will run the algorithm of exact and over-approximate to ensure the correctness and robustness properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Our primary plan is to take the onnx and PTH versions of Point- Net and make a compatible verifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' There are two main components of PCV: Noise Model and Robustness Verifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='1 Noise Model The noise Model is the first basic component of PCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' How well a model can handle noisy data, define how robust the model is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' The hybrid perturbation method adds noise into three stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' In the first stage, for PCV noise genera- tor, we can define the function that creates the adversarial examples by perturbing the original inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' The hybrid per- turbation function takes three inputs, original clean input (x), ‘epsilon’ is the element-wise perturbation amount (ϵ), and data_grad is the gradient of the loss w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='t the input (∇xJ(θ, x, y)) and η is the noise function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' After that same ϵ factor of Gaussian noise will be added to the perturbed input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Every real point will have neighbor noise points that have an impact during the convolution operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Finally, a clipping operation will reset the value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' The function then creates perturbed input as perturbed_input = original_input + epsilon ∗ sign(data_grad) = x + ϵ ∗ sign(∇xJ(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' y)) = η(perturbed_input) = Q(perturbed_input) 5 L1 Trained Network PointNet Segregator L2 [SAT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Layer Verifier UNSAT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' (AT) UNKNOWN) Ln Property Adversarial Set Input Set Property Translator Generator GeneratorPCV: A point cloud-based network verifier Figure 7: Model Performance with original validation data Figure 8: Architecture of Robustness Verifier In order to maintain the original range of the data,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' the perturbed input is clipped Q() to range [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' This hybrid method generates more robust perturbation than FGSM[8] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='2 Robustness Verifier The overall goal is to create the intended misclassification with the least amount of disruption to the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' The enemy only cares about the output classification is incor- rect, not the new classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' The concept is simple: instead of altering weights based on backpropagated gra- dients to minimize loss, the attack alters the input data to increase loss based on the same backpropagated gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' In other words, the attack maximizes the loss by using the gradient of the loss in relation to the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' To clear the confusion, we state that all the perturbation process is done in the verification dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' All models are trained with the original training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' There are multiple stages of the verification process that are handled by dif- ferent modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' After the training process, we saved a trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' The verification process starts by loading the trained model and generating the Model Output from the verification dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' It has two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' At first, we calcu- late the model Verification Accuracy and save it in a data structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' In parallel, we have to calculate the verification accuracy with perturb input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' To do that, we Calculate Loss from the model output and Reset Gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Loss Backward is an important step for resetting the model state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' In the next stage, hybrid perturbation is started.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' There are three different parameters: trained model, epsilon, and Data grad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' In the beginning, we generate signs from the data grid and change the gradient steps by multiplying the input with epsilon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' After that, add these two results and regenerate the Perturb Input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' In addition, we add an epsilon factor of random noise to perturb input and exploit the property for the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Finally, a clipped function is applied to perturb input and output of the final perturb verification dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' With that perturb input, we calculate the model output and generate Perturb Verification Accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' The robustness 6 Model Output Calculate Loss Reset Gradient Loss Backward Adversarial Set Verification Robustness Accuracy Comparator Data Grad Perturb Input Generator Random Noise Perturb PC Generator Perturb Perturb Model Verification Output AccuracyPCV: A point cloud-based network verifier 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='30 Epsilon 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='0 Accuracy Accuracy vs Epsilon Figure 9: Impact on accuracy due to increase of epsilon factors on PointNet comparator will take two verification accuracy as input and results from the impact of noise into deviation of the accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Based on the logic, if the verification results go down to the tipping point, we append that perturb dataset into Adversarial Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' We define the tipping point based on the standard of accuracy for a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' If it is an n-class classification problem, then below 50% is the safest place to count on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Because there will be at least 2 classes for a multi-class model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='3 Verifier Algorithm We have presented the core ideas for the robustness analy- sis of PointNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' The adversarial set is constructed by every input with a certain factor of perturbation for which the accuracy of output drops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' The full algorithm has two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' We present the Algorithm-1 for step by step verification process of every single input dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' It takes trained Point- Net, epsilon, and validation dataset as input and returns the threshold tipping and adversarial data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Algorithm-2 is the process of generating noisy data into three stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' The robustness verification function uses it for generating perturbed input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' HY BRIDP also takes three param- eters as input: single point cloud frame, epsilon, and the gradients generated from the original set data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' The epsilon factor is used for both steps to maximize loss and for gen- erating Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' These unique factors make the perturbation process more robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Algorithm 1 Robustness Verification in PointNet Input: PNet, ϵ, V Set Output: T {.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='}, A{.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='} for all ϵ do for all V Set_datafromV Set do data, target ← V Set_data data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='requires_grad ← TRUE output = model(data) i_pred ← output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='max(1, keepdim = True) if i_pred !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' = target then I_CC+ = f_pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='item().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='sum() end if loss = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='nll_loss(output, target) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='zero_grad() loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='backward() data_grad = data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='grad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='data p_data = HY BRID_P(data, ϵ, data_grad) output = model(p_data) f_pred ← output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='max(1, keepdim = True) if i_pred !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' = f_pred then A{} ← V Set_data else F_CC+ = f_pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='item().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='sum() end if i_acc = I_CC/len(V Set) f_acc = F_CC/len(V Set) if f_acc ≤ i_acc × 50% then T {} ← f_acc end if end for end for return T {.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='}, A{.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='} 7 PCV: A point cloud-based network verifier Algorithm 2 HYBRID_P() Perturbation process into input data Input: x, ϵ, data_grad Output: perturb_x s_dg, ← data_grad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='sign() perturb_x = x + ϵ × s_dg perturb_x = η(perturb_x) perturb_x = Q(perturb_x) return perturb_x 5 Evaluation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='1 Experimental Setup We have used PointNet[3] as the target model to verify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' There are two parts: classification and segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' In the current scope, the verification module works only for the classification part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' We trained PointNet with the Mod- elNet dataset for 20 epochs with 10 batch sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' It has 10 classes of point cloud data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Training and validation data sizes are 3991 and 908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' After training with clean data, we saved the model into ’onnx’ and ’pth’ format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Our development environment configuration was Ubuntu 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='04 with Intel Core i9, 64GB physical memory, and Titan XP GPU with 12GB integrated memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' It took nearly 4 hours to train the whole model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' For executing the robustness process with 7 different epsilon values, it took nearly 1hr 15 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='2 Experiment Results We re-sampled the input into [64, 3, 1024], where 64 is the input dimension with 1024 sampled point clouds that have 3-dimensional(x, y, z) coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' The validation function is the project’s most important outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Each call to this test function runs the ModelNet test sets in full and reports the final accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' This function, how- ever, also accepts an epsilon input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Because the validation function reports the accuracy of a model under attack from an adversary with strength ϵ, this is the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' The func- tion computes the gradient of the loss w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' the input data (data_grad), generate a perturbed image with "gra- dient_sign_p" (perturbed_data), and then tests to see if the perturbed example is adversarial for each sample in the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' The function saves and returns several successful adversarial samples in addition to verifying the model’s correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' The final step in the process is to actually run the assault.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' For each epsilon value in the ‘epsilons’ input, we conduct a whole test step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' We also reserve the final accuracy and a few successful adversarial samples for each epsilon, which will be plotted in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' As the epsilon value grows, the printed accuracies drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Also, keep in mind that the epsilon = 0 condition represents the initial test accuracy without any attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' An increase of epsilon, means an increase in step size to maximize the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' That means accuracy will be decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' We will analyze the pattern of decreasing accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' In Figure-7, the validation results in accuracy for each class with 908 validation inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Although we developed a custom PointNet model, with only 10 epochs of training, validation shows good results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' That is our base accuracy for verifying the robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' We try to find the robustness threshold T when the accuracy goes down less than 50% for PointNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='3 Results Analysis For the verification of robustness, we plot the accuracy ver- sus epsilon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' The target is to find the tipping point where the model will fail to predict the desired classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' As pre- viously stated, as epsilon increases, test accuracy should decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' This is because greater epsilons indicate that we are taking a larger step in the direction of maximum loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Even though the epsilon numbers are linearly spaced, the curve’s trend is not linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' For instance, in Figure-9, while the accuracy at epsilon = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='05 is just roughly 5% lower than epsilon = 0, the accuracy at epsilon = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='2 is 20% lower than epsilon = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Also, the model’s accuracy for a 10-class classifier between epsilon = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='25 and epsilon = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='3 hits random accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' The test accuracy drops as epsilon grows, making the per- turbations more visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' In reality, an attacker must weigh the tradeoff between accuracy degradation and perceptibil- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' At each epsilon value, we demonstrate some examples of effective adversarial examples in Figure-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' The epsilon value for each row of the figure is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' The first row shows the epsilon = 0 samples, which are the clean pho- tos that have not been altered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' The initial classification → adversarial classification is shown in the title of each im- age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' At epsilon = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='15, the perturbations become visible, and at epsilon = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='3, they are fairly visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Despite the increased noise, humans are still capable of selecting the correct class in all circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' From Figure-9, when the value of ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='20 the accuracy falls to 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='1%, which is below 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Tipping pointT for that specific adversarial input (ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='20) We have submitted our project files into the github reposi- tory: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='com/arupcsedu/PCV 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='4 Limitations and Future Work Due to complicated network architectures, we had to rewrite the PointNet and removed the segmentation part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' PCV only works for the classification module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' On top of PointNet, PointNet++ is currently state of art with the con- cept of hierarchical convolution, and PCV is not designed to verify that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' It is the first verifier for 3D vision and we plan to support n-dimensional convolution in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' In addition, any bounding box-based object detection mod- els (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=', PointRCNN[15]), has more than 3D dimension input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Height(h), Width(w), Length(l), Orientation(θ) are important dimension for real time 3D object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' 8 PCV: A point cloud-based network verifier Figure 10: Impact of noise on Point Clouds data The KITTI dataset has that support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' PCV only supports (x, y, z) coordinates of data(e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=', ModelNet or ShapeNet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' But by design, PCV can be customized with a more com- plex and robust dataset (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='g KITTI, Argoverse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' In the next phase, we plan to add verification of the segmentation module with PointNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' There are other 3D object detection models such as PointRCNN, PointNet++, VoxelNet[25] where Voxelnet has voxel 3D data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' PCV can be extended for these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' 6 Related Work In our proposal, we mentioned using ImageStar[19], a set-based framework for the verifier of CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' The set- based representation and reachability algorithm is built in NNV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' We could not use this verifier later due to some challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' One of the major complications is its imple- mentation which is available in MatLab and there are no full guidelines to use it given that it is a very new verifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' It was way complicated to convert this MatLab implemen- tation to PyTorch implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Also, there is no source available for PointNet in MatLab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Other than ImageStar we try different existing verifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' For this task we take the help of DNNV [16], an open- source tool supporting 13 neural network verifiers with extensive documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' This DNNV framework makes the life of the DNN verifier researchers and developers easier by standardizing the inputs and output formats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' For property, DSL is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' In figure 12, the architecture of this tool is explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' It takes the network in ONNX format as input, along with a property file written in Domain Specific Language DNNP and the name of the verifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' DNNV then translates the network format and property to be applica- ble for the target verifier and finally gives an output in a standard format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' To verify our PointNet model we use two verifiers Neurify and Marabou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Neurify [21] gives a tight output bound of a network for a given input range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' It is also applicable to a larger network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Marabou[11] is an extension of Reluplex [10] algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Marabou pro- vides native support for fully connected and convolutional DNNs with arbitrary piecewise-linear activation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' However, we fail to verify our Pointnet model in both ver- ifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' We got the error message “NotImplementedError: Non-2D convolutions are not supported".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' This is because DNNV currently only supports 2d convolutions and we have identified that at least one of the convolutions in our model is not 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' This is because our point cloud input has 3-dimensional coordinates as x,y,z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' None of the verifiers will run because DNNV does not support the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' We use another tool DNNF[17] which is based on falsi- fication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' A falsification is a complementary approach to verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' It tries to find out the violations of the prop- erties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' It can sometimes give output more quickly than verifiers when the property is false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' DNNF can transform the correctness problem to equivalent sets of adversarial robustness problems using reduction11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' We decide to try this tool because DNNF supports a wide range of convo- lutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' However, we also fail here to run DNNF with the same error message as DNNV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='2 0?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' y AA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='5PCV: A point cloud-based network verifier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='Reduction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='Falsifiers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='SAT/UNKNOWN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='DNNF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='ONNX ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='DNNF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='Safety ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='Robustness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='Figure 11: Architecture of DNNF [17] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='Translate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='Input ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='Property in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='DNNF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='Reduced Property ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='Simplify Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='Run Verifier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='Translate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='Output ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='SAT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='UNSAT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='UNKNOWN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='Network in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='ONNX ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='DNNV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='Figure 12: Architecture of DNNV [16] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='Conclusions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='The correctness and Robustness properties of a Machine ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='Learning model play a vital role to handle unknown and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content='corner cases in large input domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' PCV is the first Point- Cloud Network verifier for the robustness of the network and successfully verifies the model properties by gener- ating adversarial sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Our proposed hybrid perturbation technique can exploit the properties of the model and com- promise it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' These adversarial sets create a state of the art examples for any future 3D model verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' Although we have played with basic properties to verify the network, there is still significant scope to verify other non-linear properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' In the next stages, we will extend this verifier for the more complex model in real-time 3D vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} +page_content=' References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFKT4oBgHgl3EQfaC59/content/2301.11806v1.pdf'} 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a/ItE1T4oBgHgl3EQf_wb4/content/tmp_files/2301.03585v1.pdf.txt b/ItE1T4oBgHgl3EQf_wb4/content/tmp_files/2301.03585v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..482766e04f6e32cf214c868b4eae0a69f64e7bde --- /dev/null +++ b/ItE1T4oBgHgl3EQf_wb4/content/tmp_files/2301.03585v1.pdf.txt @@ -0,0 +1,1545 @@ +Refining Network Message Segmentation with +Principal Component Analysis +Stephan Kleber +Institute of Distributed Systems +Ulm University +Ulm, Germany +ORCID 0000-0001-9836-4897 +Frank Kargl +Institute of Distributed Systems +Ulm University +Ulm, Germany +ORCID 0000-0003-3800-8369 +Abstract—Reverse engineering of undocumented protocols is +a common task in security analyses of networked services. The +communication itself, captured in traffic traces, contains much +of the necessary information to perform such a protocol reverse +engineering. The comprehension of the format of unknown +messages is of particular interest for binary protocols that are +not human-readable. One major challenge is to discover probable +fields in a message as the basis for further analyses. Given a set of +messages, split into segments of bytes by an existing segmenter, +we propose a method to refine the approximation of the field +inference. We use principle component analysis (PCA) to discover +linearly correlated variance between sets of message segments. +We relocate the boundaries of the initial coarse segmentation to +more accurately match with the true fields. We perform different +evaluations of our method to show its benefit for the message +format inference and subsequent analysis tasks from literature +that depend on the message format. We can achieve a median +improvement of the message format accuracy across different +real-world protocols by up to 100 %. +Index Terms—network reconnaissance, protocol reverse engi- +neering, vulnerability research +I. INTRODUCTION +Analyzing the threat posed by botnet and malware com- +munication [4], validating the correct and secure design and +implementation of network services [29], and efficiently con- +figuring smart fuzzers [8] requires the understanding of the +involved network protocols. In case of malware and propri- +etary systems, the protocols are often undocumented and first +require protocol reverse engineering (PRE) to uncover data +exfiltration or vulnerabilities in the network services. As an +example, PRE recently played an important role in discovering +a severe vulnerability in the proprietary Apple Wireless Direct +Link (AWDL) protocol stack [25]. The vulnerability led to +a zero-click exploit [1] affecting all of Apple’s iOS-based +product lines and could be fixed due to the protocol analysis. +Samples of unknown protocols can typically be collected +from observing communication of devices implementing this +protocol. PRE can make use of these traffic traces to infer the +specification of the unknown network protocol and thus to gain +knowledge about its syntax and behavior. The approximation +of protocol fields for an unknown message syntax is required +to determine the message format, semantics of the fields’ data +representations, and message types. Among others, protocols +used for embedded systems often are optimized for efficiency +and thus transmit binary data instead of ASCII-encoded, +human-readable values. The latter are called textual protocols. +Most existing PRE approaches only support textual protocols +using techniques from natural language processing (NLP) [16, +28]. This requires repeated keywords and separators to search +for in the structure of the messages. The analysis of binary +protocols that do not exhibit such structural features is con- +siderably harder than of textual protocols [2, 5, 6, 11]. +Widespread early approaches designed for binary proto- +cols [11] use sequence alignment [3, 5]. Designed for bio- +informatics, it solves the problem of inferring structure from +a small number of sequences of amino acids. The challenge is +reduced by additional knowledge about the chemical properties +of the sequences to be aligned. For binary protocols, a large +number of messages, i. e., sequences, of one protocol is +beneficial for observing the variability of values. The lack of +conclusive properties to guide the alignment process and the +larger number of sequences to be compared pose a major ob- +stacle for applying sequence alignment to network protocols. +Therefore, more recent approaches rely on statistical variance +analysis which offers improved performance. We previously +proposed NEMESYS [10], one such statistical variance analy- +sis for the message format inference of binary protocols, and +a set of refinement methods to improve the accuracy of the +approximation of message fields as part of NEMETYL [9]. +Based on these previous works, the main contributions +of this paper are two novel methods for segmenting and +refining message formats of binary protocols: We propose a +segmentation refinement and also derive a new segmenter +that works independent of NEMESYS. While previous work +typically requires to select protocol-dependent analysis param- +eters, which is difficult for an unknown protocol, our refine- +ment and segmenter are independent of any parameters. We +use the message format quality measure FMS [10] to compare +the existing refinements [9] to the methods of this paper. The +evaluation results show that, in most cases, we can improve +the quality of the segmentation alone by about 50 % across +different protocols, and we can significantly improve analyses +relying on this segmentation, like message type identification +©2020 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including +reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or +reuse of any copyrighted component of this work in other works. +IEEE Conference on Communications and Network Security 2022. + +and semantic analysis. From the evaluation results, we deduce +which of the presented methods is suited best for different +tasks of PRE. +II. RELATED WORK +Several surveys [6, 11, 18, 23] discuss the plethora of +approaches of PRE and also have proposed to structure the +overall PRE process into different phases [6, 11]. These phases +are data collection into traces, feature extraction, message +format inference, message type identification, semantic deduc- +tion, and behavior model reconstruction. As previous work +showed, support for performing the tasks of the phases on +binary protocols is severely limited in comparison to textual +protocols [2, 5, 6, 11]. While most phases are well covered +in literature, approaches for segmenting messages have gained +only limited attention despite its relevance for all subsequent +phases. We adopt the term “segment” from previous work. A +segment is a subsequence of bytes of a message that approxi- +mates a field’s boundaries from the protocol definition in terms +of byte positions. In an optimal inference, the segments match +the true field boundaries from the unknown syntax. +Netzob [3], Discoverer [5], and others [15] deduce fields +as a by-product of sequence alignment with the already +mentioned disadvantages. Existing statistical methods either +require an already existing segmentation [3, 5, 15] or expect +field boundaries at globally fixed positions [2, 26, 27], limiting +the applicability to protocols specifically designed without +variable length fields. If meta-data and common offsets of +values in messages are available, the task is as simple as find- +ing the corresponding or correlating values in the messages. +For instance, analyzing wireless communication of a medical +device or sensor node, there is no encapsulation present from +which to extract addresses and identifiers. Thus, we argue that +no convincing method exists for accurately inferring structure +of binary protocols [11]. +Opposed to all previous approaches, we make very few +assumptions, thus, creating a more generic approach. It does +not require a specific message format or protocol structure, +like globally common byte offsets or typical field lengths. +It works without any preceding classification of messages or +the identification of flows, and thus also works for protocols +without any encapsulation like TCP, UDP, and IP1. The last +property is of particular interest when reverse engineering +proprietary wireless point-to-point protocols were no encap- +sulation is present or accessible, like such used by mobile +devices [25] and for medical devices [17, 20], simple Internet- +of-Things nodes [19], or vehicle sensors, like tire-pressure +monitoring systems [21]. +In +this +paper, +we +propose +refinements +that +enhance +NEMESYS [10] and we compare our novel segmenter to the +refinements from NEMETYL [9]. Our approach uses a concept +for the clustering of segments that we proposed earlier [12]. +1Transmission Control Protocol (RFC 793), User Datagram Protocol (RFC +768), and Internet Protocol (RFC 791) +Thus, we briefly introduce these three fundamental approaches +in the rest of this section. +We utilize the NEMESYS segmentation [10] for a first, +raw approximation of the true message structure without +knowledge of the specification. The method derives segments +from the distribution of value changes within subsequent bytes +of single messages. Unlike sequence alignment or statistical +methods, NEMESYS does not compare different messages +with each other. It is known that NEMESYS boundaries regu- +larly exhibit an off-by-one error [10]. We confirmed for all of +our test protocols that the vast majority of NEMESYS’ “near- +match” boundary errors are such off-by-one errors. We tried +to find patterns in these errors that would enable us to create +a systematic correction but could not find constant off-by-one +error rules for NEMESYS segments. Therefore, we propose +our refinement to correct this type of errors dynamically. +The approach NEMETYL [9] identifies message types from +clustering of messages by their similarity. It additionally +contains refinements to apply to the segments obtained from +NEMESYS to more correctly approximate fields. NEMETYL +proposes a simple frequency analysis to determine the most +common segment values, which point out probable field +boundaries throughout the protocol under analysis. Further- +more, NEMETYL recognizes char sequences embedded within +the binary protocol. +To obtain sets of comparable, related segments, which we +subsequently intend to analyze together, we cluster segments +based on their dissimilarity. We base our clustering approach +on our previous proposal [12] to use DBSCAN [7]. The +used measure is the Canberra dissimilarity [9] which extends +the better-known Canberra distance [14] to vectors of differ- +ing dimensions. We use the Canberra dissimilarity between +segments as affinity measure to guide the clustering. The +clustering results in concise sets of segments that overlay +best, matching each others’ values measured by the smallest +possible Canberra dissimilarity of segments. +III. BYTE-WISE SEGMENT VARIANCE ANALYSIS +The evaluation of different aspects of NEMESYS [9, 10, +12] has proven that it can yield useful approximations of the +protocol message structure of unknown protocols. It is highly +efficient and refinements could improve some shortcomings +of NEMESYS’ raw segmentation. However, none of these +proposed enhancements take variance of values within and +between messages into account. +Inter- and intra-message variance reveals details about +the message structure that is not visible otherwise and has the +potential to increase the accuracy of inferred field boundaries. +We determine which bytes vary together in sets of segments +that we interpret as data vectors to identify probable field +boundaries. We separate vector components, variance-locked +to each other, from linearly independent vector components +by Principal Component Analysis (PCA). PCA shows which +bytes from each set of segments are associated with each other +and, thus, belong to a common field. + +Raw segments +Recursive clustering +Section III-B +PCA +Section III-A +Boundary adjustment +Section III-C +Refined segments +Figure 4 +Fig. 1: PCA process overview +We first describe the core of our variance analysis based +on PCA, in Section III-A. As illustrated in Figure 1, before +we can start the PCA, we need to prepare groups of segments +by a recursive clustering step (Section III-B) and after the +PCA, we interpret its results (Section III-C) to adjust the +boundaries of the raw input segments. The whole method, +which we call PCA refinement, is embedded in a chain of other +refinements, partly from related work and partly by methods +we introduce in this paper. We locate all refinements in the +context of the processing chain in Section V. +A. Principal Component Analysis +The PCA calculates the multivariate main direction of +variance and its scale. The multivariate variance identifies +the components that vary together, i. e., linearly dependent. +Typically, PCA is applied to feature vectors and its result is +used to classify samples, represented by these vectors, based +on commonalities in the variance revealed by the PCA. In +contrast, we use the immediate result of a PCA to determine +the contribution of different components to the variance of the +vectors and thus the variance contained in a set of segments +represented by the vectors. +We first revisit the elements of the well-known generic +PCA to clarify our usage of terms and notations. The data +vectors are collected into a matrix X with each vector as +row, in our case corresponding to the bytes of one segment. +The variance of X is described by the covariance matrix C. +The equations in Figure 2 give an example for matrices X +and C derived from segments. The eigenvalues (“factors” or +“component scores” λ) and eigenvectors (“loadings” wi) of the +covariance matrix C are the foundation of the PCA. The first +X = +� +� +� +� +� +� +� +� +00 +08 +50 +00 +02 +01 +08 +90 +00 +04 +01 +08 +90 +00 +07 +01 +08 +b0 +00 +02 +02 +90 +40 +01 +02 +02 +90 +40 +01 +02 +01 +08 +80 +00 +04 +01 +08 +80 +00 +04 +� +� +� +� +� +� +� +� +(1) +C = +� +� +� +0.41 +34 +−9.71 +0.25 +−0.19 +34 +3963 +−2020 +29.14 +−53.42 +−9.71 +−2020 +1737 +−14.85 +34.85 +0.25 +29.14 +−14.85 +0.21 +−0.39 +−0.19 +−53.42 +34.85 +−0.39 +3.12 +� +� +� +(2) +Fig. 2: Example for X, with each row representing one +segment’s sequence of byte values, and the C of this X. +principal component i +λ +0 +1 +2 +3 +4 +0 +1 000 +2 000 +3 000 +4 000 +5 000 +6 000 +significant PCs +• +• +• +• +• +knee +Fig. 3: Scree graph of PCs sorted by their eigenval- +ues/component scores λi. +principal component (PC) is the highest eigenvalue λ and its +associated eigenvector wi and it intuitively states the direction +of the prevalent variance in the data. Further PC loadings +are orthogonal to the previous ones. We call the variance +at a specific byte position the notable contribution if the +ith eigenvector component wi is significantly different from +zero. According to common analysis methods, the transition +between the PCs with a significant contribution and negligible +components is marked by the knee of the scree graph of the +eigenvalues λ as illustrated in Figure 3. We determine the knee +by the Kneedle algorithm [22]. +To be able to start a PCA, two basic prerequisites must be +met. First, we need to prepare a set of segments that contain +related data, so that the PCA does not measure arbitrary vari- +ance, but only meaningfully comparable variances of segments +that represent the same kind of data. Second, we need to +determine which PCs significantly contribute to the variance +of the data, i. e., the segments represented in X. +1) Overlaying Segment Vectors and Calculating PCs: PCA +requires an existing coarse approximation of fields, i. e., initial +segments, to overlay the segments that are related in one +set. This overlaying is necessary to calculate the covariance +from it. We use the Canberra dissimilarity [9] to find the best +fitting overlay of multiple segments. Thus, we superimpose +the segments at the most probable useful offsets, resulting in +overlays of the most meaningfully comparable message parts. +While similar to the longest common subsequence [24], we +additionally allow for variations instead of requiring identical +subsequences. As Figure 4 illustrates, we next calculate the +covariance matrix C from the aligned segment data X. Then + +Segments per +initial cluster +Vectors from +byte values +Overlay segments on +Canberra dissimilarity +Covariance +matrix +Eigenvalues +and -vectors +PCA +Fig. 4: PCA preparation process performed per initial cluster. +we calculate the eigenvalues λ and eigenvectors wi of C, +which the PCA uses directly. +2) Process Overview: The PCA is embedded into a process +that prepares the segment data and interprets the PCA result. +Introduced in Figure 1 in high-level abstraction, the process +is composed of the following steps in detail: +1) Cluster the raw segments by similarity, +2) interpret the segment byte values as vectors, overlay the +vectors per cluster, calculate the covariance, and from it +the eigenvalues and eigenvectors that are required for the +subsequent PCA (summarized in Figure 4), +3) check if prerequisites for PCA are met by each cluster, +4) for clusters that fail this check, recurse from step 1, +5) for clusters that pass this check, perform the PCA, and +6) finally apply rules for which variance characteristics +quantified by the PCA indicate field boundaries. +The rest of this section explains the auxiliary algorithms for the +preparation and interpretation of the PCA. These algorithms +make use of thresholds and other parameters for which we +empirically determined suitable values (Table I). +B. Recursive Clustering +Before performing the PCA, we obtain sets of comparable, +related segments by clustering the segments [12]. To adjust +the clusters optimally for applying PCA, we recursively cluster +segments and check whether the prerequisites for a component +analysis are met by each cluster. If a cluster does not allow to +conduct a PCA, we sub-cluster it and test whether the smaller +clusters result in more suitable sets of segments. If PCA is still +not applicable to a cluster, we recurse the sub-clustering on the +respective cluster, otherwise we stop the recursion for this sub- +cluster branch and perform the PCA on it (see Section III-A). +We estimate whether a cluster is adequate for PCA by +PCA Prerequisites. The main criterion is that the variance is +systematic. We distinguish systematic from random variance +by the number of significant PCs. For a successful PCA only +a limited number of significant PCs are allowed, since only if +the variance is concentrated in a small number of PCs we can +deduce that the data is non-random. We define an absolute pp +and a relative maximum pq of allowed significant PCs. The +number of allowed significant PCs is exceeded if +|⟨λi : λi > qs⟩n +i=0| > min(pp, dim λ · pq) +with the threshold qs of any eigenvalue λi for a PC to count as +significant: qs = min(K(λ), 1 +10λ0, ps). If the condition is true +and thus the PCA will fail on the cluster, we further recurse +the sub-clustering to gain a subset of segments that then is +appropriate. We now interpret the PCA of each cluster. +C. Interpretation of the PCA +The clusters from the previous step that are suitable for PCA +are interpreted in two steps. First we apply inference rules for +field boundaries and then we determine commonly aligned +offsets within clusters that likely show additional boundaries. +For any interpretation, we use the common alignment of all +segments within a cluster, which we obtain as explained in +Section III-A1. This results in relative offsets that are common +throughout one cluster. Thus, the field-boundary inference +rules and the additional conditions for boundaries work on +this dissimilarity-aligned segments per cluster. +1) Inference Rules: The covariance shows transitions be- +tween related message parts in the byte sequences of a set of +segments provided by a cluster. To get an impression about +how the covariance matrix C represents such related parts, +regard Figure 5. The more intense the color, the stronger the +linear dependent variance at this offset. We use the loadings +w of significant PCs calculated from C to test for conditions +that represent characteristics of field boundaries. We deduce +two different rules from the typical data type representation in +byte streams of network messages. +a) Rule A: The first rule is governed by the observation +that the data of a field with a common data type exhibits a rise +in variance towards the field’s end [10]. Using the same cluster +as in Figure 5 as basis, Figure 6 illustrates this by the loadings +of the significant PCs for segments of 25 bytes length. A PC i +0 1 2 3 4 5 6 7 8 9101112131415161718192021222324 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +20 +21 +22 +23 +24 +7500 +5000 +2500 +0 +2500 +5000 +7500 +Fig. 5: Covariance matrix as heat map. Each row and each +column correspond to the relative offsets in the scope of the +overlayed and aligned set of segments of one cluster. White +lines are the relative positions of true boundaries in this set. + +1 0 1 2 3 4 5 6 7 8 9 10111213141516171819202122232425 +0.6 +0.4 +0.2 +0.0 +0.2 +0.4 +1, = 27733.1 +2, = 18219.4 +3, = 12390.1 +4, = 8538.7 +5, = 3826.7 +Fig. 6: Example for matches of rule A: significant loadings +of one cluster (with eigenvalue > 2 773). The dot denotes the +matches, the vertical lines the true boundaries. +is significant if it has an eigenvalue λi above the threshold qs +(Section III-B). We thus define a significant PC i to relevantly +contribute to the variance at a byte position k, as such with +its maximum loading mk > pr, with +mk = +n +max +i=1 ⟨|wi,k|⟩ +(3) +where n is the number of significant PCs i : λi > qs. +We now search for a considerable relative drop in the +absolute covariance after the peak at the end of a suspected +field. Thus, a field boundary is detected at byte position k if +the Boolean expression +mk−1 > pr +∧ +mk ≤ pr +∧ +mk−1 − mk +mk−1 +> pd (4) +is true, regardless of which λi is responsible for the peak. +b) Rule B: The second rule detects the start of a new +field in situations where a prolonged sequence of byte po- +sitions do not significantly vary and the next byte suddenly +peaks in its variance. This typically happens at the transition +between constant fields or such with few possible values, like +enumerations, commands, or flags. +The maximum loading m is defined as for Rule A in +Equation 3 and we introduce the thresholds pz to denote +loadings that are near zero, pb the length in bytes that a loading +needs to be below pz to trigger the rule, and pt the minimum +absolute value of a loading to constitute a notable contribution +to the variance. A variance peak after a prolonged sequence +of near constancy is thus defined to be at a byte position k +for which the following Boolean expression is true: +pb +� +j=1 +mk−j < pz +∧ +mk > pt +∧ +mk − mk−1 +mk +> pd +We apply both rules to all suitable clusters individually. We +move or add boundaries at the same relative offsets for all +segments in one cluster for which any of the rules apply. +2) Commonly Aligned Boundaries Conditions: To overlay +segments of a cluster, we dislocate the segments against each +other as described in Section III-A1, so that the sub-sequence +of one segment is at the same relative offset that is most similar +to the other segment. +If the overlaying shows a prevalent common boundary in +the aligned starts and ends of segments and if that boundary +is missing in only a minority of segments of the cluster, +we cut segments within one cluster at all relative offsets of +boundaries that are more common than their direct neighbor, +which originated from the raw segmentation or which where +detected by component analysis. The reason, we can detect +these errors in the raw segmentation, is that the alignment on +Canberra dissimilarity reveals segment relations. +IV. NULL-BYTES TRANSITIONS +The central goal of our work is to improve the accuracy of +the field inference of unknown messages. PCA requires a pre- +existing segmentation that supplies comparable data vectors. +We observed that null bytes, produced by unset fields, unused +most significant bytes of numbers, or null terminated strings, +denote probable field boundaries in binary protocols [10]. +Applied as a refinement for NEMESYS, we improve its raw +segments by relocating inferred boundaries near the beginning +and end of sequences of nulls: (1) If the last bytes before the +nulls fulfill the character heuristic [9], we assume it is a null- +termination for the string and allocate the nulls to the end of +the character segment; (2) otherwise, we assign the null-bytes +to the following segment, assuming they are the unset most +significant bytes of a number value. +As an alternative, we propose a novel standalone segmenter +that is not based on NEMESYS to compare this as a simpler +foundation to apply our PCA afterwards. Similar to the null- +bytes refinement for NEMESYS, the segments that our so- +called Null-Bytes Segmenter yields are the subsequences of the +messages which are delimited by null bytes. The segmentation +is only coarse but still adequate to prepare the fine-grained +field inference by PCA. +V. IMPLEMENTATION +We implemented a proof-of-concept in Python 3.2 We re- +quire numerous thresholds and other parameters throughout +our method, for which Table I provides an overview. To +empirically determine universal values of the fixed parameters +and thresholds our approach relies on, we used traces of +the binary network protocols DHCP, DNS, NBNS, NTP, and +SMB.3 These traces are publicly available.4 +Both, our novel method based on PCA for refining the +NEMESYS segmenter and our standalone Null-Bytes Seg- +menter, process raw segments in a chain of refinements. This +2https://github.com/vs-uulm/nemesys +3Dynamic Host Configuration Protocol (RFC 2131), Domain Name System +(RFC 1035), NetBIOS Name Service (RFC 1002), Network Time Protocol +(RFC 958), and Server Message Block +4DHCP, NBNS, NTP, and SMB extracted from http://download.netresec. +com/pcap/smia-2011/; DNS extracted from https://ictf.cs.ucsb.edu/archive/ +2010/dumps/ictf2010pcap.tar.gz + +concept is similar to the original segmenter’s and the refined +version of NEMETYL. Table II provides an overview of the +applied refinements from literature and our own approaches. +The refinement methods introduced by previous work are +the simple char detection heuristic in NEMESYS, which we +call MergeCharsv1, and the advanced char detection heuristic +in NEMETYL, MergeCharsv2. NEMETYL counts the most +frequent segment values and crops these from larger segments. +We call this refinement CropDistinct. NEMETYL also intro- +duced the splitting of the first segment of each message into +fixed chunks, which we denote SplitFixedv1. +The primary contributions of this paper are two novel meth- +ods for the refinement chain. The first one are the Null-Bytes +Refinement and Segmenter Section IV, which we abbreviate +by NullBytes, and the second one is the application of Principal +Component Analysis to guide segment refinements, PCA (see +Section III-A). In preparation of PCA, we propose Entropy- +Merge for merging of NEMESYS segments if they have similar +local entropy. We also add slight improvements for the splitting +of the first segment of each message, SplitFixedv2, and the +handling of cropping char segments, CropChar, which is based +on NEMETYL’s MergeCharsv2. +Using these refinement methods as elements for a process- +ing chain, we apply our improvements in two different ways +that we call NEMEPCA and NullPCA. We mark the placement +of our contributions in the novel processing chains by a bold +font in Table II. +VI. EVALUATION +Using our proof-of-concept implementation presented in +Section V, we evaluate our approach. We evaluate the quality +of the inferred segment boundaries with seven representative +binary network protocols. Beyond this directly measurable +effect on the segmentation, we show the impact of the refined +segments that our method yields by applying analyses from +previous work using the improved segments as starting point. +Thus, we perform message type identification as previous work +proposed in conjunction with segments from NEMESYS [9]. +Furthermore, we classify field data types using an existing +method, which we also proposed to work with segments from +NEMESYS [12]. +TABLE I: Parameters and empirically determined values for +our algorithms. +Parameter +Task +Value +Scree threshold +Sub-cluster +qs +Scree minimum +Sub-cluster +ps = 10 +Maximum significant principals +Sub-cluster +pp = 4 +Significant principals ratio +Sub-cluster +pq = 0.5 +Length difference ratio +Sub-cluster +pl = 0.5 +Minimum cluster size +Sub-cluster +pc = 6 +Significant w-contribution threshold +Field inference A +pr = 0.1 +Signific. loading-difference threshold +Field inference A +pd = 0.98 +Near-zero threshold +Field inference B +pz = 0.05 +Near-zero minimum length +Field inference B +pb = 4 +Notable w-contribution threshold +Field inference B +pt = 0.005 +Additionally to the protocols DHCP, DNS, NBNS, NTP, +and SMB that we used to select the parameters for our +algorithm (Section V), we also use our own traces of the +two proprietary protocols Apple Wireless Direct Link (AWDL) +and Auto Unlock (AU). AWDL is a Wi-Fi-based link-layer +protocol for peer-to-peer communication. AU implements a +proprietary distance bounding protocol.5 Both protocols were +undocumented until they recently have been manually reverse +engineered. The reverse-engineered specification of AWDL, +including a dissector, is publicly available [25], and we had +access to a private Wireshark dissector of the AU protocol. +Thus, both protocols are typical PRE use cases while we have +ground truth available. As the source of the ground truth, we +parse the Wireshark dissectors’ output for each message. All +evaluated protocols are binary, while DNS, DHCP, SMB, and +AWDL also contain embedded char sequences. The binary +fields of DNS, NBNS, and NTP have fixed length, while +DHCP, SMB, AWDL, and AU use a mix of fixed and variable- +length fields. DHCP, DNS, NBNS, SMB, AWDL, and AU +support varying numbers of fields in different messages, while +NTP has a fixed structure. Thus, our set of traces represents +a wide variety of protocol properties. +We compare the results of the three approaches NEMETYL- +refined baseline, NEMEPCA, and NullPCA which we de- +scribed in Section V. The baseline is the refinement as +described in NEMETYL and the other two are applications +with and without NEMESYS (Section IV), respectively, in +conjunction with our novel analysis method using PCA, in- +troduced in Section III. +A. Message Segmentation +The immediate effect we expect of our refinement is that +the segment boundaries will more accurately match the field +boundaries of the protocol specification. To measure this, we +use the Format Match Score (FMS) that we proposed together +with NEMESYS [10]. We apply our refinements to each of our +test protocols, calculate the FMS, and discuss the results in this +section. The FMS is a measure of correctness of the inference +of a message format. By the FMS, we can quantitatively +compare the quality of different inference methods. The FMS +is calculated for each message individually and we therefore +calculate the median of all FMS’ for one trace. +5https://support.apple.com/en-us/HT206995 +TABLE II: Refinement overview for the compared approaches. +Our contributions are printed in bold font. +NEMESYS +NEMETYL +NEMEPCA +NullPCA +Original [10] +Baseline [9] +[10] + Sec. III +Sec. IV + III +NEMESYS +NEMESYS +NEMESYS +– +MergeCharsv1 +MergeCharsv2 +EntropyMerge +NullBytes +CropDistinct +NullBytes +CropChars +SplitFixedv1 +CropChars +PCA +PCA +CropDistinct +CropDistinct +SplitFixedv2 +SplitFixedv2 + +Summarized in Table III, the NEMETYL [9] baseline per- +forms worst for DHCP and SMB, while our PCA refinement, +in contrast, stays lower for DHCP and AU, however at a +reasonable absolute value of above 0.35 and still improving +on NEMETYL’s DHCP results of below of 0.30. For all +other protocols, NEMEPCA clearly outperforms the baseline +except for NTP where the FMS values are closeby. The results +also show that traces of 1 000 and 100 messages are almost +identical, confirming that decent results can be produced with +even small traces. Compared to the baseline, our NEMEPCA +yields better results for 100 than for 1 000 DHCP, SMB, and +AWDL messages, showing that it is effective in extracting +more structural information even from small traces than the +previous approaches. +NEMESYS requires to select a value for σ that is dependent +of the field lengths expected for a protocol. The results for +the baseline show that the quality is significantly higher on +average for σ = 0.9. Our goal is to become independent from +this parameter, since it is difficult to determine it correctly. Per- +forming our NEMEPCA refinements on NEMESYS segments +with different σ values, we observe that the results are almost +identical for both σ values. The only protocol that declines +in quality is AU while all other protocols increase their field +correctness in terms of the FMS. Assuming that we know +nothing about the protocol that helps to select the optimal σ +value and thus blindly selecting σ = 1.2 for the baseline, we +improve the results by almost 100 % using NEMEPCA due to +its robustness against σ changes. +Finally, we used our novel Null-Bytes Segmenter that works +without NEMESYS and applied the PCA refinement to it as +described in Section V. The resulting FMS values on average +are considerably lower than those of NEMEPCA, but similar +to the baseline. This shows the advantage of using NEMESYS +as a heuristic method and that the complex effort of refining +its segmentation is worthwhile. +TABLE III: Comparison of message segmentation quality +using the median values of FMS per protocol trace. +Note: Numbers printed in bold in the protocol rows are the worst cases +discussed in the text and the bold median values at the bottom are the +mainly discussed comparison. +FMS median +baseline +NEMEPCA +NullPCA +trace +msg.s +σ 0.9 +σ 1.2 +σ 0.9 +σ 1.2 +DHCP +100 +0.29 +0.23 +0.38 +0.37 +0.21 +DHCP +1 000 +0.30 +0.28 +0.38 +0.35 +0.29 +DNS +100 +0.56 +0.48 +0.74 +0.74 +0.74 +DNS +1 000 +0.51 +0.49 +0.90 +0.90 +0.87 +NBNS +100 +0.52 +0.29 +0.66 +0.66 +0.57 +NBNS +1 000 +0.55 +0.32 +0.68 +0.68 +0.56 +NTP +100 +0.58 +0.72 +0.66 +0.71 +0.53 +NTP +1 000 +0.59 +0.71 +0.59 +0.71 +0.52 +SMB +100 +0.36 +0.29 +0.59 +0.61 +0.47 +SMB +1 000 +0.39 +0.30 +0.56 +0.52 +0.36 +AWDL +100 +0.45 +0.35 +0.62 +0.60 +0.35 +AWDL +768 +0.40 +0.33 +0.54 +0.52 +0.46 +AU +123 +0.60 +0.56 +0.41 +0.38 +0.23 +median +0.51 +0.33 +0.59 +0.61 +0.47 +B. Message Type Identification +We apply the baseline segmentation to identify message +types and compare these results with applying the NEMETYL +message type identification on top of our segment refinements. +Table IV shows the results measured in classification precision +and recall. They where calculated exactly as described in the +NEMETYL paper [9]. For simplicity, we only use σ = 1.2 for +the NEMESYS-based analyses: the baseline and NEMEPCA. +While the precision is almost optimal for all of the compared +approaches, the previous and our novel ones, Our NEMEPCA +can reach or outperform the baseline only for few traces. +NullPCA has a precision that is on average similar to +the others but with a recall that is notably lower. However, +NullPCA is able to outperform the precision for single protocol +traces of NTP and SMB that are particularly difficult for both, +the baseline and NEMEPCA. This is due to the structure of +these protocols that contain sequences of null bytes separating +segments which denote the message type. +Our method cannot provide groundbreaking improvements +for message type identification. This shows that a more ac- +curate field approximation is not necessarily improving the +message type identification. NullPCA is an alternative only +if the protocol for the most part contains segments that are +clearly separated by null bytes. +C. Field Data Type Clustering +The second analysis based on the refined segments that +we use as evaluation aspect is the clustering of field data +types from NEMESYS’ message segments [12]. Table V shows +our evaluation results by measuring precision, recall, the +number of segments that are considered noise by the clustering +algorithm, and the number of unknown segments. Segments +are considered unknown if no match to any field from the +ground truth is possible that would allow to determine the +field type for the segment. Thus, this value denotes one aspect +of the deviation of the heuristic segmentation from the true +field structure of the protocol. +TABLE IV: Message type identification quality measured by +precision (P) and recall (R). +baseline +NEMEPCA +NullPCA +trace +msg.s +P +R +P +R +P +R +DHCP +100 +0.94 +0.10 +0.98 +0.11 +0.96 +0.15 +DHCP +1 000 +1.00 +0.58 +0.99 +0.77 +1.00 +0.49 +DNS +100 +1.00 +0.45 +0.88 +0.41 +1.00 +0.59 +DNS +1 000 +1.00 +1.00 +1.00 +0.24 +1.00 +0.54 +NBNS +100 +0.99 +0.68 +0.98 +0.88 +0.99 +0.47 +NBNS +1 000 +0.78 +0.89 +0.80 +0.83 +0.89 +0.35 +NTP +100 +0.20 +0.87 +0.77 +0.18 +0.91 +0.14 +NTP +1 000 +0.52 +0.69 +0.51 +0.85 +0.52 +0.94 +SMB +100 +1.00 +0.28 +1.00 +0.28 +1.00 +0.25 +SMB +1 000 +0.89 +0.75 +0.92 +0.71 +1.00 +0.75 +AWDL +100 +1.00 +1.00 +1.00 +0.64 +1.00 +0.86 +AWDL +768 +1.00 +0.52 +1.00 +0.44 +1.00 +0.51 +AU +123 +1.00 +0.26 +1.00 +0.18 +1.00 +0.16 +median +1.00 +0.68 +0.98 +0.59 +1.00 +0.51 + +TABLE V: Field data type clustering quality in precision (P), recall (R), noise, and number of unknown segments (unk.). +baseline +NEMEPCA +NullPCA +trace +msg.s +P +R +noise +unk. +P +R +noise +unk. +P +R +noise +unk. +DHCP +100 +0.86 +0.51 +87 +187 +0.71 +0.24 +125 +79 +0.83 +0.30 +121 +75 +DHCP +1 000 +0.86 +0.19 +275 +972 +0.95 +0.47 +166 +192 +0.87 +0.78 +29 +93 +DNS +100 +0.88 +0.21 +10 +23 +0.97 +0.50 +27 +4 +1.00 +0.53 +31 +5 +DNS +1 000 +0.87 +0.21 +31 +30 +0.99 +0.97 +20 +5 +1.00 +0.92 +50 +6 +NBNS +100 +0.99 +0.85 +1 +19 +0.99 +0.52 +13 +19 +0.99 +0.89 +5 +4 +NBNS +1 000 +0.99 +0.80 +12 +34 +0.99 +0.58 +4 +14 +1.00 +0.94 +5 +290 +NTP +100 +0.81 +0.75 +93 +25 +0.99 +0.15 +108 +105 +knee detection failed +NTP +1 000 +0.69 +0.30 +2 128 +189 +0.89 +0.21 +731 +1 105 +0.44 +0.84 +36 +1 105 +SMB +100 +0.84 +0.12 +128 +181 +0.77 +0.22 +85 +80 +0.80 +0.20 +98 +72 +SMB +1 000 +0.37 +0.02 +692 +1 549 +0.67 +0.34 +268 +664 +0.89 +0.20 +439 +551 +AWDL +100 +0.74 +0.05 +270 +393 +0.63 +0.10 +192 +206 +0.73 +0.17 +22 +193 +AWDL +768 +0.80 +0.16 +261 +2 205 +0.45 +0.01 +929 +703 +0.84 +0.50 +44 +1 235 +AU +123 +1.00 +0.06 +662 +352 +0.99 +0.57 +193 +121 +0.99 +0.45 +89 +880 +median +0.86 +0.26 +90 +108 +0.96 +0.41 +97 +80 +0.89 +0.78 +36 +75 +Comparing the baseline to NEMEPCA, the precision stays +similar for most protocols, but we can increase the recall +regarding the median of all protocols by 57 %. The number of +unknown segments are reduced considerably as expected due +to improved segment accuracy. As an exception, the increased +unknowns of NTP account for the low recall of the traces. The +larger AWDL trace yields a high amount of noise and thus +the lowest precision and recall of all traces. The protocol is +highly complex and eludes our heuristic. However, on average, +we can improve precision and recall while reducing unknown +segments significantly. +Our other proposal, NullPCA, shows tripled recall and re- +duced noise and unknown segments compared to the baseline. +This comes at the cost of a slightly decreased precision, +however. Also, the usage of null bytes as primary segment +detection mechanism leads to the loss of details in the bound- +aries of tightly packed fields that NEMESYS can discern. +VII. CONCLUSION +In this paper, we presented a chain of refinements for two +segmenters, NEMESYS and our simple null-byte transition +detection. Besides the static rule-based algorithms Entropy- +Merge, NullBytes, CropChars, CropDistinct, and SplitFixed, +we introduce the novel, dynamic PCA method. It measures +the byte-wise variance of segment contents to increase the +accuracy of the field boundary detection in unknown bi- +nary protocols over previous work. Opposed to its common +application in classification tasks, we utilize PCA directly +to determine linearly dependent variance and use a novel +interpretation of these results to detect field boundaries from +a rough preliminary segmentation. +Our evaluation shows improved field inference quality of +up to 100 % for most protocols. Moreover, it renders any +parameter selection by guessing unnecessary. PCA is effective +in extracting more structural information even from small +traces than the previous approaches. Thus, our method notably +improves the field accuracy as measured by FMS over the +previous segmenters and their refinements. +As we discuss within our evaluation, our simple Null-Bytes +Segmenter, combined with PCA, provides slightly less accu- +racy than refined NEMESYS segments. Despite the original +assumption of PCA that it can correct off-by-one errors in +particular, the PCA refinement still works well enough for +segments generated only from the slicing of null-bytes. How- +ever, while more complex, using NEMESYS segments as basis +for further analyses outperforms this simpler segmenter. Thus, +we conclude that the improved inference accuracy makes the +relatively complex refinement of segments from NEMESYS +worthwhile. +Encrypted messages cannot be inferred by our method di- +rectly and require obtaining of plain-text traces. Furthermore, +PCA can only detect linearly dependent correlations and fails +for random and non-linearly correlated message values. A +larger set of input data improves the accuracy only slightly, +while the exponential memory complexity of our refinement +limits the practical applicability to traces of a little over +one thousand messages. Thus, small traces that contain high +variability of message values are preferable for our method. +We propose for future work to find more sophisticated rule sets +for deducing boundaries from relations between PCs. This will +improve the accuracy of the field boundary detection. +The inference of the message format is a crucial task in +protocol reverse engineering. Often performed during security +assessments, it provides knowledge about the placement of +field boundaries in protocol messages, which is necessary to +correlate information to field values. Among other use cases, +it helps to determine portions of the message to inspect, e. g., +by Fuzz testing, to identify features for the fingerprinting +of protocols and to train intrusion detection systems. 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Sija, Young-Hoon Goo, Kyu-Seok Shim, +Huru Hasanova, and Myung-Sup Kim. “A Survey of +Automatic Protocol Reverse Engineering Approaches, +Methods, and Tools on the Inputs and Outputs View”. +In: Security and Communication Networks 10.1 (Feb. +2018). Hindawi. +[24] +Temple F. Smith and Michael S. Waterman. “Identifica- +tion of Common Molecular Subsequences”. In: Journal +of Molecular Biology 147.1 (Mar. 1981). Elsevier. +[25] +Milan Stute, David Kreitschmann, and Matthias Hol- +lick. “One Billion Apples’ Secret Sauce: Recipe for +the Apple Wireless Direct Link Ad hoc Protocol”. In: +MobiCom ’18. 2018. +[26] +Fanghui Sun, Shen Wang, Chunrui Zhang, and Hongli +Zhang. “Unsupervised field segmentation of unknown +protocol messages”. In: Computer Communications 146 +(Oct. 2019). +[27] +Antonio Trifilo, Stefan Burschka, and Ernst Biersack. +“Traffic to Protocol Reverse Engineering”. In: CISDA. +2009. +[28] +Yipeng Wang, Xiao-Chun Yun, Muhammad Zubair +Shafiq, Liyan Wang, Alex X. Liu, Zhibin Zhang, Dan- +feng Yao, Yongzheng Zhang, and Li Guo. “A Semantics + +Aware Approach to Automated Reverse Engineering +Unknown Protocols”. In: ICNP. 2012. +[29] +Shameng Wen, Qingkun Meng, Chao Feng, and Chao- +jing Tang. “Protocol Vulnerability Detection Based on +Network Traffic Analysis and Binary Reverse Engineer- +ing”. In: PLOS ONE 12.10 (Oct. 2017). Public Library +of Science. + diff --git a/ItE1T4oBgHgl3EQf_wb4/content/tmp_files/load_file.txt b/ItE1T4oBgHgl3EQf_wb4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9dc2fbd219009f7c170eb3d9e559b53b041b81a3 --- /dev/null +++ b/ItE1T4oBgHgl3EQf_wb4/content/tmp_files/load_file.txt @@ -0,0 +1,761 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf,len=760 +page_content='Refining Network Message Segmentation with Principal Component Analysis Stephan Kleber Institute of Distributed Systems Ulm University Ulm, Germany ORCID 0000-0001-9836-4897 Frank Kargl Institute of Distributed Systems Ulm University Ulm, Germany ORCID 0000-0003-3800-8369 Abstract—Reverse engineering of undocumented protocols is a common task in security analyses of networked services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The communication itself, captured in traffic traces, contains much of the necessary information to perform such a protocol reverse engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The comprehension of the format of unknown messages is of particular interest for binary protocols that are not human-readable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' One major challenge is to discover probable fields in a message as the basis for further analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Given a set of messages, split into segments of bytes by an existing segmenter, we propose a method to refine the approximation of the field inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' We use principle component analysis (PCA) to discover linearly correlated variance between sets of message segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' We relocate the boundaries of the initial coarse segmentation to more accurately match with the true fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' We perform different evaluations of our method to show its benefit for the message format inference and subsequent analysis tasks from literature that depend on the message format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' We can achieve a median improvement of the message format accuracy across different real-world protocols by up to 100 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Index Terms—network reconnaissance, protocol reverse engi- neering, vulnerability research I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' INTRODUCTION Analyzing the threat posed by botnet and malware com- munication [4], validating the correct and secure design and implementation of network services [29], and efficiently con- figuring smart fuzzers [8] requires the understanding of the involved network protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' In case of malware and propri- etary systems, the protocols are often undocumented and first require protocol reverse engineering (PRE) to uncover data exfiltration or vulnerabilities in the network services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' As an example, PRE recently played an important role in discovering a severe vulnerability in the proprietary Apple Wireless Direct Link (AWDL) protocol stack [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The vulnerability led to a zero-click exploit [1] affecting all of Apple’s iOS-based product lines and could be fixed due to the protocol analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Samples of unknown protocols can typically be collected from observing communication of devices implementing this protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' PRE can make use of these traffic traces to infer the specification of the unknown network protocol and thus to gain knowledge about its syntax and behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The approximation of protocol fields for an unknown message syntax is required to determine the message format, semantics of the fields’ data representations, and message types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Among others, protocols used for embedded systems often are optimized for efficiency and thus transmit binary data instead of ASCII-encoded, human-readable values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The latter are called textual protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Most existing PRE approaches only support textual protocols using techniques from natural language processing (NLP) [16, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' This requires repeated keywords and separators to search for in the structure of the messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The analysis of binary protocols that do not exhibit such structural features is con- siderably harder than of textual protocols [2, 5, 6, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Widespread early approaches designed for binary proto- cols [11] use sequence alignment [3, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Designed for bio- informatics, it solves the problem of inferring structure from a small number of sequences of amino acids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The challenge is reduced by additional knowledge about the chemical properties of the sequences to be aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' For binary protocols, a large number of messages, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=', sequences, of one protocol is beneficial for observing the variability of values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The lack of conclusive properties to guide the alignment process and the larger number of sequences to be compared pose a major ob- stacle for applying sequence alignment to network protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Therefore, more recent approaches rely on statistical variance analysis which offers improved performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' We previously proposed NEMESYS [10], one such statistical variance analy- sis for the message format inference of binary protocols, and a set of refinement methods to improve the accuracy of the approximation of message fields as part of NEMETYL [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Based on these previous works, the main contributions of this paper are two novel methods for segmenting and refining message formats of binary protocols: We propose a segmentation refinement and also derive a new segmenter that works independent of NEMESYS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' While previous work typically requires to select protocol-dependent analysis param- eters, which is difficult for an unknown protocol, our refine- ment and segmenter are independent of any parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' We use the message format quality measure FMS [10] to compare the existing refinements [9] to the methods of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The evaluation results show that, in most cases, we can improve the quality of the segmentation alone by about 50 % across different protocols, and we can significantly improve analyses relying on this segmentation, like message type identification ©2020 IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Personal use of this material is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' IEEE Conference on Communications and Network Security 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' and semantic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' From the evaluation results, we deduce which of the presented methods is suited best for different tasks of PRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' RELATED WORK Several surveys [6, 11, 18, 23] discuss the plethora of approaches of PRE and also have proposed to structure the overall PRE process into different phases [6, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' These phases are data collection into traces, feature extraction, message format inference, message type identification, semantic deduc- tion, and behavior model reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' As previous work showed, support for performing the tasks of the phases on binary protocols is severely limited in comparison to textual protocols [2, 5, 6, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' While most phases are well covered in literature, approaches for segmenting messages have gained only limited attention despite its relevance for all subsequent phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' We adopt the term “segment” from previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' A segment is a subsequence of bytes of a message that approxi- mates a field’s boundaries from the protocol definition in terms of byte positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' In an optimal inference, the segments match the true field boundaries from the unknown syntax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Netzob [3], Discoverer [5], and others [15] deduce fields as a by-product of sequence alignment with the already mentioned disadvantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Existing statistical methods either require an already existing segmentation [3, 5, 15] or expect field boundaries at globally fixed positions [2, 26, 27], limiting the applicability to protocols specifically designed without variable length fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' If meta-data and common offsets of values in messages are available, the task is as simple as find- ing the corresponding or correlating values in the messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' For instance, analyzing wireless communication of a medical device or sensor node, there is no encapsulation present from which to extract addresses and identifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Thus, we argue that no convincing method exists for accurately inferring structure of binary protocols [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Opposed to all previous approaches, we make very few assumptions, thus, creating a more generic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' It does not require a specific message format or protocol structure, like globally common byte offsets or typical field lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' It works without any preceding classification of messages or the identification of flows, and thus also works for protocols without any encapsulation like TCP, UDP, and IP1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The last property is of particular interest when reverse engineering proprietary wireless point-to-point protocols were no encap- sulation is present or accessible, like such used by mobile devices [25] and for medical devices [17, 20], simple Internet- of-Things nodes [19], or vehicle sensors, like tire-pressure monitoring systems [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' In this paper, we propose refinements that enhance NEMESYS [10] and we compare our novel segmenter to the refinements from NEMETYL [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Our approach uses a concept for the clustering of segments that we proposed earlier [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' 1Transmission Control Protocol (RFC 793), User Datagram Protocol (RFC 768), and Internet Protocol (RFC 791) Thus, we briefly introduce these three fundamental approaches in the rest of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' We utilize the NEMESYS segmentation [10] for a first, raw approximation of the true message structure without knowledge of the specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The method derives segments from the distribution of value changes within subsequent bytes of single messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Unlike sequence alignment or statistical methods, NEMESYS does not compare different messages with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' It is known that NEMESYS boundaries regu- larly exhibit an off-by-one error [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' We confirmed for all of our test protocols that the vast majority of NEMESYS’ “near- match” boundary errors are such off-by-one errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' We tried to find patterns in these errors that would enable us to create a systematic correction but could not find constant off-by-one error rules for NEMESYS segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Therefore, we propose our refinement to correct this type of errors dynamically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The approach NEMETYL [9] identifies message types from clustering of messages by their similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' It additionally contains refinements to apply to the segments obtained from NEMESYS to more correctly approximate fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' NEMETYL proposes a simple frequency analysis to determine the most common segment values, which point out probable field boundaries throughout the protocol under analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Further- more, NEMETYL recognizes char sequences embedded within the binary protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' To obtain sets of comparable, related segments, which we subsequently intend to analyze together, we cluster segments based on their dissimilarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' We base our clustering approach on our previous proposal [12] to use DBSCAN [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The used measure is the Canberra dissimilarity [9] which extends the better-known Canberra distance [14] to vectors of differ- ing dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' We use the Canberra dissimilarity between segments as affinity measure to guide the clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The clustering results in concise sets of segments that overlay best, matching each others’ values measured by the smallest possible Canberra dissimilarity of segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' BYTE-WISE SEGMENT VARIANCE ANALYSIS The evaluation of different aspects of NEMESYS [9, 10, 12] has proven that it can yield useful approximations of the protocol message structure of unknown protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' It is highly efficient and refinements could improve some shortcomings of NEMESYS’ raw segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' However, none of these proposed enhancements take variance of values within and between messages into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Inter- and intra-message variance reveals details about the message structure that is not visible otherwise and has the potential to increase the accuracy of inferred field boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' We determine which bytes vary together in sets of segments that we interpret as data vectors to identify probable field boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' We separate vector components, variance-locked to each other, from linearly independent vector components by Principal Component Analysis (PCA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' PCA shows which bytes from each set of segments are associated with each other and, thus, belong to a common field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Raw segments Recursive clustering Section III-B PCA Section III-A Boundary adjustment Section III-C Refined segments Figure 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' 1: PCA process overview We first describe the core of our variance analysis based on PCA, in Section III-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' As illustrated in Figure 1, before we can start the PCA, we need to prepare groups of segments by a recursive clustering step (Section III-B) and after the PCA, we interpret its results (Section III-C) to adjust the boundaries of the raw input segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The whole method, which we call PCA refinement, is embedded in a chain of other refinements, partly from related work and partly by methods we introduce in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' We locate all refinements in the context of the processing chain in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Principal Component Analysis The PCA calculates the multivariate main direction of variance and its scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The multivariate variance identifies the components that vary together, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=', linearly dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Typically, PCA is applied to feature vectors and its result is used to classify samples, represented by these vectors, based on commonalities in the variance revealed by the PCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' In contrast, we use the immediate result of a PCA to determine the contribution of different components to the variance of the vectors and thus the variance contained in a set of segments represented by the vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' We first revisit the elements of the well-known generic PCA to clarify our usage of terms and notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The data vectors are collected into a matrix X with each vector as row, in our case corresponding to the bytes of one segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The variance of X is described by the covariance matrix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The equations in Figure 2 give an example for matrices X and C derived from segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The eigenvalues (“factors” or “component scores” λ) and eigenvectors (“loadings” wi) of the covariance matrix C are the foundation of the PCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The first X = � � � � � � � � 00 08 50 00 02 01 08 90 00 04 01 08 90 00 07 01 08 b0 00 02 02 90 40 01 02 02 90 40 01 02 01 08 80 00 04 01 08 80 00 04 � � � � � � � � (1) C = � � � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='41 34 −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='19 34 3963 −2020 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='14 −53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='42 −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='71 −2020 1737 −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='85 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='25 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='14 −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='39 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='19 −53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='42 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='85 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='39 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='12 � � � (2) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' 2: Example for X, with each row representing one segment’s sequence of byte values, and the C of this X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' principal component i λ 0 1 2 3 4 0 1 000 2 000 3 000 4 000 5 000 6 000 significant PCs knee Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' 3: Scree graph of PCs sorted by their eigenval- ues/component scores λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' principal component (PC) is the highest eigenvalue λ and its associated eigenvector wi and it intuitively states the direction of the prevalent variance in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Further PC loadings are orthogonal to the previous ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' We call the variance at a specific byte position the notable contribution if the ith eigenvector component wi is significantly different from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' According to common analysis methods, the transition between the PCs with a significant contribution and negligible components is marked by the knee of the scree graph of the eigenvalues λ as illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' We determine the knee by the Kneedle algorithm [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' To be able to start a PCA, two basic prerequisites must be met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' First, we need to prepare a set of segments that contain related data, so that the PCA does not measure arbitrary vari- ance, but only meaningfully comparable variances of segments that represent the same kind of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Second, we need to determine which PCs significantly contribute to the variance of the data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=', the segments represented in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' 1) Overlaying Segment Vectors and Calculating PCs: PCA requires an existing coarse approximation of fields, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=', initial segments, to overlay the segments that are related in one set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' This overlaying is necessary to calculate the covariance from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' We use the Canberra dissimilarity [9] to find the best fitting overlay of multiple segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Thus, we superimpose the segments at the most probable useful offsets, resulting in overlays of the most meaningfully comparable message parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' While similar to the longest common subsequence [24], we additionally allow for variations instead of requiring identical subsequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' As Figure 4 illustrates, we next calculate the covariance matrix C from the aligned segment data X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Then Segments per initial cluster Vectors from byte values Overlay segments on Canberra dissimilarity Covariance matrix Eigenvalues and -vectors PCA Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' 4: PCA preparation process performed per initial cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' we calculate the eigenvalues λ and eigenvectors wi of C, which the PCA uses directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' 2) Process Overview: The PCA is embedded into a process that prepares the segment data and interprets the PCA result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Introduced in Figure 1 in high-level abstraction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' the process is composed of the following steps in detail: 1) Cluster the raw segments by similarity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' 2) interpret the segment byte values as vectors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' overlay the vectors per cluster,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' calculate the covariance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' and from it the eigenvalues and eigenvectors that are required for the subsequent PCA (summarized in Figure 4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' 3) check if prerequisites for PCA are met by each cluster,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' 4) for clusters that fail this check,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' recurse from step 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' 5) for clusters that pass this check,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' perform the PCA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' and 6) finally apply rules for which variance characteristics quantified by the PCA indicate field boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The rest of this section explains the auxiliary algorithms for the preparation and interpretation of the PCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' These algorithms make use of thresholds and other parameters for which we empirically determined suitable values (Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Recursive Clustering Before performing the PCA, we obtain sets of comparable, related segments by clustering the segments [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' To adjust the clusters optimally for applying PCA, we recursively cluster segments and check whether the prerequisites for a component analysis are met by each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' If a cluster does not allow to conduct a PCA, we sub-cluster it and test whether the smaller clusters result in more suitable sets of segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' If PCA is still not applicable to a cluster, we recurse the sub-clustering on the respective cluster, otherwise we stop the recursion for this sub- cluster branch and perform the PCA on it (see Section III-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' We estimate whether a cluster is adequate for PCA by PCA Prerequisites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The main criterion is that the variance is systematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' We distinguish systematic from random variance by the number of significant PCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' For a successful PCA only a limited number of significant PCs are allowed, since only if the variance is concentrated in a small number of PCs we can deduce that the data is non-random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' We define an absolute pp and a relative maximum pq of allowed significant PCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The number of allowed significant PCs is exceeded if |⟨λi : λi > qs⟩n i=0| > min(pp, dim λ · pq) with the threshold qs of any eigenvalue λi for a PC to count as significant: qs = min(K(λ), 1 10λ0, ps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' If the condition is true and thus the PCA will fail on the cluster, we further recurse the sub-clustering to gain a subset of segments that then is appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' We now interpret the PCA of each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Interpretation of the PCA The clusters from the previous step that are suitable for PCA are interpreted in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' First we apply inference rules for field boundaries and then we determine commonly aligned offsets within clusters that likely show additional boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' For any interpretation, we use the common alignment of all segments within a cluster, which we obtain as explained in Section III-A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' This results in relative offsets that are common throughout one cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Thus, the field-boundary inference rules and the additional conditions for boundaries work on this dissimilarity-aligned segments per cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' 1) Inference Rules: The covariance shows transitions be- tween related message parts in the byte sequences of a set of segments provided by a cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' To get an impression about how the covariance matrix C represents such related parts, regard Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The more intense the color, the stronger the linear dependent variance at this offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' We use the loadings w of significant PCs calculated from C to test for conditions that represent characteristics of field boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' We deduce two different rules from the typical data type representation in byte streams of network messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' a) Rule A: The first rule is governed by the observation that the data of a field with a common data type exhibits a rise in variance towards the field’s end [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Using the same cluster as in Figure 5 as basis, Figure 6 illustrates this by the loadings of the significant PCs for segments of 25 bytes length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' A PC i 0 1 2 3 4 5 6 7 8 9101112131415161718192021222324 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 7500 5000 2500 0 2500 5000 7500 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' 5: Covariance matrix as heat map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Each row and each column correspond to the relative offsets in the scope of the overlayed and aligned set of segments of one cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' White lines are the relative positions of true boundaries in this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' 1 0 1 2 3 4 5 6 7 8 9 10111213141516171819202122232425 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='4 1, = 27733.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='1 2, = 18219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='4 3, = 12390.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='1 4, = 8538.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='7 5, = 3826.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' 6: Example for matches of rule A: significant loadings of one cluster (with eigenvalue > 2 773).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The dot denotes the matches, the vertical lines the true boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' is significant if it has an eigenvalue λi above the threshold qs (Section III-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' We thus define a significant PC i to relevantly contribute to the variance at a byte position k, as such with its maximum loading mk > pr, with mk = n max i=1 ⟨|wi,k|⟩ (3) where n is the number of significant PCs i : λi > qs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' We now search for a considerable relative drop in the absolute covariance after the peak at the end of a suspected field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Thus, a field boundary is detected at byte position k if the Boolean expression mk−1 > pr ∧ mk ≤ pr ∧ mk−1 − mk mk−1 > pd (4) is true, regardless of which λi is responsible for the peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' b) Rule B: The second rule detects the start of a new field in situations where a prolonged sequence of byte po- sitions do not significantly vary and the next byte suddenly peaks in its variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' This typically happens at the transition between constant fields or such with few possible values, like enumerations, commands, or flags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The maximum loading m is defined as for Rule A in Equation 3 and we introduce the thresholds pz to denote loadings that are near zero, pb the length in bytes that a loading needs to be below pz to trigger the rule, and pt the minimum absolute value of a loading to constitute a notable contribution to the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' A variance peak after a prolonged sequence of near constancy is thus defined to be at a byte position k for which the following Boolean expression is true: pb � j=1 mk−j < pz ∧ mk > pt ∧ mk − mk−1 mk > pd We apply both rules to all suitable clusters individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' We move or add boundaries at the same relative offsets for all segments in one cluster for which any of the rules apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' 2) Commonly Aligned Boundaries Conditions: To overlay segments of a cluster, we dislocate the segments against each other as described in Section III-A1, so that the sub-sequence of one segment is at the same relative offset that is most similar to the other segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' If the overlaying shows a prevalent common boundary in the aligned starts and ends of segments and if that boundary is missing in only a minority of segments of the cluster, we cut segments within one cluster at all relative offsets of boundaries that are more common than their direct neighbor, which originated from the raw segmentation or which where detected by component analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The reason, we can detect these errors in the raw segmentation, is that the alignment on Canberra dissimilarity reveals segment relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' NULL-BYTES TRANSITIONS The central goal of our work is to improve the accuracy of the field inference of unknown messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' PCA requires a pre- existing segmentation that supplies comparable data vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' We observed that null bytes, produced by unset fields, unused most significant bytes of numbers, or null terminated strings, denote probable field boundaries in binary protocols [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Applied as a refinement for NEMESYS, we improve its raw segments by relocating inferred boundaries near the beginning and end of sequences of nulls: (1) If the last bytes before the nulls fulfill the character heuristic [9], we assume it is a null- termination for the string and allocate the nulls to the end of the character segment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' (2) otherwise, we assign the null-bytes to the following segment, assuming they are the unset most significant bytes of a number value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' As an alternative, we propose a novel standalone segmenter that is not based on NEMESYS to compare this as a simpler foundation to apply our PCA afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Similar to the null- bytes refinement for NEMESYS, the segments that our so- called Null-Bytes Segmenter yields are the subsequences of the messages which are delimited by null bytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The segmentation is only coarse but still adequate to prepare the fine-grained field inference by PCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' IMPLEMENTATION We implemented a proof-of-concept in Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='2 We re- quire numerous thresholds and other parameters throughout our method, for which Table I provides an overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' To empirically determine universal values of the fixed parameters and thresholds our approach relies on, we used traces of the binary network protocols DHCP, DNS, NBNS, NTP, and SMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='3 These traces are publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='4 Both, our novel method based on PCA for refining the NEMESYS segmenter and our standalone Null-Bytes Seg- menter, process raw segments in a chain of refinements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' This 2https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='com/vs-uulm/nemesys 3Dynamic Host Configuration Protocol (RFC 2131), Domain Name System (RFC 1035), NetBIOS Name Service (RFC 1002), Network Time Protocol (RFC 958), and Server Message Block 4DHCP, NBNS, NTP, and SMB extracted from http://download.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='netresec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' com/pcap/smia-2011/;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' DNS extracted from https://ictf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='ucsb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='edu/archive/ 2010/dumps/ictf2010pcap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='tar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='gz concept is similar to the original segmenter’s and the refined version of NEMETYL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Table II provides an overview of the applied refinements from literature and our own approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The refinement methods introduced by previous work are the simple char detection heuristic in NEMESYS, which we call MergeCharsv1, and the advanced char detection heuristic in NEMETYL, MergeCharsv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' NEMETYL counts the most frequent segment values and crops these from larger segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' We call this refinement CropDistinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' NEMETYL also intro- duced the splitting of the first segment of each message into fixed chunks, which we denote SplitFixedv1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The primary contributions of this paper are two novel meth- ods for the refinement chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The first one are the Null-Bytes Refinement and Segmenter Section IV, which we abbreviate by NullBytes, and the second one is the application of Principal Component Analysis to guide segment refinements, PCA (see Section III-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' In preparation of PCA, we propose Entropy- Merge for merging of NEMESYS segments if they have similar local entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' We also add slight improvements for the splitting of the first segment of each message, SplitFixedv2, and the handling of cropping char segments, CropChar, which is based on NEMETYL’s MergeCharsv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Using these refinement methods as elements for a process- ing chain, we apply our improvements in two different ways that we call NEMEPCA and NullPCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' We mark the placement of our contributions in the novel processing chains by a bold font in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' EVALUATION Using our proof-of-concept implementation presented in Section V, we evaluate our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' We evaluate the quality of the inferred segment boundaries with seven representative binary network protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Beyond this directly measurable effect on the segmentation, we show the impact of the refined segments that our method yields by applying analyses from previous work using the improved segments as starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Thus, we perform message type identification as previous work proposed in conjunction with segments from NEMESYS [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Furthermore, we classify field data types using an existing method, which we also proposed to work with segments from NEMESYS [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' TABLE I: Parameters and empirically determined values for our algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Parameter Task Value Scree threshold Sub-cluster qs Scree minimum Sub-cluster ps = 10 Maximum significant principals Sub-cluster pp = 4 Significant principals ratio Sub-cluster pq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='5 Length difference ratio Sub-cluster pl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='5 Minimum cluster size Sub-cluster pc = 6 Significant w-contribution threshold Field inference A pr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='1 Signific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' loading-difference threshold Field inference A pd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='98 Near-zero threshold Field inference B pz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='05 Near-zero minimum length Field inference B pb = 4 Notable w-contribution threshold Field inference B pt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='005 Additionally to the protocols DHCP, DNS, NBNS, NTP, and SMB that we used to select the parameters for our algorithm (Section V), we also use our own traces of the two proprietary protocols Apple Wireless Direct Link (AWDL) and Auto Unlock (AU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' AWDL is a Wi-Fi-based link-layer protocol for peer-to-peer communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' AU implements a proprietary distance bounding protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='5 Both protocols were undocumented until they recently have been manually reverse engineered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The reverse-engineered specification of AWDL, including a dissector, is publicly available [25], and we had access to a private Wireshark dissector of the AU protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Thus, both protocols are typical PRE use cases while we have ground truth available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' As the source of the ground truth, we parse the Wireshark dissectors’ output for each message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' All evaluated protocols are binary, while DNS, DHCP, SMB, and AWDL also contain embedded char sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The binary fields of DNS, NBNS, and NTP have fixed length, while DHCP, SMB, AWDL, and AU use a mix of fixed and variable- length fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' DHCP, DNS, NBNS, SMB, AWDL, and AU support varying numbers of fields in different messages, while NTP has a fixed structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Thus, our set of traces represents a wide variety of protocol properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' We compare the results of the three approaches NEMETYL- refined baseline, NEMEPCA, and NullPCA which we de- scribed in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The baseline is the refinement as described in NEMETYL and the other two are applications with and without NEMESYS (Section IV), respectively, in conjunction with our novel analysis method using PCA, in- troduced in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Message Segmentation The immediate effect we expect of our refinement is that the segment boundaries will more accurately match the field boundaries of the protocol specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' To measure this, we use the Format Match Score (FMS) that we proposed together with NEMESYS [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' We apply our refinements to each of our test protocols, calculate the FMS, and discuss the results in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The FMS is a measure of correctness of the inference of a message format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' By the FMS, we can quantitatively compare the quality of different inference methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The FMS is calculated for each message individually and we therefore calculate the median of all FMS’ for one trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' 5https://support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='apple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='com/en-us/HT206995 TABLE II: Refinement overview for the compared approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Our contributions are printed in bold font.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' NEMESYS NEMETYL NEMEPCA NullPCA Original [10] Baseline [9] [10] + Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' III Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' IV + III NEMESYS NEMESYS NEMESYS – MergeCharsv1 MergeCharsv2 EntropyMerge NullBytes CropDistinct NullBytes CropChars SplitFixedv1 CropChars PCA PCA CropDistinct CropDistinct SplitFixedv2 SplitFixedv2 Summarized in Table III, the NEMETYL [9] baseline per- forms worst for DHCP and SMB, while our PCA refinement, in contrast, stays lower for DHCP and AU, however at a reasonable absolute value of above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='35 and still improving on NEMETYL’s DHCP results of below of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' For all other protocols, NEMEPCA clearly outperforms the baseline except for NTP where the FMS values are closeby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The results also show that traces of 1 000 and 100 messages are almost identical, confirming that decent results can be produced with even small traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Compared to the baseline, our NEMEPCA yields better results for 100 than for 1 000 DHCP, SMB, and AWDL messages, showing that it is effective in extracting more structural information even from small traces than the previous approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' NEMESYS requires to select a value for σ that is dependent of the field lengths expected for a protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The results for the baseline show that the quality is significantly higher on average for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Our goal is to become independent from this parameter, since it is difficult to determine it correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Per- forming our NEMEPCA refinements on NEMESYS segments with different σ values, we observe that the results are almost identical for both σ values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The only protocol that declines in quality is AU while all other protocols increase their field correctness in terms of the FMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Assuming that we know nothing about the protocol that helps to select the optimal σ value and thus blindly selecting σ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='2 for the baseline, we improve the results by almost 100 % using NEMEPCA due to its robustness against σ changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Finally, we used our novel Null-Bytes Segmenter that works without NEMESYS and applied the PCA refinement to it as described in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The resulting FMS values on average are considerably lower than those of NEMEPCA, but similar to the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' This shows the advantage of using NEMESYS as a heuristic method and that the complex effort of refining its segmentation is worthwhile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' TABLE III: Comparison of message segmentation quality using the median values of FMS per protocol trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Note: Numbers printed in bold in the protocol rows are the worst cases discussed in the text and the bold median values at the bottom are the mainly discussed comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' FMS median baseline NEMEPCA NullPCA trace msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='s σ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='9 σ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='2 σ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='9 σ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='2 DHCP 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='21 DHCP 1 000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='29 DNS 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='74 DNS 1 000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='87 NBNS 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='57 NBNS 1 000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='56 NTP 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='53 NTP 1 000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='71 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='47 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Message Type Identification We apply the baseline segmentation to identify message types and compare these results with applying the NEMETYL message type identification on top of our segment refinements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Table IV shows the results measured in classification precision and recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' They where calculated exactly as described in the NEMETYL paper [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' For simplicity, we only use σ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='2 for the NEMESYS-based analyses: the baseline and NEMEPCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' While the precision is almost optimal for all of the compared approaches, the previous and our novel ones, Our NEMEPCA can reach or outperform the baseline only for few traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' NullPCA has a precision that is on average similar to the others but with a recall that is notably lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' However, NullPCA is able to outperform the precision for single protocol traces of NTP and SMB that are particularly difficult for both, the baseline and NEMEPCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' This is due to the structure of these protocols that contain sequences of null bytes separating segments which denote the message type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Our method cannot provide groundbreaking improvements for message type identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' This shows that a more ac- curate field approximation is not necessarily improving the message type identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' NullPCA is an alternative only if the protocol for the most part contains segments that are clearly separated by null bytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Field Data Type Clustering The second analysis based on the refined segments that we use as evaluation aspect is the clustering of field data types from NEMESYS’ message segments [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Table V shows our evaluation results by measuring precision, recall, the number of segments that are considered noise by the clustering algorithm, and the number of unknown segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Segments are considered unknown if no match to any field from the ground truth is possible that would allow to determine the field type for the segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Thus, this value denotes one aspect of the deviation of the heuristic segmentation from the true field structure of the protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' TABLE IV: Message type identification quality measured by precision (P) and recall (R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' baseline NEMEPCA NullPCA trace msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='s P R P R P R DHCP 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} 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1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='06 662 352 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='57 193 121 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='45 89 880 median 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='26 90 108 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='41 97 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content='78 36 75 Comparing the baseline to NEMEPCA, the precision stays similar for most protocols, but we can increase the recall regarding the median of all protocols by 57 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The number of unknown segments are reduced considerably as expected due to improved segment accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' As an exception, the increased unknowns of NTP account for the low recall of the traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The larger AWDL trace yields a high amount of noise and thus the lowest precision and recall of all traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The protocol is highly complex and eludes our heuristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' However, on average, we can improve precision and recall while reducing unknown segments significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Our other proposal, NullPCA, shows tripled recall and re- duced noise and unknown segments compared to the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' This comes at the cost of a slightly decreased precision, however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Also, the usage of null bytes as primary segment detection mechanism leads to the loss of details in the bound- aries of tightly packed fields that NEMESYS can discern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' CONCLUSION In this paper, we presented a chain of refinements for two segmenters, NEMESYS and our simple null-byte transition detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Besides the static rule-based algorithms Entropy- Merge, NullBytes, CropChars, CropDistinct, and SplitFixed, we introduce the novel, dynamic PCA method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' It measures the byte-wise variance of segment contents to increase the accuracy of the field boundary detection in unknown bi- nary protocols over previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Opposed to its common application in classification tasks, we utilize PCA directly to determine linearly dependent variance and use a novel interpretation of these results to detect field boundaries from a rough preliminary segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Our evaluation shows improved field inference quality of up to 100 % for most protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Moreover, it renders any parameter selection by guessing unnecessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' PCA is effective in extracting more structural information even from small traces than the previous approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Thus, our method notably improves the field accuracy as measured by FMS over the previous segmenters and their refinements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' As we discuss within our evaluation, our simple Null-Bytes Segmenter, combined with PCA, provides slightly less accu- racy than refined NEMESYS segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Despite the original assumption of PCA that it can correct off-by-one errors in particular, the PCA refinement still works well enough for segments generated only from the slicing of null-bytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' How- ever, while more complex, using NEMESYS segments as basis for further analyses outperforms this simpler segmenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Thus, we conclude that the improved inference accuracy makes the relatively complex refinement of segments from NEMESYS worthwhile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Encrypted messages cannot be inferred by our method di- rectly and require obtaining of plain-text traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Furthermore, PCA can only detect linearly dependent correlations and fails for random and non-linearly correlated message values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' A larger set of input data improves the accuracy only slightly, while the exponential memory complexity of our refinement limits the practical applicability to traces of a little over one thousand messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Thus, small traces that contain high variability of message values are preferable for our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' We propose for future work to find more sophisticated rule sets for deducing boundaries from relations between PCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' This will improve the accuracy of the field boundary detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' The inference of the message format is a crucial task in protocol reverse engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Often performed during security assessments, it provides knowledge about the placement of field boundaries in protocol messages, which is necessary to correlate information to field values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Among other use cases, it helps to determine portions of the message to inspect, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=', by Fuzz testing, to identify features for the fingerprinting of protocols and to train intrusion detection systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' Recent security analyses [13, 25] have shown the value that this kind of method adds to the arsenal of a security analyst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' ACKNOWLEDGMENT We would like to thank Milan Stute for his support and the AWDL traces, as well as Steffen Klee for providing us with a Wireshark dissector and traces for Apple’s Auto Unlock (AU) protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQf_wb4/content/2301.03585v1.pdf'} +page_content=' REFERENCES [1] Ian Beer.' metadata={'source': 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b/JtE4T4oBgHgl3EQfhw0J/content/tmp_files/2301.05127v1.pdf.txt @@ -0,0 +1,3618 @@ +Distributed local spline simulator for wave propagation +Xu Guoa,∗, Yaomeng Lia,∗, Yunfeng Xiongb,∗∗ +aGeotechnical and Structural Engineering Center, Shandong +University, Jinan, 250061, Shandong, China +bSchool of Mathematical Sciences, Beijing Normal University, 100871, Beijing, China +Abstract +Numerical simulation of wave propagation in elastic media faces the chal- +lenges arising from increasing demand of high resolution in modern 3-D imag- +ing applications, which requires a balance between efficiency and accuracy +in addition to being friendly to the distributed high-performance computing +environment. In this paper, we propose a distributed local spline simulator +(LOSS) for solving the wave equation. LOSS uses patched cubic B-splines to +represent the wavefields and attains an accurate evaluation of spatial deriva- +tives with linear complexity. In order to link the adjacent patches, a perfectly +matched boundary condition is introduced to give a closure of local spline +coefficients. Owing to the rapid decay property of the local wavelets in dual +space, it can recover the global spline as accurately as possible only at the +cost of local communications among adjacent neighbors. Several typical nu- +merical examples, including 2-D acoustic wave equation and P- and S- wave +propagation in 3-D homogenous or heterogenous media, are provided to val- +idate its convergence, accuracy and parallel scalability. +Keywords: +Wave equation, Spline collocation method, Artificial boundary +condition, Parallel and distributed computing +2020 MSC: , 74J05, 65D07, 65M22, 65Y05, 68W15 +∗These authors contribute equally to this paper +∗∗To whom correspondence +Email address: yfxiong@bnu.edu.cn (Yunfeng Xiong) +Preprint submitted to Elsevier +January 13, 2023 +arXiv:2301.05127v1 [math.NA] 12 Jan 2023 + +1. Introduction +The wave propagation in elastic media plays an essential role in the field +of geological imaging techniques [1, 2, 3] and new-trend neuroimaging [4]. +Nowadays, a large collection of efficient numerical techniques is available +for both forward and inverse wavefield modelings, including the widely used +staggered-grid finite difference method [5, 6], the optimal difference method +[7], the pseudo-spectral method [8, 9, 10, 11]. In spite of their great success +and extensive applications, these standard techniques face new challenges by +tremendous memory demand and significant computational cost especially +for 3-D problems, arising from the increasing need for high resolution in mod- +ern imaging applications, e.g., the teleseismic datasets for waveform inver- +sion and deep lithospheric structures [2, 12] or the sub-millimetre-resolution +in brain and surrounding tissue [4]. Thus, it urgently calls for discretized +methods to account for sharp variations of solutions induced by material dis- +continuities accurately [13] and to be friendly to large-scale high-performance +computing environment [14]. +In recent years, the spectral element method (SEM) and discontinu- +ous Galerkin method (DG) [15, 16] have gained an increasing attention +[12, 13, 17, 18, 19] as they take advantage of both flexibility of the finite +element method (FEM) in resolving multiscale phenomena [14] and high ac- +curacy of the spectral method. A significant merit of SEM and DG over FEM +is that the mass matrix is exactly diagonal by construction, which drasti- +cally simplifies the implementation and the temporal integration [13]. More- +over, the assembly of the stiffness matrix can be performed in an element- +by-element manner, thereby greatly facilitating the parallelization in a dis- +tributed computing environment [12]. +But the evaluation of the stiffness +matrix at the elemental level has a relatively high computational cost due to +the matrix multiplications involved, e.g., the complexity scales as O(n4 +l ) for +SEM in three dimensions with nl the polynomial degree used to present the +functions in each direction [13]. It may somehow pose a limitation on nl to +achieve a trade-off in accuracy and efficiency. +As forward wavefield modelings have to be simulated thousands of times +in waveform-fitting imaging, reducing the computational complexity of nu- +merical solvers is always a central issue especially for 3-D problems [1]. Thus +it is natural to seek a local polynomial basis that can calculate the spatial +derivatives of wavefields accurately with relatively low computational cost. +To achieve this, we propose a distributed local spline simulator (LOSS) for +2 + +solving the wave propagation, using the local cubic B-spline wavelet as the +basis [20]. The name LOSS comes from the semi-Lagrangian methods in +computational fluid dynamics [21, 22] and kinetic theory [23, 24, 25], where +the cubic spline has been ubiquitously applied for interpolating the advec- +tion and is believed to strike the best balance between accuracy and cost +[24]. The cubic spline achieves spatial fourth-order convergence [26] and its +construction can be realized by the standard sweeping method, where the +complexity scales linearly with respect to mesh size [27]. For these reasons, +the most recent semi-Lagrangian methods are even capable to resolve the +kinetic-type equation in full 6-D phase space, e.g., the Vlasov equation [28] +and quantum Wigner equation [29]. +In structural mechanics, the cubic spline has been also used to solve +the wave propagation in cracked rod [30] and the wave-structure interaction +[31] within the framework of FEM, while the spline coefficients are globally +dependent in principle. A major advantage of LOSS lies in its distributed +construction with only local communication cost. This is based on a key +observation that the wavelet basis decay exponentially in the dual space +[20, 22], so that the vanished off-diagonal elements in the inverse coefficient +matrix can be truncated. As a consequence, a perfectly matched boundary +condition (PMBC) can be introduced to give a closure of patched spline +coefficients and allows local splines to recover the global one as accurately +as possible [29]. Since only local communications in adjacent neighbors are +needed, LOSS is expected to be suitable for the computational clusters with +high-latency network, known as the Beowulf machines [17]. +We will also +show that the natural boundary conditions on two ends of splines are fully +compatible with the absorbing perfectly matched layers (PML) [32, 33, 34, 35] +in outer domain. At present, LOSS is readily implemented in the standard +architecture and may potentially alleviate both the memory limitation and +the computational burden for high-resolution wave propagation. +The rest of this paper is organized as follows. In Section 2, we briefly re- +view the background of the elastic wave propagation. Section 3 illustrates the +formulation of LOSS in both serial and parallel settings, as well as the expo- +nential integrator for the auxiliary differential equation form of the perfectly +matched layer (ADE-PML). In Section 4, we provide a series of benchmark +by simulating 2-D acoustic wave equation to test the convergence and ac- +curacy of LOSS, as well as its compatibility with ADE-PML. The P- and +S- wave propagation in 3-D homogenous or heterogenous media will also be +investigated to validate the performance and parallel scalability of LOSS. +3 + +Finally, conclusions and discussions are drawn in Section 5. +2. Background +The dynamics of wave propagation is governed by three sets of equations +with x ∈ Rd, d ≤ 3 [10]. The first set is the conservation of linear momentum: +ρ(x) ∂2 +∂t2ui(x, t) = +∂ +∂xj +σij(x, t) + fi(x, t), +i, j = 1, . . . , 3, +(2.1) +where σij are the components of the stress tensor, ui are the components of +the displacement vector, ρ is the mass density and fi are components of the +body forces per unit (source term). The summation over repeated indices j +is assumed in Eq. (2.1). The second set is the definition of strain tensor εij, +which can be obtained in terms of the displacement components as +εij(x, t) = 1 +2 +� ∂ +∂xi +uj(x, t) + ∂ +∂xj +ui(x, t) +� +, +i, j = 1, . . . , 3. +(2.2) +The constitutive equation reads that +σij(x, t) = MP(x)εkk(x, t)δij + 2MS(x)(εij(x, t) − εkk(x, t)δij), +(2.3) +where the summation over the repeated indices k is assumed, δij is the Kro- +necker symbol, MP(x) = c2 +P(x)ρ(x) and MS(x) = c2 +S(x)ρ(x) are moduli +with cP(x) and cS(x) the P- and S-velocities, respectively. +For instance, the velocity-stress form of the full elastic wave equation in +3-D media involves 15 wavefields. The conservation of momentum is +ρ ∂ +∂t +� +� +v1 +v2 +v3 +� +� = +� +� +∂ +∂x1 +0 +0 +0 +∂ +∂x2 +0 +0 +0 +∂ +∂x3 +� +� +� +� +σ11 +σ22 +σ33 +� +� + +� +� +∂ +∂x2 +∂ +∂x3 +0 +∂ +∂x1 +0 +∂ +∂x3 +0 +∂ +∂x1 +∂ +∂x2 +� +� +� +� +σ12 +σ13 +σ23 +� +� + +� +� +f1 +f2 +f3 +� +� . +The definition of strain tensor is given by +∂ +∂t +� +� +ε11 +ε22 +ε33 +� +� = +� +� +∂ +∂x1 +0 +0 +0 +∂ +∂x2 +0 +0 +0 +∂ +∂x3 +� +� +� +� +v1 +v2 +v3 +� +� , +∂ +∂t +� +� +ε12 +ε13 +ε23 +� +� = 1 +2 +� +� +∂ +∂x2 +∂ +∂x1 +0 +∂ +∂x3 +0 +∂ +∂x1 +0 +∂ +∂x3 +∂ +∂x2 +� +� +� +� +v1 +v2 +v3 +� +� . +4 + +And the stress-strain relation reads +� +� +� +� +� +� +� +� +σ11 +σ22 +σ33 +σ12 +σ13 +σ23 +� +� +� +� +� +� +� +� += ρ +� +� +� +� +� +� +� +� +c2 +P +c2 +P − 2c2 +S +c2 +P − 2c2 +S +0 +0 +0 +c2 +P − 2c2 +S +c2 +P +c2 +P − 2c2 +S +0 +0 +0 +c2 +P − 2c2 +S +c2 +P − 2c2 +S +c2 +P +0 +0 +0 +0 +0 +0 +2c2 +S +0 +0 +0 +0 +0 +0 +2c2 +S +0 +0 +0 +0 +0 +0 +2c2 +S +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +ε11 +ε22 +ε33 +ε12 +ε13 +ε23 +� +� +� +� +� +� +� +� +. +In some situations, the P- wave propagation can be approximated by the +acoustic wave equation based on the acoustic media assumption [36], so that +the wavefield is described by a scalar function instead of a vector, +ρ(x) ∂2 +∂t2u(x, t) − ∇ · +� +ρ(x)c2 +P(x)∇u(x, t) +� += f(x, t). +(2.4) +Taking its two-dimensional case as an example. By introducing the velocities +v1 = +∂ +∂xu, v3 = +∂ +∂zu and the scalar pressure field σ(x, z, t), it can be cast into +velocity-stress form, +∂v1(x, z, t) +∂t += − +1 +ρ(x, z) +∂σ(x, z, t) +∂x +, +∂v3(x, z, t) +∂t += − +1 +ρ(x, z) +∂σ(x, z, t) +∂z +, +∂σ(x, z, t) +∂t += −ρ(x, z)c2 +P(x, z) +�∂v1(x, z, t) +∂x ++ ∂v3(x, z, t) +∂z +� +. +(2.5) +It is seen that the complexity for solving the wave equation lies in calculations +of first-order spatial derivatives of wavefields. +3. Distributed Local spline simulator +As a powerful tool for curve fitting, the cubic spline has been applied +for solving PDEs under the framework of FEM [26, 30, 31]. +The spline +expansion is essentially global as it requires solving global algebraic equations +with tridiagonal coefficient matrice. Nonetheless, we will show that the cubic +spline can be reconstructed by imposing effective inner boundary conditions +on the junctions of local patches [25, 29]. +5 + +The spline collocation method will be derived for solving the strong form +of the wave equation in unidimensional space in Section 3.1, while multidi- +mensional wavefields can be constructed by the tensor product of unidimen- +sional splines successively. For brevity, a uniform grid mesh will be adopted +hereafter, but the idea is straightforward to be generalized to the non-uniform +grid due to the scaling property of wavelets [30]. It follows by the parallel set- +ting of LOSS in Section 3.2, where the global spline is distributed into several +local patches. The junctions are shared by adjacent nodes and the patched +splines are linked by PMBCs. In the meantime, the natural boundary condi- +tions imposed on both ends are fully compatible with ADE-PML, where the +stiffness can be largely alleviated by the usage of exponential integrators as +discussed in Section 3.3. +3.1. Spline collocation method +Without loss of generality, we adopt a uniform grid mesh for the domain +[xmin, xmax] with N + 1 points xmin = x0 ≤ x1 ≤ · · · ≤ xN = xmax. Denote by +Bi(x) the cubic B-spline with compact support over four grid points [24], +Bi(x) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +(x − xi−2)3 +6h3 +, +x ∈ [xi−2, xi−1], +− (x − xi−1)3 +2h3 ++ (x − xi−1)2 +2h2 ++ (x − xi−1) +2h ++ 1 +6, +x ∈ [xi−1, xi], +− (xi+1 − x)3 +2h3 ++ (xi+1 − x)2 +2h2 ++ (xi+1 − x) +2h ++ 1 +6, +x ∈ [xi, xi+1], +(xi+2 − x)3 +6h3 +, +x ∈ [xi+1, xi+2], +0, +otherwise, +(3.1) +implying Bi−1, Bi, Bi+1, Bi+2 overlap a grid interval (xi, xi+1) [22], and +Bi−1(xi) = 1 +6, +Bi(xi) = 2 +3, +Bi+1(xi) = 1 +6. +(3.2) +Now the velocity wavefield can be expanded by N + 3 splines with N + 3 +coefficients �v(t) = (�v−1(t), . . . , �vN+1(t))T +vi(t) ≈ +N+1 +� +i=−1 +�vi(t)Bi(x), +(3.3) +6 + +where vi(t) is short for v(xi, t). +In order to determine the coefficients, it +suggests imposing the natural boundary conditions on two ends to mini- +mize the effect of boundary constraints [31], namely, +∂2 +∂x2v(x0, t) = 0 and +∂2 +∂x2v(xN, t) = 0. By omitting the time variable for brevity, it has that +1 +h2�v−1 − 2 +h2�v0 + 1 +h2�v1 = 0, +1 +h2�vN−1 − 2 +h2�vN + 1 +h2�vN+1 = 0. +(3.4) +Combining with the relation (3.2), it remains to solve the algebraic equation +by the sweeping method with complexity O(N) [27], +A +� +� +� +� +� +� +� +� +� +�v−1 +�v0 +�v1 +... +�vN +�vN+1 +� +� +� +� +� +� +� +� +� += 1 +6 +� +� +� +� +� +� +� +� +� +� +� +6 +h2 +− 12 +h2 +6 +h2 +0 +· · · +0 +1 +4 +1 +0 +... +0 +1 +4 +1 +... +... +... +... +... +... +... +... +0 +1 +4 +1 +0 +0 +0 +6 +h2 +− 12 +h2 +6 +h2 +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +�v−1 +�v0 +�v1 +... +�vN +�vN+1 +� +� +� +� +� +� +� +� +� += +� +� +� +� +� +� +� +� +� +0 +v0 +v1 +... +vN +0 +� +� +� +� +� +� +� +� +� +. +(3.5) +Once the spline coefficients are obtained, the spatial first-order derivatives +can be directly approximated by +∂ +∂xv(xi, t) ≈ − 1 +2h�vi−1(t) + 1 +2h�vi+1(t) +(3.6) +as +∂ +∂xBi(x) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +(x − xi−2)2 +2h3 +, +x ∈ [xi−2, xi−1], +− 3(x − xi−1)2 +2h3 ++ (x − xi−1) +h2 ++ 1 +2h, +x ∈ [xi−1, xi], +3(xi+1 − x)2 +2h3 +− (xi+1 − x) +h2 +− 1 +2h, +x ∈ [xi, xi+1], +− (xi+2 − x)2 +2h3 +, +x ∈ [xi+1, xi+2], +0, +otherwise. +(3.7) +3.2. Distributed local spline +For distributed parallelization, the spline needs to be decomposed into +several pieces and stored in multiple processors. Here we simply divide N +1 +7 + +grid points on a line into p uniform parts, with M = N/p, +v0 < v1 < · · · < vM−1 +the first processor +< vM +shared +< · · · < v(p−1)M +shared +< v(p−1)M+1 < · · · < vpM +p-th processor +, +The grid points vM, v2M, . . . , v(p−1)M are shared by the adjacent patches. +Recall that our target is to recover the global B-spline by the local spline +coefficients �v(l) for l-th piece without global communications, that is, +�v(l) = (�v(l) +−1, . . . , �v(l) +M+1) = (�v−1+(l−1)M, . . . , �v(l−1)M+M+1), +l = 1, . . . , p. +(3.8) +This can be realized by imposing effective Hermite boundary conditions +on two ends of local splines (see Figure 1(b)) [24, 29], +∂v +∂x +��� +x=x(l−1)M += φ(l) +L , +∂v +∂x +��� +x=xlM += φ(l) +R , +(3.9) +which is equivalent to +− 1 +2h�v(l) +−1 + 1 +2h�v(l) +1 = φ(l) +L , +− 1 +2h�v(l) +M−1 + 1 +2h�v(l) +M+1 = φ(l) +R . +(3.10) +Thus all the coefficients �v(l) = (�v(l) +−1, . . . , �v(l) +M+1) can be obtained straightfor- +wardly by solving the algebraic equation +A(l) +M +� +� +� +� +� +� +� +� +� +� +�v(l) +−1 +�v(l) +0 +�v(l) +1... +�v(l) +M +�v(l) +M+1 +� +� +� +� +� +� +� +� +� +� += 1 +6 +� +� +� +� +� +� +� +� +� +� +� +− 3 +h +0 +3 +h +0 +· · · +0 +1 +4 +1 +0 +... +0 +1 +4 +1 +... +... +... +... +... +... +... +... +0 +1 +4 +1 +0 +0 +0 +− 3 +h +0 +3 +h +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +�v(l) +−1 +�v(l) +0 +�v(l) +1... +�v(l) +M +�v(l) +M+1 +� +� +� +� +� +� +� +� +� +� += +� +� +� +� +� +� +� +� +� +� +φ(l) +L +v(l) +0 +v(l) +1... +v(l) +M +φ(l) +R +� +� +� +� +� +� +� +� +� +� +. +(3.11) +where (M + 3) × (M + 3) coefficient matrix A(l) +M has an explicit LU decom- +position, namely, A(l) +M = LU, +L = +� +� +� +� +� +� +� +� +� +� +� +1 +0 +0 +· · · +· · · +0 +− h +3 +1 +0 +... +... +0 +l1 +1 +... +... +0 +0 +l2 +... +... +... +... +lM +1 +0 +0 +0 +· · · +−3lM +h +3lM+1 +h +1 +� +� +� +� +� +� +� +� +� +� +� +(3.12) +8 + +and +U = 1 +6 +� +� +� +� +� +� +� +� +� +� +� +− 3 +h +0 +3 +h +0 +· · · +· · · +0 +0 +d1 +2 +0 +... +... +0 +0 +d2 +1 +... +... +0 +0 +0 +d3 +... +... +... +... +0 +dM+1 +0 +0 +0 +· · · +0 +0 +3dM+2 +h +� +� +� +� +� +� +� +� +� +� +� +, +(3.13) +with +d1 = 4, +l1 = 1/4, +d2 = 4 − 2l1 = 7/2, +li = 1/di, +di+1 = 4 − li, +i = 2, . . . , M + 1, +lM+1 = 1/(dMdM+1), +dM+2 = 1 − lM+1. +(3.14) +Now the solution of Eq. (3.11) should be equivalent to that of Eq. (3.5) by +choosing appropriate φ(l) +L and φ(l) +R . Denote by (bij) = A−1, −1 ≤ i, j ≤ pM+1, +thus the solution �vi of Eq. (3.5) can be represented by +�vi = biivi + +i−1 +� +j=−1 +bijvj + +pM+1 +� +j=i+1 +bijvj, +i = −1, . . . , pM + 1. +(3.15) +Using the constraints (3.8), it directly solves φ(l) +L and φ(l) +R by +φ(l) +L = − 1 +2h�v(l) +−1 + 1 +2h�v(l) +1 = − 1 +2h�v−1+(l−1)M + 1 +2h�v1+(l−1)M, +φ(l) +R = − 1 +2h�v(l) +M−1 + 1 +2h�v(l) +M+1 = − 1 +2h�vM−1+(l−1)M + 1 +2h�vM+1+(l−1)M, +(3.16) +where �vi are given by Eq. (3.15). +At first glance, it still requires the information from all other pieces. +Fortunately, there is a key observation that the non-diagonal elements bij +decays exponentially away from the main diagonal bii due to the rapid decay +of the wavelet basis in its dual space [20, 22], which is clearly visualized in +Figure 1(a). Therefore, it allows us to truncate bij when |i − j| ≥ nnb for +sufficiently large nnb, +�vi ≈ biivi + +i−1 +� +j=i−(nnb−1) +bijvj + +i+nnb−1 +� +j=i+1 +bijvj, +i = −1, . . . , pM + 1. +(3.17) +9 + +0 +10 +20 +30 +40 +50 +60 +-35 +-30 +-25 +-20 +-15 +-10 +-5 +0 +(a) The element |bij| of A−1 in log10 scale. +10 +20 +30 +40 +50 +60 +10 +20 +30 +40 +50 +60 +-12 +-10 +-8 +-6 +-4 +-2 +0 +PMBC +PMBC +PMBC +(b) Approximation of A−1 by (A(l) +M )−1. +Figure 1: Since the non-diagonal elements bij in A−1 decays exponentially away from +the main diagonal bii, the coefficients �v = A−1(v0, . . . , vN)τ can be approximated by +�v(l) = (A(l) +M )−1(v(l) +0 , . . . , v(l) +M )τ when PMBCs are adopted. +Now using the truncated stencils (3.17), it has that +�vlM−1 ≈ +(lM−1)+nnb−1 +� +j=(lM−1)−nnb+1 +blM−1,jvj = +nnb−2 +� +j=−nnb +blM−1,lM+jvlM+j, +�vlM+1 ≈ +(lM+1)+nnb−1 +� +j=(lM+1)−nnb+1 +blM+1,jvj = +nnb +� +j=−nnb+2 +blM+1,lM+jvlM+j. +(3.18) +By further adding four more terms in Eq. (3.18) to complete the summations +from −nnb to nnb, it yields that +− 1 +2h�vlM−1 + 1 +2h�vlM+1 ≈ +� +− 1 +2hblM−1,lM + 1 +2hblM+1,lM +� +vlM +shared by adjacent two processors += +−1 +� +j=−nnb +� +− 1 +2hblM−1,lM+j + 1 +2hblM+1,lM+j +� +vlM+j +stored in left processor ++ +nnb +� +j=1 +� +− 1 +2hblM−1,lM+j + 1 +2hblM+1,lM+j +� +vlM+j +stored in right processor +. +10 + +Thus it arrives at the formulation of PMBC for 1 ≤ l ≤ p − 1, +φ(l) +R = φ(l+1) +L +≈ 1 +2c0,lvlM + +nnb +� +j=1 +c− +j,lvlM−j +stored in left processor ++ 1 +2c0,lvlM + +nnb +� +j=1 +c+ +j,lvlM+j +stored in right processor +, +(3.19) +where c0,l = − blM−1,lM +2h ++ blM+1,lM +2h +and +c+ +j,l = −blM−1,lM+j +2h ++ blM+1,lM+j +2h +, +c− +j,l = −blM−1,lM−j +2h ++ blM+1,lM−j +2h +. (3.20) +Finally, the effective boundary conditions φ(1) +L and φ(p) +R on two ends should +match the natural boundary conditions of the global cubic spline, yielding +φ(1) +L = �v1 − �v−1 +2h +≈ +nnb +� +j=0 +c− +j,0vj, +c− +j,0 = −b−1,j + b1,j +2h +, +(3.21) +φ(p) +R = �vN+1 − �vN−1 +2h +≈ +nnb +� +j=0 +c+ +j,pvN−j, +c+ +j,p = −bN−1,N−j + bN+1,N−j +2h +. (3.22) +3.3. Exponential integrator for the wave equation and ADE-PML +PML is usually adopted in finite computational domain to model wave +propagation in unbounded media [32, 33, 34, 35], which suggests adding a +layer of the thickness L outside to attenuate the wave and avoid its artificial +reflection. It will be shown that the natural boundary condition of cubic +spline can be imposed on the differential equation form of PML (known as +ADE-PML [35]), and the resulting rigid dynamics can be solved efficiently +by exponential integrators [37, 11]. +For convenience, we use the 2-D acoustic wave equation to illustrate the +basic idea of ADE-PML. It starts by taking the Fourier transform of the +acoustic wave equation in the tensile coordinates (t → ω) and adding two +stretching terms 1/sx and 1/sz [35], +iω�v1(x, z, ω) = −1 +ρ +1 +sx +� +∂σ +∂x(x, z, ω), +(3.23) +iω�v3(x, z, ω) = −1 +ρ +1 +sz +� +∂σ +∂z (x, z, ω), +(3.24) +iω�σ(x, z, ω) = −ρc2 +P +� +1 +sx +� +∂v1 +∂x (x, z, ω) + 1 +sz +� +∂v3 +∂z (x, z, ω) +� +, +(3.25) +11 + +where �· denotes the Fourier transform, and si (i = x, z) are introduced to +present a complex stretching of the coordinate system, +si = ki + +di +αi + iω, +s−1 +i += 1 +ki +− di +k2 +i +1 +di +ki + αi + iω, +i = x, z. +(3.26) +Here αi, di and ki are flexible parameters that adjust the absorbing effect, +di = d0 +� i +L +�2 +, +ki = 1 + (kmax − 1)m, +αi = αmax +� +1 − i +L +�p +(3.27) +and d0 = −3cP,max log(R)/2L, with cP,max equal to the maximal velocity of +the pressure wave. R is the theoretical reflection coefficient of the target, +αmax = πf0 with f0 as the main frequency of the hypocenter, and kmax is +the cut-off wavenumber [35]. Here it suffices to take m = 1 and p = 1 for +simplicity. +For the first equation 3.23 of � +∂σ +∂x(x, z, ω), it is equivalent to +1 +sx +� +∂σ +∂x(x, z, ω) = 1 +kx +� +∂σ +∂x(x, z, ω) − dx +k2 +x +1 +dx +kx + αx + iω +� +∂σ +∂x(x, z, ω). +(3.28) +By introducing a memory variable �Ψx(x, z, ω), +�Ψx(x, z, ω) = −dx +k2 +x +1 +dx +kx + αx + iω +� +∂σ +∂x(x, z, ω), +(3.29) +it arrives at an auxiliary differential equation by taking inverse Fourier trans- +form �Ψx(x, z, ω) → Ψx(x, z, t), +∂Ψx +∂t (x, z, t) + +�dx +kx ++ αx +� +Ψx(x, z, t) = −dx +k2 +x +∂σ(x, z, t) +∂x +. +(3.30) +Similarly, we can define memory variables Ψz(x, z, t) for � +∂σ +∂z (x, z, ω) in +Eq. 3.24 and Φx(x, z, t), Φz(x, z, t) for � +∂v1 +∂x (x, z, ω), � +∂v3 +∂z (x, z, ω) in Eq. 3.25, re- +spectively, yielding the ADE-PML for velocity components outside the com- +12 + +putational domain, +(A) +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +∂v1 +∂t = −1 +ρ +� 1 +kx +∂σ +∂x + Ψx +� +, +∂v3 +∂t = −1 +ρ +� 1 +kz +∂σ +∂z + Ψz +� +, +∂Ψx +∂t += − +�dx +kx ++ αx +� +Ψx − dx +k2 +x +∂σ +∂x, +∂Ψz +∂t = − +�dz +kz ++ αz +� +Ψz − dz +k2 +z +∂σ +∂z , +(3.31) +and those for the stress component, +(B) +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +∂σ +∂t = −ρc2 +P +� 1 +kx +∂v1 +∂x + Φx + 1 +kz +∂v3 +∂z + Φz +� +, +∂Φx +∂t = − +�dx +kx ++ αx +� +Φx − dx +k2 +x +∂v1 +∂x , +∂Φz +∂t = − +�dz +kz ++ αz +� +Φz − dz +k2 +z +∂v3 +∂z . +(3.32) +The stiff terms in the auxiliary differential equations might pose severe +limitation on the time step when using explicit numerical integrators. For- +tunately, this can be alleviated by the exponential integrator [37, 11]. +For the time interval [tn, tn+1] with ∆t = tn+1 − tn, it starts from the +variation-of-constant formula of Eq. (3.30), +Ψx(x, tn+1) = e−( dx +kx +αx)∆tΨx(x, t) − dx +k2 +x +� ∆t +0 +e−( dx +kx +αx)(∆t−τ)∂σ +∂x(x, tn + τ)dτ. +When the Euler approximation ∂σ +∂x(x, t + τ) ≈ ∂σ +∂x(x, t) is used, it yields [37], +Ψx(x, tn+1) ≈ e−( dx +kx +αx)∆tΨx(x, tn) − dx +k2 +x +1 − e−( dx +kx +αx)∆t +( dx +kx + αx) +∂σ +∂x(x, tn). +(3.33) +The other three auxiliary equations for Φz, Ψx and Ψz can be tackled in a +similar way. Combining with the temporal finite difference scheme for v1, v3 +and σ, one can obtain the non-splitting exponential Euler scheme for ADE- +PML. Because the exact flow of the stiff term in the auxiliary dynamics (3.30) +is exploited, it can largely alleviate the restriction on the time step. +13 + +Alternatively, one can utilize the exponential operator splitting scheme, +which also utilizes the exact stiff flow of the auxiliary equations. The basic +idea is alternating update of velocities and stress based on the splitting of +two subproblems (3.31) and (3.32), which is similar to our short-memory +operator splitting for time-fractional constant-Q wave equation [11]. +One can first solve (A) exactly by assuming that σ, Φx and Φz are in- +variant in small time step. For instance, when ∂σ +∂x is invariant in [tn, tn+1], it +yields the exact solution of Ψx, +Ψx(x, tn+1) = e−( dx +kx +αx)∆tΨx(x, tn) − dx +k2 +x +1 − e−( dx +kx +αx)∆t +( dx +kx + αx) +∂σ +∂x(x, tn). +(3.34) +In addition, since +� tn+1 +tn +∂Ψx +∂t dt = − +�dx +kx ++ αx +� � tn+1 +tn +Ψx(x, t)dt − dx +k2 +x +� tn+1 +tn +∂σ +∂x(x, t)dt += − +�dx +kx ++ αx +� � tn+1 +tn +Ψx(x, t)dt − ∆tdx +k2 +x +∂σ +∂x(x, tn), +it further yields +� tn+1 +tn +Ψx(x, t)dt = − +1 +( dx +kx + αx) +� +Ψ(x, tn+1) − Ψ(x, tn) + ∆tdx +k2 +x +∂σ +∂x(x, tn) +� +. +Substituting it into Eq. (3.31), then the velocity v1 can be solved by +v1(x, tn+1) =v1(x, tn) − +1 +ρ(x) +�∆t +kx +∂σ +∂x(x, tn) + +� tn+1 +tn +Ψx(x, t)dt +� +=v1(x, tn) + +1 +ρ(x) +kx +(dx + αxkx)(Ψx(x, tn+1) − Ψx(x, tn)) +− ∆t +ρ(x) +αx +(dx + αxkx) +∂σ +∂x(x, tn). +(3.35) +The solution of v3 and Φz can be obtained in the same way. Besides, one can +also solve the subsystem (B) exactly when v1, v3, Ψx and Ψz are assumed +to be invariant in small time step. Specifically, when the Strang splitting is +used, say, half step evolution of (A) + full step evolution of (B) + half step +evolution of (A), it is expected to achieve global second-order convergence as +two exact flows are exploited. +14 + +4. Numerical experiments +From this section, several benchmarks have been performed to evaluate +the performance of LOSS. In the first example, we made a series of bench- +marks on 2-D acoustic wave equation in homogenous media to test the conver- +gence of the spline collocation method, where the ADE-PML associated with +the natural boundary condition was adopted. In particular, the influence of +several key parameters, including the layer thickness L, the reflection coeffi- +cient R and the cut-off wavenumber kmax, were carefully studied. After that, +we used LOSS to simulate 3-D wave propagation in either a homogenous me- +dia or a double-layer media activated by the impulse of a Ricker-type wavelet +history. These typical examples may validate the performance of LOSS when +the coefficients are either smooth or of a large variation, as well as its parallel +scalability. +To evaluate the errors of LOSS, we adopted two metrics: the relative l2- +error ε2(t) and the relative maximal error ε∞(t), where vnum +3 +and vref +3 +denoted +the numerical and reference velocity, respectively. +ε2(t) = (� +i |vnum +3 +(xi, t) − vref +3 (xi, t)|2)1/2 +maxx |vref +3 (x, t)| +, +ε∞(t) = maxx |vnum +3 +(x, t) − vref +3 | +maxx |vref +3 (x, t)| +, +All the 2-D simulations were realized by MATLAB, while all the 3-D +simulations performed via our Fortran implementations ran on the platform: +AMD Ryzen 7950X (4.50GHz, 64MB Cache, 16 Cores, 32 Threads) with +64GB Memory (4800Mhz). The parallelization was realized by the Message +Passing Interface (MPI). +4.1. 2-D acoustic wave equation +First we need to validate the convergence of the spline collocation method. +The model parameters were set as: the wave speed cP = 50m/s and the +density ρ = 1kg/m3. The computational domain was [−5, 5]2 (10m×10m). +The Strang operator splitting was adopted with time step ∆t = 0.0001s +and the final time was T = 0.2s. The parameters of PML were given by: the +thickness L = 50, the reflection coefficient R = 10−6, the cut-off wavenumber +kmax = 1 and fP = cP/L, amax = πfP. +Five groups of spline simulations under the grid size N = Nx × Nz = +322, 642, 1282, 2562, 5122 were performed, with zero initial velocity and the +initial pressure +σ(x, z, 0) = e−5(x2+z2). +(4.1) +15 + +The reference solutions were produced by the Fourier spectral method (FSM) +with a Nx × Nz = 2562 grid, where the domain was extended to 27m×27m +with 7122 grid points to avoid the reflection of waves. +The snapshots of vibrational velocity wavefield v3 and the distribution +of numerical errors are visualized in Figures 3 and 4, respectively. +From +Figure 3, the velocity wavefield propagates inside the domain before t = +0.1s, and it begins to leave the domain when t > 0.1s. +Fortunately, the +penetrating wavefields are successfully attenuated by ADE-PML and the +artificial reflection is almost negligible, as clearly observed in Figure 4. +The time evolution of l2-errors ε2(t) are recorded in Table 1 and the +convergence rate is given in Figure 2. The slope of the dashed line is −4, +which perfectly matches the theoretical fourth-order convergence. In addi- +tion, according to Table 1 and Figure 2, the accuracy of ADE-PML can be +significantly improved under a finer grid, and the convergence rate is close +to 4. This indicates the performance of ADE-PML depends heavily on the +accuracy of the underlying numerical solver. +5 +5.5 +6 +6.5 +7 +7.5 +8 +8.5 +9 +-25 +-20 +-15 +-10 +-5 +0 +5 +(a) When wave propagates in the domain. +5 +5.5 +6 +6.5 +7 +7.5 +8 +8.5 +9 +-25 +-20 +-15 +-10 +-5 +0 +5 +(b) When wave leaves the domain. +Figure 2: 2-D acoustic wave equation: The fourth-order convergence of the spline colloca- +tion method is verified for ADE-PML. +4.2. The performance of ADE-PML under spline collocation method +As mentioned above, the performance of ADE-PML relies on three param- +eters: the thickness L, the reflection coefficient R and the cut-off wavenumber +kmax [33, 34, 35]. Therefore, their influence on the absorbing effect, as well +16 + +-5 +0 +5 +x +-5 +-4 +-3 +-2 +-1 +0 +1 +2 +3 +4 +5 +z +-4 +-3 +-2 +-1 +0 +1 +2 +3 +4 +10-3 +(a) t = 0.02s. +-5 +0 +5 +x +-5 +-4 +-3 +-2 +-1 +0 +1 +2 +3 +4 +5 +z +-2 +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +2 +10-3 +(b) t = 0.06s. +-5 +0 +5 +x +-5 +-4 +-3 +-2 +-1 +0 +1 +2 +3 +4 +5 +z +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +10-3 +(c) t = 0.10s. +-5 +0 +5 +x +-5 +-4 +-3 +-2 +-1 +0 +1 +2 +3 +4 +5 +z +-1 +-0.5 +0 +0.5 +1 +10-3 +(d) t = 0.12s. +-5 +0 +5 +x +-5 +-4 +-3 +-2 +-1 +0 +1 +2 +3 +4 +5 +z +-2.5 +-2 +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +2 +2.5 +10-4 +(e) t = 0.16s. +-5 +0 +5 +x +-5 +-4 +-3 +-2 +-1 +0 +1 +2 +3 +4 +5 +z +-2.5 +-2 +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +2 +2.5 +10-5 +(f) t = 0.20s. +Figure 3: 2-D acoustic wave equation: The snapshot of vibration velocity wavefields v3. +The wave begins to be absorbed by PML at t = 0.1s. +Table 1: 2-D acoustic wave equation: The l2-error ε2(t) at different instants. The nu- +merical accuracy of the spline collocation method and ADE-PML can be systematically +improved by refining the grid mesh. +Time +Grid +N = 322 +N = 642 +N = 1282 +N = 2562 +N = 5122 +When wave propagates inside the domain +0.02s +7.634 ×10−1 +5.663 ×10−2 +3.382 ×10−3 +2.057 ×10−4 +1.276 ×10−5 +0.04s +3.634×10−1 +2.197×10−2 +1.264×10−3 +7.640×10−5 +4.737×10−6 +0.06s +4.476×10−1 +3.638×10−2 +2.167×10−3 +1.309×10−4 +8.118×10−6 +When wave is absorbed by PML outside the domain +0.10s +2.413×10−1 +3.099×10−2 +1.469×10−3 +8.802×10−5 +5.466×10−6 +0.15s +2.186×10−1 +1.038×10−2 +3.837×10−4 +2.329×10−5 +1.445×10−6 +0.20s +2.441×10−1 +1.779×10−2 +1.080×10−3 +6.611×10−5 +4.238×10−6 +as the compatibility with natural boundary conditions for the cubic spline, +deserves a careful investigation. +Among them, the most important parameter is the layer thickness L. +17 + +-5 +0 +5 +x +-5 +-4 +-3 +-2 +-1 +0 +1 +2 +3 +4 +5 +z +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +10-7 +(a) t = 0.02s. +-5 +0 +5 +x +-5 +-4 +-3 +-2 +-1 +0 +1 +2 +3 +4 +5 +z +-3 +-2 +-1 +0 +1 +2 +3 +10-7 +(b) t = 0.06s. +-5 +0 +5 +x +-5 +-4 +-3 +-2 +-1 +0 +1 +2 +3 +4 +5 +z +-4 +-3 +-2 +-1 +0 +1 +2 +3 +4 +10-7 +(c) t = 0.10s. +-5 +0 +5 +x +-5 +-4 +-3 +-2 +-1 +0 +1 +2 +3 +4 +5 +z +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +10-7 +(d) t = 0.12s. +-5 +0 +5 +x +-5 +-4 +-3 +-2 +-1 +0 +1 +2 +3 +4 +5 +z +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +10-8 +(e) t = 0.16s. +-5 +0 +5 +x +-5 +-4 +-3 +-2 +-1 +0 +1 +2 +3 +4 +5 +z +-4 +-3 +-2 +-1 +0 +1 +2 +3 +4 +10-11 +(f) t = 0.20s. +Figure 4: 2-D acoustic wave equation: Visualization of numerical errors in the vibrational +velocity wavefields v3. +Intuitively speaking, increasing the layer thickness improves the absorbing +effect of PML, at the cost of storing more wavefields and higher computa- +tional costs. Therefore, it is expected to use absorbing layers as fewer as +possible to maintain the accuracy. +To evaluate its effect, we have performed a series of benchmarks under +different L and grids. +The l2-errors at t = 0.2s, as recorded in Table 2, +were adopted to measure the accuracy of ADE-PML. The convergence with +respect to the mesh size is plotted in Figure 5. Clearly, the accuracy of ADE- +PML can be systematically improved as N increases, and the convergence +rate matches the theoretical order 4 when L is sufficiently large. +It is observed that too small L may result in a dramatic reduction in +accuracy for sufficiently large N, while its influence becomes negligible when +Nx = Nz ≤ 128 because the discretization errors dominate. Therefore, the +choice of L should match the size of grid mesh. When a coarse grid is used, +the thickness might be not too large for the sake of efficiency. By contrast, +18 + +5 +5.5 +6 +6.5 +7 +7.5 +8 +8.5 +9 +-25 +-20 +-15 +-10 +-5 +0 +5 +(a) t=0.1s. +5 +5.5 +6 +6.5 +7 +7.5 +8 +8.5 +9 +-25 +-20 +-15 +-10 +-5 +0 +5 +(b) t=0.15s. +5 +5.5 +6 +6.5 +7 +7.5 +8 +8.5 +9 +-25 +-20 +-15 +-10 +-5 +0 +5 +(c) t=0.20s. +Figure 5: 2-D PML: The convergence of l2-errors under different boundary layer thick- +nesses L. It is observed that L = 40-50 might be a good choice to strike a balance in +accuracy and efficiency. +Table 2: 2-D PML: The l2-errors at t = 0.2s under different boundary layer thicknesses L +and grid meshes. The errors induced by PML are negligible when L ≥ 50. +L +Grid +N = 322 +N = 642 +N = 1282 +N = 2562 +N = 5122 +10 +1.405×10−1 +1.089×10−2 +6.785×10−4 +4.155×10−5 +1.453×10−5 +20 +1.394×10−1 +1.077×10−2 +6.785×10−4 +4.155×10−5 +2.593×10−6 +30 +1.394×10−1 +1.077×10−2 +6.787×10−4 +4.155×10−5 +2.587×10−6 +40 +1.394×10−1 +1.077×10−2 +6.786×10−4 +4.154×10−5 +2.581×10−6 +50 +1.394×10−1 +1.077×10−2 +6.786×10−4 +4.157×10−5 +2.580×10−6 +60 +1.394×10−1 +1.077×10−2 +6.786×10−4 +4.156×10−5 +2.581×10−6 +70 +1.394×10−1 +1.077×10−2 +6.786×10−4 +4.154×10−5 +2.579×10−6 +80 +1.394×10−1 +1.077×10−2 +6.786×10−4 +4.155×10−5 +2.577×10−6 +90 +1.394×10−1 +1.077×10−2 +6.786×10−4 +4.155×10−5 +2.578×10−6 +100 +1.394×10−1 +1.077×10−2 +6.786×10−4 +4.155×10−5 +2.580×10−6 +when a fine grid mesh is used, one should use a thicker absorbing layer to +ensure the accuracy. Based on the results in Figure 5, L = 40 to 50 might be +a good choice to strike a balance in accuracy and efficiency, while too large +L might not bring in evident improvements as the accuracy is still limited +by the numerical solver under the specified grid mesh. +Second, we studied the effect of the reflection coefficient R on the absorb- +ing boundary. The mesh size was fixed to be Nx × Nz = 5122 with the layer +thickness L = 50. The time evolution of the l2-errors are plotted in Figure +6(a), corresponding to the data in Table 3. It is observed that R = 10−6 +achieves l2-error about 10−6, whereas it seems to reach the limitation of ac- +19 + +curacy as the errors vary slightly when R further decreases. These findings +coincide with the observation made in [32, 35] as the choice of R should +match the thickness L of PML. +0.06 +0.08 +0.1 +0.12 +0.14 +0.16 +0.18 +0.2 +-6 +-5 +-4 +-3 +-2 +-1 +0 +(a) Different reflection coefficients R. +0.06 +0.08 +0.1 +0.12 +0.14 +0.16 +0.18 +0.2 +-6 +-5 +-4 +-3 +-2 +-1 +0 +1 +2 +3 +(b) Different cut-off wavenumber kmax. +Figure 6: 2-D PML: Time evolution of l2-errors under different reflection coefficients R +and cut-off wavenumber kmax. The mesh size is fixed to be N = 5122. From numerical +perspective, The choice of R should match the thickness L of PML, and cut-off wavenumber +kmax = 1 achieves the optimal performance. +Table 3: 2-D PML: The l2-errors under different reflection coefficients R, with Nx = Nz = +512. Here R = 10−6 is suggested to ensure the effectiveness of PML. +R +Time +0.06s +0.08s +0.10s +0.12s +0.14s +0.16s +0.18s +0.20s +10−1 +3.11×10−6 +2.73×10−6 +2.52×10−6 +4.99×10−5 +8.15×10−5 +1.00×10−4 +1.38×10−4 +4.32×10−4 +10−2 +3.11×10−6 +2.73×10−6 +2.52×10−6 +2.66×10−5 +3.99×10−5 +5.40×10−5 +6.28×10−5 +5.97×10−5 +10−3 +3.11×10−6 +2.73×10−6 +2.52×10−6 +1.07×10−5 +1.80×10−5 +2.28×10−5 +3.24×10−5 +2.98×10−5 +10−4 +3.11×10−6 +2.73×10−6 +2.52×10−6 +3.77×10−6 +7.43×10−6 +8.54×10−6 +1.48×10−5 +1.44×10−5 +10−5 +3.11×10−6 +2.73×10−6 +2.52×10−6 +1.51×10−6 +2.76×10−6 +3.20×10−6 +7.67×10−6 +8.13×10−6 +10−6 +3.11×10−6 +3.11×10−6 +2.53×10−6 +1.16×10−6 +1.80×10−6 +2.40×10−6 +5.92×10−6 +5.98×10−6 +10−7 +3.11×10−6 +2.73×10−6 +2.53×10−6 +1.13×10−6 +1.77×10−6 +2.39×10−6 +5.58×10−6 +5.58×10−6 +10−8 +3.11×10−6 +2.73×10−6 +2.53×10−6 +1.12×10−6 +1.77×10−6 +2.39×10−6 +5.51×10−6 +5.48×10−6 +10−9 +3.11×10−6 +2.73×10−6 +2.53×10−6 +1.12×10−6 +1.77×10−6 +2.39×10−6 +5.51×10−6 +5.48×10−6 +10−10 +3.11×10−6 +2.73×10−6 +2.53×10−6 +1.12×10−6 +1.78×10−6 +2.39×10−6 +5.53×10−6 +5.51×10−6 +Finally, for the cut-off wavenumber kmax, we also plot the time evolution of +the l2-errors under different kmax in Figure 6(b), with data collected in Table +4. The mesh size was fixed to be Nx ×Nz = 5122 and the layer thickness was +again set as L = 50. The results show that ADE-PML can work only when +kmax = 1. In fact, just as pointed out in [34], the sharp variations of the +profile of the kx and kz functions might augment the reflection coefficient of +waves impinging on the boundary, so that too large kmax is not recommended. +20 + +Table 4: 2-D PML: The l2-errors under different cut-off wavenumber kmax, with Nx = +Nz = 512. The ADE-PML may only work when kmax = 1. +kmax +Time +0.06s +0.08s +0.10s +0.12s +0.14s +0.16s +0.18s +0.20s +0.5 +3.11×10−6 +1.30×10−5 +1.02×10−3 +5.61×10−3 +1.31×10−2 +1.37×10−2 +1.29×10−2 +9.34×10−3 +1 +3.11×10−6 +2.73×10−6 +2.52×10−6 +1.51×10−6 +2.76×10−6 +3.20×10−6 +7.67×10−6 +8.13×10−6 +6 +3.11×10−6 +5.52×10−4 +7.95×10−2 +1.73×10−1 +2.07×10−1 +2.61×10−1 +5.37×10−1 +5.89×10−1 +11 +3.11×10−6 +6.96×10−4 +1.68×10−1 +2.87×10−1 +4.04×10−1 +3.71×10−1 +1.03 +1.05 +16 +3.11×10−6 +7.27×10−4 +1.94×10−1 +2.90×10−1 +4.21×10−1 +3.64×10−1 +1.09 +9.91×10−1 +19 +3.11×10−6 +3.11×10−6 +2.01×10−1 +2.90×10−1 +4.21×10−1 +3.76×10−1 +1.09 +9.59×10−1 +4.3. Elastic wave propagation in 3-D homogenous media +To study the wave equation in 3-D homogenous media, we set the ini- +tial velocity and stress tensors at equilibrium state and simulated the wave +propagation activated by source functions with a Ricker-type wavelet history, +f1(x, t) = f2(x, t) = f3(x, t) = A(x)fr(t), +(4.2) +where the amplitude A(x) was a Gaussian profile +A(x) = e−(x−x0)2−(y−y0)2−(z−z0)2 +(4.3) +and the Ricker wavelet was given by +fr(t) = (1 − 2(πfP(t − dr)2))e−(πfP (t−dr))2, +(4.4) +where fP was the peak frequency and dr was the temporal delay. Here we +chose fP = 100Hz, dr = 0 and the centre position x0 = 0, y0 = 0, z0 = 10. +The model parameters in a homogenous media were set as follows: The group +velocities for P-wave and S-wave were cP = 2.614km/s and cS = 0.802km/s, +respectively, and the mass density was ρ = 2.2kg/m3. +The computational domain was [−40, 40]3 (80km × 80km × 80km). Four +groups of simulations have been performed under N = Nx × Ny × Nz = +1293, 2573, 3853, 5133. The Strang splitting was adopted with time step ∆t = +0.005s and the final instant was T = 10s (2000 steps in total). The reference +solution was produced by FSM with a fine grid mesh N = 5123 for testing +the convergence of LOSS. +As a pretreatment in LOSS, the domain was evenly decomposed into +4 × 4 × 4 patches, thereby achieving a perfect balance in overload. Here +PMBCs were assembled by truncating the neighborhoods at nnb = 20 [29]. A +visualization of the domain decomposition in 3-D space is presented in Figure +21 + +9(a), when the whole computational domain is decomposed into 2 × 2 × 2 +cells. For the calculations of first-order spatial derivatives, it only requires +the local spline expansion in one direction, where PMBCs are assembled at +the shared faces. For instance, when one needs the first-order derivatives in +x-direction, it only requires the communications in 1 ↔ 2, 3 ↔ 4, 5 ↔ 6 and +7 ↔ 8, thereby greatly reducing the communication cost. +7 +7.2 +7.4 +7.6 +7.8 +8 +8.2 +8.4 +8.6 +8.8 +9 +-10 +-9 +-8 +-7 +-6 +-5 +-4 +-3 +-2 +-1 +0 +7 +7.2 +7.4 +7.6 +7.8 +8 +8.2 +8.4 +8.6 +8.8 +9 +-4 +-2 +0 +2 +4 +6 +8 +(a) Convergence with respect to Nz. (left: maximal error ε∞(10), right: l2-error ε2(10)) +-40 +-30 +-20 +-10 +0 +10 +20 +30 +40 +-4 +-3 +-2 +-1 +0 +1 +2 +3 +10-5 +-40 +-30 +-20 +-10 +0 +10 +20 +30 +40 +-0.15 +-0.1 +-0.05 +0 +0.05 +0.1 +0.15 +0.2 +(b) v3(0, 0, z) in the homogenuous media (left) and the relative errors (right) at t = 10s. +Figure 7: Wave propagation in 3-D homogenous media: A comparison of the vi- +brational wavefield v3(0, 0, z) at t = 10s. The convergence of LOSS is verified for +smooth data and coefficients. +The relative maximal error ε∞(t) and l2-error ε2(t) were again adopted +to measure the numerical accuracy and the spatial convergence is plotted +in Figure 7(a). Clearly, the convergence rate coincides with the theoreti- +cal fourth order. We also investigate the influence of the parameter nnb in +assembling PMBCs. As shown in Figure 7(a), the errors induced by trunca- +tion in PMBC seem negligible even when nnb = 10, which accords with our +observations made in [29]. +The snapshots of the vibrational wave-field v3 of the P- and S- wave are +given in Figure 8, where three sectional drawings are provided to visualize +the wave propagation in the homogenous media. To further demonstrate the +errors of LOSS, we plot the synthetic data v3(0, 0, z) at t = 10s in Figure +22 + +(a) v3(x, y, z) at t = 4s. (left: FSM, N = 5123, right: LOSS, N = 5133) +(b) v3(x, y, z) at t = 6s. (left: FSM, N = 5123, right: LOSS, N = 5133) +(c) v3(x, y, z) at t = 8s. (left: FSM, N = 5123, right: LOSS, N = 5133) +(d) v3(x, y, z) at t = 10s. (left: FSM, N = 5123, right: LOSS, N = 5133) +Figure 8: Wave propagation in 3-D homogenous media: A comparison of snapshots +of wavefield v3(x, y, z). +23 + +4 +2 +40 +30 +0 +20 +-2 +2 +10. +-4 +0 +-10 +-6 +-10 +-10 +0 +0 +10 +-8 +10 +20 +20 +30 +30 +-10 +y +40 +40 +×10~540 +0 +30 +20 +2 +10. +0 +-5 +-10 +-10 +-10 +0 +0 +10 +10 +20 +20 +-10 +30 +30 +y +40 +40 +×10-53 +2 +1 +40 +0 +30 +-1 +20 +2 +-2 +10. +-3 +0 +-4 +-10 +-10 +-10 +0 +0 +-5 +10 +10 +20 +20 +-6 +30 +30 +y +40 +40 +×10~53 +2 +1 +40 +0 +30 +-1 +20 +2 +-2 +10. +-3 +0 +-4 +-10 +-10 +-10 +0 +0 +-5 +10 +10 +20 +20 +-6 +30 +30 +y +40 +40 +×10~52 +1 +40 +0 +30 +20 +-1 +2 +10. +-2 +0 +-3 +-10 +-10 +-10 +0 +0 +-4 +10 +10 +20 +20 +30 +30 +-5 +y +40 +40 +×10-52 +1 +40 +0 +30 +20 +-1 +2 +10. +-2 +0 +-3 +-10 +-10 +-10 +0 +0 +-4 +10 +10 +20 +20 +30 +30 +-5 +y +40 +40 +×10~540 +0 +30 +20 +-1 +2 +10. +0 +-2 +-10 +-10 +-10 +0 +0 +-3 +10 +10 +20 +20 +30 +30 +-4 +y +40 +40 +×10~540 +0 +30 +20 +1 +2 +10. +0 +-2 +-10 +-10 +-10 +0 +-3 +0 +10 +10 +20 +20 +30 +30 +-4 +y +40 +40 +×10~57(b). Small oscillations are observed around the reference solution when too +coarse grid mesh is used (N = 2573). Nonetheless, they could be suppressed +to a large extent when the finer grid mesh was adopted. +4.4. Elastic wave propagation in a 3-D double-layer media +As a more challenging example, we also studied the wave propagation in a +double-layer structure, with the same computational domain [−40, 40]3 and +the same source impulse as adopted in Section 4.3. The model parameters in a +heterogenous media, iven in Figure 9(b), were set to be: The group velocities +for the upper layer 402×[0, 40] were cP = 2.614km/s and cS = 0.802km/s and +the mass density was ρ = 2.2kg/m3, while for the deeper layer 402 ×[−40, 0], +cP = 5.228km/s, cS = 1.604km/s and ρ = 2.5kg/m3. +(a) Domain decomposition in 3-D space. +(b) A double-layer heterogeneous structure. +Figure 9: An illustration of domain decomposition of 3-D space and a double-layer het- +erogeneous structure. +Seven groups of simulations have been performed under N = Nx × Ny × +Nz = 1293, 2573, 3853, 5133, 6413, 7693 and 5132 × 1025. The Strang splitting +was adopted with time step ∆t = 0.005s and the final instant was T = 10s +(2000 steps in total). The reference solution was still produced by FSM with +a fine grid mesh N = 6403. +The snapshots of the vibrational wavefield v3 of the P- and S-wave are +given in Figure 12. Unlike the propagation in the homogenous media, the +scattering of wave occurs when it enters a different media, resulting in the +reflection of wavepackets. From the synthetic data in Figures 10(b) and 10(c), +the superposition of the impulse and reflected wave is clearly observed. +The convergence of LOSS is still verified in Figure 10(a), albeit with +an evident reduction in the order. +This is caused by the sharp variation +of the model parameters. As visualized in Figures 10(b) and 10(c), small +24 + +8 +7 +40 +30 ~ +4 +20 ~ +5 +10 ~ +6 +2 +-0 +3 +-10 ~ +-20 ~ +1 +-40 +-30 +-20 +-40 +2 +40 +0 +20 +0 +20 +-20 +40 +y +-4040 +30 +20 +10- +cp = 2.614km/s, cs = 0.802km/s,p = 2.2kg/m3 +2 +0 +-10 +-20 +-40 +-30 +cp = 5.228km/s, cs = 1.604km/s, p = 2.5kg/m3 +-20 +-40 +40 +0 +20 +0 +20 +-20 +40 +y +-407 +7.5 +8 +8.5 +9 +9.5 +-5 +-4.5 +-4 +-3.5 +-3 +-2.5 +-2 +-1.5 +-1 +-0.5 +0 +7 +7.5 +8 +8.5 +9 +9.5 +-2 +-1 +0 +1 +2 +3 +4 +5 +6 +7 +(a) Convergence with respect to Nz (left: maximal error ε∞(10), right: l2-error ε2(10)). +-40 +-30 +-20 +-10 +0 +10 +20 +30 +40 +-6 +-5 +-4 +-3 +-2 +-1 +0 +1 +2 +3 +4 +10-5 +-40 +-30 +-20 +-10 +0 +10 +20 +30 +40 +-0.2 +-0.15 +-0.1 +-0.05 +0 +0.05 +0.1 +0.15 +(b) v3(0, 0, z) in the heterogenuous media (left) and the relative errors (right) at t = 6s. +-40 +-30 +-20 +-10 +0 +10 +20 +30 +40 +-4 +-3 +-2 +-1 +0 +1 +2 +3 +10-5 +-40 +-30 +-20 +-10 +0 +10 +20 +30 +40 +-0.2 +-0.15 +-0.1 +-0.05 +0 +0.05 +0.1 +0.15 +0.2 +0.25 +(c) v3(0, 0, z) in the heterogenuous media (left) and the relative errors (right) at t = 10s. +Figure 10: Wave propagation in 3-D heterogeneous media: A comparison of the +vibrational data v3(0, 0, z) at t = 6s and t = 10s. The reduction in the conver- +gence order is observed when model parameters have sharp variations, and small +oscillations are produced by the spline construction. Nevertheless, the errors can +be diminished as a finer grid mesh is adopted. +oscillations in the reflected waves are produced by the spline construction +under a coarse grid mesh (Nx ≤ 385). Fortunately, such phenomenon can +be alleviated when a finer grid mesh was adopted. In particular, we tried +to refine the grid mesh in z-direction, where there was a sharp variation in +density and group velocities, and found that it succeeded in further reducing +the errors (see Table 5). This manifests that LOSS is still capable to deal +25 + +with the multi-layer geological model, albeit the grid mesh must be refined +near the discontinuity of model parameters. +Table 5: The memory requirement for storing nine wavefields (in both serial setting +and 4 × 4 × 4 decomposition) and the numerical errors up to T = 10s (with +∆t = 0.005s, 2000 steps). +Grid +1293 +2573 +5133 +6413 +5132 × 1025 +7693 +Memory (Serial) +0.14GB +1.14GB +9.05GB +17.66GB +18.09GB +30.49GB +Memory (MPI) +0.17GB +1.23GB +9.43GB +18.25GB +18.64GB +31.33GB +ε2(t = 10) +77.845 +26.078 +8.320 +5.742 +5.334 +4.985 +ε∞(t = 10) +0.785 +0.300 +0.100 +0.075 +0.083 +0.067 +Finally, we tested the complexity of LOSS by recording the computational +time under N = NxNyNz = 1293, 2573, 5133, 6413, 7693. As shown in Figure +11(a), the complexity of LOSS is almost proportional to the mesh size N. +Besides, we also calculated the speedup ratio of LOSS in Figure 11(b) by +performing the simulations under the fixed grid mesh N = 2413 and time +step ∆t = 0.005s. For serial realization, it required 11.85 hours to reach the +final time T = 10s (2000 steps in total). By contrast, when the domain was +decomposed into 4 × 4 × 4 patches and 32 cores (64 tasks) were used, it only +took 0.85 hour and the speedup ratio was about 43.71%. In fact, the storage +of shared grids is relatively small compared with the total memory require- +ment (see Table 5). Thus it is expected that LOSS can achieve even higher +speedup ratio when the latency caused by the Hyper-Threading technique is +precluded. +6.5 +7 +7.5 +8 +8.5 +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +2 +2.5 +(a) Computational complexity. +5 +10 +15 +20 +25 +30 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +(b) Speedup ratio. +Figure 11: The complexity of LOSS is almost proportional to NxNyNz. In addition, it +achieves a speedup ratio about 43.71% using 32 cores. The platform is AMD Ryzen 7950X +(4.50GHz, 64MB Cache, 16 Cores, 32 Threads) with 64GB Memory (4800Mhz). +26 + +(a) v3(x, y, z) at t = 4s. (left: FSM, N = 5123, right: LOSS, N = 5132 × 1025) +(b) v3(x, y, z) at t = 6s. (left: FSM, N = 5123, right: LOSS, N = 5132 × 1025) +(c) v3(x, y, z) at t = 8s. (left: FSM, N = 5123, right: LOSS, N = 5132 × 1025) +(d) v3(x, y, z) at t = 10s. (left: FSM, N = 5123, right: LOSS, N = 5132 × 1025) +Figure 12: Wave propagation in 3-D double-layer media: A comparison of snapshots +of wavefield v3(x, y, z). +27 + +5 +40 +30 +0 +20 +2 +10. +0 +-5 +-10 +-10 +-10 +0 +0 +10 +10 +20 +20 +30 +30 +-10 +y +40 +40 +×10~540 +30 +0 +20 +2 +10. +0 +-5 +-10 +-10 +-10 +0 +0 +10 +10 +20 +20 +30 +30 +-10 +y +40 +40 +×10~53 +2 +40 +1 +30 +0 +20 +-1 +2 +10. +-2 +0 +-3 +-10 +-4 +-10 +-10 +0 +0 +-5 +10 +10 +20 +20 +-6 +30 +30 +y +40 +40 +×10~53 +2 +1 +40 +0 +30 +-1 +20 +2 +-2 +10. +-3 +0 +-4 +-10 +-10 +-10 +0 +0 +-5 +10 +10 +20 +20 +-6 +30 +30 +y +40 +40 +×10~52 +40 +1 +30 +0 +20 +-1 +2 +10. +-2 +0 +-3 +-10 +-10 +-10 +0 +0 +-4 +10 +10 +20 +20 +-5 +30 +30 +y +40 +40 +×10~53 +2 +40 +1 +30 +0 +20 +-1 +2 +10. +-2 +0 +-3 +-10 +-10 +-10 +0 +0 +-4 +10 +10 +20 +20 +-5 +30 +30 +y +40 +40 +×10~52 +1 +40 +30 +0 +20 +2 +-1 +10. +0 +-2 +-10 +-10 +-10 +0 +0 +-3 +10 +10 +20 +20 +30 +30 +-4 +y +40 +40 +×10~51 +40 +0 +30 +20 +-1 +2 +10. +0 +-2 +-10 +-10 +-10 +0 +-3 +0 +10 +10 +20 +20 +30 +30 +-4 +y +40 +40 +×10~55. Conclusions and discussions +This paper proposes the distributed local spline simulator (LOSS) for +solving the elastic wave propagation, where the wavefields are expanded by +patched local cubic B-splines and the first-order spatial derivatives can be +calculated accurately with low complexity. A perfectly matched boundary +condition (PMBC) is introduced by exploiting the exponential decay prop- +erty of the wavelet basis in its dual space. In this manner, the local spline is +able to recover the global spline as accurately as possible with only local com- +munication costs, thereby greatly facilitating the distributed parallelization. +Several typical 2-D and 3-D examples are provided to validate the accuracy, +efficiency and parallel scalability of LOSS. +For brevity, we only discuss the implementation under a uniform grid +mesh, but a nonuniform grid is much more desirable when the model pa- +rameters have discontinuities or large variations. Actually, the settings of +PMBCs can be readily generalized to the non-uniform grid owing to the +scaling and flexibility of wavelet basis, and the implementation of LOSS on +a structure-driven grid mesh may be a topic of our future work. +Acknowledgement +This research was supported by the National Natural Science Foundation +of China (Nos. 12271303). +X. Guo was partially supported by the Nat- +ural Science Foundation of Shandong Province for Excellent Youth Schol- +ars (Nos. ZR2020YQ02) and the Taishan Scholars Program of Shandong +Province of China (Nos. tsqn201909044). Y. Xiong was partially supported +by the National Natural Science Foundation of China (No. 1210010642) and +the Fundamental Research Funds for the Central Universities (Nos. 310421125). +References +[1] J. Virieux, S. Operto, An overview of full-waveform inversion in explo- +ration geophysics, Geophysics 74(6) (2009). +[2] J. Tromp, Seismic wavefield imaging of Earth’s interior across scales, +Nat. Rev. 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Anal. 43 (3) +(2005) 1069–1090. +32 + diff --git a/JtE4T4oBgHgl3EQfhw0J/content/tmp_files/load_file.txt b/JtE4T4oBgHgl3EQfhw0J/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e3cd36c9dd55f0099b01051db22427ed562d2a4d --- /dev/null +++ b/JtE4T4oBgHgl3EQfhw0J/content/tmp_files/load_file.txt @@ -0,0 +1,1187 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf,len=1186 +page_content='Distributed local spline simulator for wave propagation Xu Guoa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Yaomeng Lia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Yunfeng Xiongb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='∗∗ aGeotechnical and Structural Engineering Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Shandong University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Jinan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 250061,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Shandong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' China bSchool of Mathematical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Beijing Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 100871,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' China Abstract Numerical simulation of wave propagation in elastic media faces the chal- lenges arising from increasing demand of high resolution in modern 3-D imag- ing applications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' which requires a balance between efficiency and accuracy in addition to being friendly to the distributed high-performance computing environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' In this paper, we propose a distributed local spline simulator (LOSS) for solving the wave equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' LOSS uses patched cubic B-splines to represent the wavefields and attains an accurate evaluation of spatial deriva- tives with linear complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' In order to link the adjacent patches, a perfectly matched boundary condition is introduced to give a closure of local spline coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Owing to the rapid decay property of the local wavelets in dual space, it can recover the global spline as accurately as possible only at the cost of local communications among adjacent neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Several typical nu- merical examples, including 2-D acoustic wave equation and P- and S- wave propagation in 3-D homogenous or heterogenous media, are provided to val- idate its convergence, accuracy and parallel scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Keywords: Wave equation, Spline collocation method, Artificial boundary condition, Parallel and distributed computing 2020 MSC: , 74J05, 65D07, 65M22, 65Y05, 68W15 ∗These authors contribute equally to this paper ∗∗To whom correspondence Email address: yfxiong@bnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='cn (Yunfeng Xiong) Preprint submitted to Elsevier January 13, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='05127v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='NA] 12 Jan 2023 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Introduction The wave propagation in elastic media plays an essential role in the field of geological imaging techniques [1, 2, 3] and new-trend neuroimaging [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Nowadays, a large collection of efficient numerical techniques is available for both forward and inverse wavefield modelings, including the widely used staggered-grid finite difference method [5, 6], the optimal difference method [7], the pseudo-spectral method [8, 9, 10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' In spite of their great success and extensive applications, these standard techniques face new challenges by tremendous memory demand and significant computational cost especially for 3-D problems, arising from the increasing need for high resolution in mod- ern imaging applications, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=', the teleseismic datasets for waveform inver- sion and deep lithospheric structures [2, 12] or the sub-millimetre-resolution in brain and surrounding tissue [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Thus, it urgently calls for discretized methods to account for sharp variations of solutions induced by material dis- continuities accurately [13] and to be friendly to large-scale high-performance computing environment [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' In recent years, the spectral element method (SEM) and discontinu- ous Galerkin method (DG) [15, 16] have gained an increasing attention [12, 13, 17, 18, 19] as they take advantage of both flexibility of the finite element method (FEM) in resolving multiscale phenomena [14] and high ac- curacy of the spectral method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' A significant merit of SEM and DG over FEM is that the mass matrix is exactly diagonal by construction, which drasti- cally simplifies the implementation and the temporal integration [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' More- over, the assembly of the stiffness matrix can be performed in an element- by-element manner, thereby greatly facilitating the parallelization in a dis- tributed computing environment [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' But the evaluation of the stiffness matrix at the elemental level has a relatively high computational cost due to the matrix multiplications involved, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=', the complexity scales as O(n4 l ) for SEM in three dimensions with nl the polynomial degree used to present the functions in each direction [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' It may somehow pose a limitation on nl to achieve a trade-off in accuracy and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' As forward wavefield modelings have to be simulated thousands of times in waveform-fitting imaging, reducing the computational complexity of nu- merical solvers is always a central issue especially for 3-D problems [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Thus it is natural to seek a local polynomial basis that can calculate the spatial derivatives of wavefields accurately with relatively low computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' To achieve this, we propose a distributed local spline simulator (LOSS) for 2 solving the wave propagation, using the local cubic B-spline wavelet as the basis [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The name LOSS comes from the semi-Lagrangian methods in computational fluid dynamics [21, 22] and kinetic theory [23, 24, 25], where the cubic spline has been ubiquitously applied for interpolating the advec- tion and is believed to strike the best balance between accuracy and cost [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The cubic spline achieves spatial fourth-order convergence [26] and its construction can be realized by the standard sweeping method, where the complexity scales linearly with respect to mesh size [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' For these reasons, the most recent semi-Lagrangian methods are even capable to resolve the kinetic-type equation in full 6-D phase space, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=', the Vlasov equation [28] and quantum Wigner equation [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' In structural mechanics, the cubic spline has been also used to solve the wave propagation in cracked rod [30] and the wave-structure interaction [31] within the framework of FEM, while the spline coefficients are globally dependent in principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' A major advantage of LOSS lies in its distributed construction with only local communication cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' This is based on a key observation that the wavelet basis decay exponentially in the dual space [20, 22], so that the vanished off-diagonal elements in the inverse coefficient matrix can be truncated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' As a consequence, a perfectly matched boundary condition (PMBC) can be introduced to give a closure of patched spline coefficients and allows local splines to recover the global one as accurately as possible [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Since only local communications in adjacent neighbors are needed, LOSS is expected to be suitable for the computational clusters with high-latency network, known as the Beowulf machines [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' We will also show that the natural boundary conditions on two ends of splines are fully compatible with the absorbing perfectly matched layers (PML) [32, 33, 34, 35] in outer domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' At present, LOSS is readily implemented in the standard architecture and may potentially alleviate both the memory limitation and the computational burden for high-resolution wave propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' In Section 2, we briefly re- view the background of the elastic wave propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Section 3 illustrates the formulation of LOSS in both serial and parallel settings, as well as the expo- nential integrator for the auxiliary differential equation form of the perfectly matched layer (ADE-PML).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' In Section 4, we provide a series of benchmark by simulating 2-D acoustic wave equation to test the convergence and ac- curacy of LOSS, as well as its compatibility with ADE-PML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The P- and S- wave propagation in 3-D homogenous or heterogenous media will also be investigated to validate the performance and parallel scalability of LOSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 3 Finally, conclusions and discussions are drawn in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Background The dynamics of wave propagation is governed by three sets of equations with x ∈ Rd, d ≤ 3 [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The first set is the conservation of linear momentum: ρ(x) ∂2 ∂t2ui(x, t) = ∂ ∂xj σij(x, t) + fi(x, t), i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' , 3, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='1) where σij are the components of the stress tensor, ui are the components of the displacement vector, ρ is the mass density and fi are components of the body forces per unit (source term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The summation over repeated indices j is assumed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The second set is the definition of strain tensor εij, which can be obtained in terms of the displacement components as εij(x, t) = 1 2 � ∂ ∂xi uj(x, t) + ∂ ∂xj ui(x, t) � , i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' , 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='2) The constitutive equation reads that σij(x, t) = MP(x)εkk(x, t)δij + 2MS(x)(εij(x, t) − εkk(x, t)δij), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='3) where the summation over the repeated indices k is assumed, δij is the Kro- necker symbol, MP(x) = c2 P(x)ρ(x) and MS(x) = c2 S(x)ρ(x) are moduli with cP(x) and cS(x) the P- and S-velocities, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' For instance, the velocity-stress form of the full elastic wave equation in 3-D media involves 15 wavefields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The conservation of momentum is ρ ∂ ∂t � � v1 v2 v3 � � = � � ∂ ∂x1 0 0 0 ∂ ∂x2 0 0 0 ∂ ∂x3 � � � � σ11 σ22 σ33 � � + � � ∂ ∂x2 ∂ ∂x3 0 ∂ ∂x1 0 ∂ ∂x3 0 ∂ ∂x1 ∂ ∂x2 � � � � σ12 σ13 σ23 � � + � � f1 f2 f3 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The definition of strain tensor is given by ∂ ∂t � � ε11 ε22 ε33 � � = � � ∂ ∂x1 0 0 0 ∂ ∂x2 0 0 0 ∂ ∂x3 � � � � v1 v2 v3 � � , ∂ ∂t � � ε12 ε13 ε23 � � = 1 2 � � ∂ ∂x2 ∂ ∂x1 0 ∂ ∂x3 0 ∂ ∂x1 0 ∂ ∂x3 ∂ ∂x2 � � � � v1 v2 v3 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 4 And the stress-strain relation reads � � � � � � � � σ11 σ22 σ33 σ12 σ13 σ23 � � � � � � � � = ρ � � � � � � � � c2 P c2 P − 2c2 S c2 P − 2c2 S 0 0 0 c2 P − 2c2 S c2 P c2 P − 2c2 S 0 0 0 c2 P − 2c2 S c2 P − 2c2 S c2 P 0 0 0 0 0 0 2c2 S 0 0 0 0 0 0 2c2 S 0 0 0 0 0 0 2c2 S � � � � � � � � � � � � � � � � ε11 ε22 ε33 ε12 ε13 ε23 � � � � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' In some situations, the P- wave propagation can be approximated by the acoustic wave equation based on the acoustic media assumption [36], so that the wavefield is described by a scalar function instead of a vector, ρ(x) ∂2 ∂t2u(x, t) − ∇ · � ρ(x)c2 P(x)∇u(x, t) � = f(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='4) Taking its two-dimensional case as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' By introducing the velocities v1 = ∂ ∂xu, v3 = ∂ ∂zu and the scalar pressure field σ(x, z, t), it can be cast into velocity-stress form, ∂v1(x, z, t) ∂t = − 1 ρ(x, z) ∂σ(x, z, t) ∂x , ∂v3(x, z, t) ∂t = − 1 ρ(x, z) ∂σ(x, z, t) ∂z , ∂σ(x, z, t) ∂t = −ρ(x, z)c2 P(x, z) �∂v1(x, z, t) ∂x + ∂v3(x, z, t) ∂z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5) It is seen that the complexity for solving the wave equation lies in calculations of first-order spatial derivatives of wavefields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Distributed Local spline simulator As a powerful tool for curve fitting, the cubic spline has been applied for solving PDEs under the framework of FEM [26, 30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The spline expansion is essentially global as it requires solving global algebraic equations with tridiagonal coefficient matrice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Nonetheless, we will show that the cubic spline can be reconstructed by imposing effective inner boundary conditions on the junctions of local patches [25, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 5 The spline collocation method will be derived for solving the strong form of the wave equation in unidimensional space in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='1, while multidi- mensional wavefields can be constructed by the tensor product of unidimen- sional splines successively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' For brevity, a uniform grid mesh will be adopted hereafter, but the idea is straightforward to be generalized to the non-uniform grid due to the scaling property of wavelets [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' It follows by the parallel set- ting of LOSS in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='2, where the global spline is distributed into several local patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The junctions are shared by adjacent nodes and the patched splines are linked by PMBCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' In the meantime, the natural boundary condi- tions imposed on both ends are fully compatible with ADE-PML, where the stiffness can be largely alleviated by the usage of exponential integrators as discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Spline collocation method Without loss of generality, we adopt a uniform grid mesh for the domain [xmin, xmax] with N + 1 points xmin = x0 ≤ x1 ≤ · · · ≤ xN = xmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Denote by Bi(x) the cubic B-spline with compact support over four grid points [24], Bi(x) = � � � � � � � � � � � � � � � � � � � � � � � � � � � (x − xi−2)3 6h3 , x ∈ [xi−2, xi−1], − (x − xi−1)3 2h3 + (x − xi−1)2 2h2 + (x − xi−1) 2h + 1 6, x ∈ [xi−1, xi], − (xi+1 − x)3 2h3 + (xi+1 − x)2 2h2 + (xi+1 − x) 2h + 1 6, x ∈ [xi, xi+1], (xi+2 − x)3 6h3 , x ∈ [xi+1, xi+2], 0, otherwise, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='1) implying Bi−1, Bi, Bi+1, Bi+2 overlap a grid interval (xi, xi+1) [22], and Bi−1(xi) = 1 6, Bi(xi) = 2 3, Bi+1(xi) = 1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='2) Now the velocity wavefield can be expanded by N + 3 splines with N + 3 coefficients �v(t) = (�v−1(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' , �vN+1(t))T vi(t) ≈ N+1 � i=−1 �vi(t)Bi(x), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='3) 6 where vi(t) is short for v(xi, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' In order to determine the coefficients, it suggests imposing the natural boundary conditions on two ends to mini- mize the effect of boundary constraints [31], namely, ∂2 ∂x2v(x0, t) = 0 and ∂2 ∂x2v(xN, t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' By omitting the time variable for brevity, it has that 1 h2�v−1 − 2 h2�v0 + 1 h2�v1 = 0, 1 h2�vN−1 − 2 h2�vN + 1 h2�vN+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='4) Combining with the relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='2), it remains to solve the algebraic equation by the sweeping method with complexity O(N) [27], A � � � � � � � � � �v−1 �v0 �v1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' �vN �vN+1 � � � � � � � � � = 1 6 � � � � � � � � � � � 6 h2 − 12 h2 6 h2 0 · · 0 1 4 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 0 1 4 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 0 1 4 1 0 0 0 6 h2 − 12 h2 6 h2 � � � � � � � � � � � � � � � � � � � � �v−1 �v0 �v1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' �vN �vN+1 � � � � � � � � � = � � � � � � � � � 0 v0 v1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' vN 0 � � � � � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5) Once the spline coefficients are obtained, the spatial first-order derivatives can be directly approximated by ∂ ∂xv(xi, t) ≈ − 1 2h�vi−1(t) + 1 2h�vi+1(t) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='6) as ∂ ∂xBi(x) = � � � � � � � � � � � � � � � � � � � � � � � � � � � (x − xi−2)2 2h3 , x ∈ [xi−2, xi−1], − 3(x − xi−1)2 2h3 + (x − xi−1) h2 + 1 2h, x ∈ [xi−1, xi], 3(xi+1 − x)2 2h3 − (xi+1 − x) h2 − 1 2h, x ∈ [xi, xi+1], − (xi+2 − x)2 2h3 , x ∈ [xi+1, xi+2], 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='7) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Distributed local spline For distributed parallelization, the spline needs to be decomposed into several pieces and stored in multiple processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Here we simply divide N +1 7 grid points on a line into p uniform parts, with M = N/p, v0 < v1 < · · · < vM−1 the first processor < vM shared < · · · < v(p−1)M shared < v(p−1)M+1 < · · · < vpM p-th processor , The grid points vM, v2M, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' , v(p−1)M are shared by the adjacent patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Recall that our target is to recover the global B-spline by the local spline coefficients �v(l) for l-th piece without global communications, that is, �v(l) = (�v(l) −1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' , �v(l) M+1) = (�v−1+(l−1)M, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' , �v(l−1)M+M+1), l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' , p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='8) This can be realized by imposing effective Hermite boundary conditions on two ends of local splines (see Figure 1(b)) [24, 29], ∂v ∂x ��� x=x(l−1)M = φ(l) L , ∂v ∂x ��� x=xlM = φ(l) R , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='9) which is equivalent to − 1 2h�v(l) −1 + 1 2h�v(l) 1 = φ(l) L , − 1 2h�v(l) M−1 + 1 2h�v(l) M+1 = φ(l) R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='10) Thus all the coefficients �v(l) = (�v(l) −1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' , �v(l) M+1) can be obtained straightfor- wardly by solving the algebraic equation A(l) M � � � � � � � � � � �v(l) −1 �v(l) 0 �v(l) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' �v(l) M �v(l) M+1 � � � � � � � � � � = 1 6 � � � � � � � � � � � − 3 h 0 3 h 0 · · 0 1 4 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 0 1 4 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 0 1 4 1 0 0 0 − 3 h 0 3 h � � � � � � � � � � � � � � � � � � � � � �v(l) −1 �v(l) 0 �v(l) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' �v(l) M �v(l) M+1 � � � � � � � � � � = � � � � � � � � � � φ(l) L v(l) 0 v(l) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' v(l) M φ(l) R � � � � � � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='11) where (M + 3) × (M + 3) coefficient matrix A(l) M has an explicit LU decom- position, namely, A(l) M = LU, L = � � � � � � � � � � � 1 0 0 · · · · 0 − h 3 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 0 l1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 0 0 l2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' lM 1 0 0 0 · · −3lM h 3lM+1 h 1 � � � � � � � � � � � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='12) 8 and U = 1 6 � � � � � � � � � � � − 3 h 0 3 h 0 · · · · 0 0 d1 2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 0 0 d2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 0 0 0 d3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 0 dM+1 0 0 0 · · 0 0 3dM+2 h � � � � � � � � � � � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='13) with d1 = 4, l1 = 1/4, d2 = 4 − 2l1 = 7/2, li = 1/di, di+1 = 4 − li, i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' , M + 1, lM+1 = 1/(dMdM+1), dM+2 = 1 − lM+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='14) Now the solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='11) should be equivalent to that of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5) by choosing appropriate φ(l) L and φ(l) R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Denote by (bij) = A−1, −1 ≤ i, j ≤ pM+1, thus the solution �vi of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5) can be represented by �vi = biivi + i−1 � j=−1 bijvj + pM+1 � j=i+1 bijvj, i = −1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' , pM + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='15) Using the constraints (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='8), it directly solves φ(l) L and φ(l) R by φ(l) L = − 1 2h�v(l) −1 + 1 2h�v(l) 1 = − 1 2h�v−1+(l−1)M + 1 2h�v1+(l−1)M, φ(l) R = − 1 2h�v(l) M−1 + 1 2h�v(l) M+1 = − 1 2h�vM−1+(l−1)M + 1 2h�vM+1+(l−1)M, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='16) where �vi are given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' At first glance, it still requires the information from all other pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Fortunately, there is a key observation that the non-diagonal elements bij decays exponentially away from the main diagonal bii due to the rapid decay of the wavelet basis in its dual space [20, 22], which is clearly visualized in Figure 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Therefore, it allows us to truncate bij when |i − j| ≥ nnb for sufficiently large nnb, �vi ≈ biivi + i−1 � j=i−(nnb−1) bijvj + i+nnb−1 � j=i+1 bijvj, i = −1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' , pM + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='17) 9 0 10 20 30 40 50 60 35 30 25 20 15 10 5 0 (a) The element |bij| of A−1 in log10 scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 10 20 30 40 50 60 10 20 30 40 50 60 12 10 8 6 4 2 0 PMBC PMBC PMBC (b) Approximation of A−1 by (A(l) M )−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Figure 1: Since the non-diagonal elements bij in A−1 decays exponentially away from the main diagonal bii, the coefficients �v = A−1(v0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' , vN)τ can be approximated by �v(l) = (A(l) M )−1(v(l) 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' , v(l) M )τ when PMBCs are adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Now using the truncated stencils (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='17), it has that �vlM−1 ≈ (lM−1)+nnb−1 � j=(lM−1)−nnb+1 blM−1,jvj = nnb−2 � j=−nnb blM−1,lM+jvlM+j, �vlM+1 ≈ (lM+1)+nnb−1 � j=(lM+1)−nnb+1 blM+1,jvj = nnb � j=−nnb+2 blM+1,lM+jvlM+j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='18) By further adding four more terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='18) to complete the summations from −nnb to nnb, it yields that − 1 2h�vlM−1 + 1 2h�vlM+1 ≈ � − 1 2hblM−1,lM + 1 2hblM+1,lM � vlM shared by adjacent two processors = −1 � j=−nnb � − 1 2hblM−1,lM+j + 1 2hblM+1,lM+j � vlM+j stored in left processor + nnb � j=1 � − 1 2hblM−1,lM+j + 1 2hblM+1,lM+j � vlM+j stored in right processor .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 10 Thus it arrives at the formulation of PMBC for 1 ≤ l ≤ p − 1, φ(l) R = φ(l+1) L ≈ 1 2c0,lvlM + nnb � j=1 c− j,lvlM−j stored in left processor + 1 2c0,lvlM + nnb � j=1 c+ j,lvlM+j stored in right processor , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='19) where c0,l = − blM−1,lM 2h + blM+1,lM 2h and c+ j,l = −blM−1,lM+j 2h + blM+1,lM+j 2h , c− j,l = −blM−1,lM−j 2h + blM+1,lM−j 2h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='20) Finally, the effective boundary conditions φ(1) L and φ(p) R on two ends should match the natural boundary conditions of the global cubic spline, yielding φ(1) L = �v1 − �v−1 2h ≈ nnb � j=0 c− j,0vj, c− j,0 = −b−1,j + b1,j 2h , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='21) φ(p) R = �vN+1 − �vN−1 2h ≈ nnb � j=0 c+ j,pvN−j, c+ j,p = −bN−1,N−j + bN+1,N−j 2h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='22) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Exponential integrator for the wave equation and ADE-PML PML is usually adopted in finite computational domain to model wave propagation in unbounded media [32, 33, 34, 35], which suggests adding a layer of the thickness L outside to attenuate the wave and avoid its artificial reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' It will be shown that the natural boundary condition of cubic spline can be imposed on the differential equation form of PML (known as ADE-PML [35]), and the resulting rigid dynamics can be solved efficiently by exponential integrators [37, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' For convenience, we use the 2-D acoustic wave equation to illustrate the basic idea of ADE-PML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' It starts by taking the Fourier transform of the acoustic wave equation in the tensile coordinates (t → ω) and adding two stretching terms 1/sx and 1/sz [35], iω�v1(x, z, ω) = −1 ρ 1 sx � ∂σ ∂x(x, z, ω), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='23) iω�v3(x, z, ω) = −1 ρ 1 sz � ∂σ ∂z (x, z, ω), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='24) iω�σ(x, z, ω) = −ρc2 P � 1 sx � ∂v1 ∂x (x, z, ω) + 1 sz � ∂v3 ∂z (x, z, ω) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='25) 11 where �· denotes the Fourier transform, and si (i = x, z) are introduced to present a complex stretching of the coordinate system, si = ki + di αi + iω, s−1 i = 1 ki − di k2 i 1 di ki + αi + iω, i = x, z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='26) Here αi, di and ki are flexible parameters that adjust the absorbing effect, di = d0 � i L �2 , ki = 1 + (kmax − 1)m, αi = αmax � 1 − i L �p (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='27) and d0 = −3cP,max log(R)/2L, with cP,max equal to the maximal velocity of the pressure wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' R is the theoretical reflection coefficient of the target, αmax = πf0 with f0 as the main frequency of the hypocenter, and kmax is the cut-off wavenumber [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Here it suffices to take m = 1 and p = 1 for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' For the first equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='23 of � ∂σ ∂x(x, z, ω), it is equivalent to 1 sx � ∂σ ∂x(x, z, ω) = 1 kx � ∂σ ∂x(x, z, ω) − dx k2 x 1 dx kx + αx + iω � ∂σ ∂x(x, z, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='28) By introducing a memory variable �Ψx(x, z, ω), �Ψx(x, z, ω) = −dx k2 x 1 dx kx + αx + iω � ∂σ ∂x(x, z, ω), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='29) it arrives at an auxiliary differential equation by taking inverse Fourier trans- form �Ψx(x, z, ω) → Ψx(x, z, t), ∂Ψx ∂t (x, z, t) + �dx kx + αx � Ψx(x, z, t) = −dx k2 x ∂σ(x, z, t) ∂x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='30) Similarly, we can define memory variables Ψz(x, z, t) for � ∂σ ∂z (x, z, ω) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='24 and Φx(x, z, t), Φz(x, z, t) for � ∂v1 ∂x (x, z, ω), � ∂v3 ∂z (x, z, ω) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='25, re- spectively, yielding the ADE-PML for velocity components outside the com- 12 putational domain, (A) � � � � � � � � � � � � � � � � � � � � � � � � � ∂v1 ∂t = −1 ρ � 1 kx ∂σ ∂x + Ψx � , ∂v3 ∂t = −1 ρ � 1 kz ∂σ ∂z + Ψz � , ∂Ψx ∂t = − �dx kx + αx � Ψx − dx k2 x ∂σ ∂x, ∂Ψz ∂t = − �dz kz + αz � Ψz − dz k2 z ∂σ ∂z , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='31) and those for the stress component, (B) � � � � � � � � � � � � � � � � � ∂σ ∂t = −ρc2 P � 1 kx ∂v1 ∂x + Φx + 1 kz ∂v3 ∂z + Φz � , ∂Φx ∂t = − �dx kx + αx � Φx − dx k2 x ∂v1 ∂x , ∂Φz ∂t = − �dz kz + αz � Φz − dz k2 z ∂v3 ∂z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='32) The stiff terms in the auxiliary differential equations might pose severe limitation on the time step when using explicit numerical integrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' For- tunately, this can be alleviated by the exponential integrator [37, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' For the time interval [tn, tn+1] with ∆t = tn+1 − tn, it starts from the variation-of-constant formula of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='30), Ψx(x, tn+1) = e−( dx kx +αx)∆tΨx(x, t) − dx k2 x � ∆t 0 e−( dx kx +αx)(∆t−τ)∂σ ∂x(x, tn + τ)dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' When the Euler approximation ∂σ ∂x(x, t + τ) ≈ ∂σ ∂x(x, t) is used, it yields [37], Ψx(x, tn+1) ≈ e−( dx kx +αx)∆tΨx(x, tn) − dx k2 x 1 − e−( dx kx +αx)∆t ( dx kx + αx) ∂σ ∂x(x, tn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='33) The other three auxiliary equations for Φz, Ψx and Ψz can be tackled in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Combining with the temporal finite difference scheme for v1, v3 and σ, one can obtain the non-splitting exponential Euler scheme for ADE- PML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Because the exact flow of the stiff term in the auxiliary dynamics (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='30) is exploited, it can largely alleviate the restriction on the time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 13 Alternatively, one can utilize the exponential operator splitting scheme, which also utilizes the exact stiff flow of the auxiliary equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The basic idea is alternating update of velocities and stress based on the splitting of two subproblems (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='31) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='32), which is similar to our short-memory operator splitting for time-fractional constant-Q wave equation [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' One can first solve (A) exactly by assuming that σ, Φx and Φz are in- variant in small time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' For instance, when ∂σ ∂x is invariant in [tn, tn+1], it yields the exact solution of Ψx, Ψx(x, tn+1) = e−( dx kx +αx)∆tΨx(x, tn) − dx k2 x 1 − e−( dx kx +αx)∆t ( dx kx + αx) ∂σ ∂x(x, tn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='34) In addition, since � tn+1 tn ∂Ψx ∂t dt = − �dx kx + αx � � tn+1 tn Ψx(x, t)dt − dx k2 x � tn+1 tn ∂σ ∂x(x, t)dt = − �dx kx + αx � � tn+1 tn Ψx(x, t)dt − ∆tdx k2 x ∂σ ∂x(x, tn), it further yields � tn+1 tn Ψx(x, t)dt = − 1 ( dx kx + αx) � Ψ(x, tn+1) − Ψ(x, tn) + ∆tdx k2 x ∂σ ∂x(x, tn) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Substituting it into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='31), then the velocity v1 can be solved by v1(x, tn+1) =v1(x, tn) − 1 ρ(x) �∆t kx ∂σ ∂x(x, tn) + � tn+1 tn Ψx(x, t)dt � =v1(x, tn) + 1 ρ(x) kx (dx + αxkx)(Ψx(x, tn+1) − Ψx(x, tn)) − ∆t ρ(x) αx (dx + αxkx) ∂σ ∂x(x, tn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='35) The solution of v3 and Φz can be obtained in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Besides, one can also solve the subsystem (B) exactly when v1, v3, Ψx and Ψz are assumed to be invariant in small time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Specifically, when the Strang splitting is used, say, half step evolution of (A) + full step evolution of (B) + half step evolution of (A), it is expected to achieve global second-order convergence as two exact flows are exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 14 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Numerical experiments From this section, several benchmarks have been performed to evaluate the performance of LOSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' In the first example, we made a series of bench- marks on 2-D acoustic wave equation in homogenous media to test the conver- gence of the spline collocation method, where the ADE-PML associated with the natural boundary condition was adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' In particular, the influence of several key parameters, including the layer thickness L, the reflection coeffi- cient R and the cut-off wavenumber kmax, were carefully studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' After that, we used LOSS to simulate 3-D wave propagation in either a homogenous me- dia or a double-layer media activated by the impulse of a Ricker-type wavelet history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' These typical examples may validate the performance of LOSS when the coefficients are either smooth or of a large variation, as well as its parallel scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' To evaluate the errors of LOSS, we adopted two metrics: the relative l2- error ε2(t) and the relative maximal error ε∞(t), where vnum 3 and vref 3 denoted the numerical and reference velocity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' ε2(t) = (� i |vnum 3 (xi, t) − vref 3 (xi, t)|2)1/2 maxx |vref 3 (x, t)| , ε∞(t) = maxx |vnum 3 (x, t) − vref 3 | maxx |vref 3 (x, t)| , All the 2-D simulations were realized by MATLAB, while all the 3-D simulations performed via our Fortran implementations ran on the platform: AMD Ryzen 7950X (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='50GHz, 64MB Cache, 16 Cores, 32 Threads) with 64GB Memory (4800Mhz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The parallelization was realized by the Message Passing Interface (MPI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 2-D acoustic wave equation First we need to validate the convergence of the spline collocation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The model parameters were set as: the wave speed cP = 50m/s and the density ρ = 1kg/m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The computational domain was [−5, 5]2 (10m×10m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The Strang operator splitting was adopted with time step ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='0001s and the final time was T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The parameters of PML were given by: the thickness L = 50, the reflection coefficient R = 10−6, the cut-off wavenumber kmax = 1 and fP = cP/L, amax = πfP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Five groups of spline simulations under the grid size N = Nx × Nz = 322, 642, 1282, 2562, 5122 were performed, with zero initial velocity and the initial pressure σ(x, z, 0) = e−5(x2+z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='1) 15 The reference solutions were produced by the Fourier spectral method (FSM) with a Nx × Nz = 2562 grid, where the domain was extended to 27m×27m with 7122 grid points to avoid the reflection of waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The snapshots of vibrational velocity wavefield v3 and the distribution of numerical errors are visualized in Figures 3 and 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' From Figure 3, the velocity wavefield propagates inside the domain before t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='1s, and it begins to leave the domain when t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Fortunately, the penetrating wavefields are successfully attenuated by ADE-PML and the artificial reflection is almost negligible, as clearly observed in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The time evolution of l2-errors ε2(t) are recorded in Table 1 and the convergence rate is given in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The slope of the dashed line is −4, which perfectly matches the theoretical fourth-order convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' In addi- tion, according to Table 1 and Figure 2, the accuracy of ADE-PML can be significantly improved under a finer grid, and the convergence rate is close to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' This indicates the performance of ADE-PML depends heavily on the accuracy of the underlying numerical solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 9 25 20 15 10 5 0 5 (a) When wave propagates in the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 9 25 20 15 10 5 0 5 (b) When wave leaves the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Figure 2: 2-D acoustic wave equation: The fourth-order convergence of the spline colloca- tion method is verified for ADE-PML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The performance of ADE-PML under spline collocation method As mentioned above, the performance of ADE-PML relies on three param- eters: the thickness L, the reflection coefficient R and the cut-off wavenumber kmax [33, 34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Therefore, their influence on the absorbing effect, as well 16 5 0 5 x 5 4 3 2 1 0 1 2 3 4 5 z 4 3 2 1 0 1 2 3 4 10-3 (a) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='02s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 5 0 5 x 5 4 3 2 1 0 1 2 3 4 5 z 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 2 10-3 (b) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='06s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 5 0 5 x 5 4 3 2 1 0 1 2 3 4 5 z 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 10-3 (c) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='10s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 5 0 5 x 5 4 3 2 1 0 1 2 3 4 5 z 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 1 10-3 (d) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='12s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 5 0 5 x 5 4 3 2 1 0 1 2 3 4 5 z 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 10-4 (e) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='16s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 5 0 5 x 5 4 3 2 1 0 1 2 3 4 5 z 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 10-5 (f) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='20s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Figure 3: 2-D acoustic wave equation: The snapshot of vibration velocity wavefields v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The wave begins to be absorbed by PML at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Table 1: 2-D acoustic wave equation: The l2-error ε2(t) at different instants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The nu- merical accuracy of the spline collocation method and ADE-PML can be systematically improved by refining the grid mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Time Grid N = 322 N = 642 N = 1282 N = 2562 N = 5122 When wave propagates inside the domain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='02s 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='634 ×10−1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='663 ×10−2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='382 ×10−3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='057 ×10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='276 ×10−5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='04s 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='634×10−1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='197×10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='264×10−3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='640×10−5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='737×10−6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='06s 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='476×10−1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='638×10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='167×10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='309×10−4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='118×10−6 When wave is absorbed by PML outside the domain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='10s 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='413×10−1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='099×10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='469×10−3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='802×10−5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='466×10−6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='15s 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='186×10−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='038×10−2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='837×10−4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='329×10−5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='445×10−6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='20s 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='441×10−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='779×10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='080×10−3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='611×10−5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='238×10−6 as the compatibility with natural boundary conditions for the cubic spline, deserves a careful investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Among them, the most important parameter is the layer thickness L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 17 5 0 5 x 5 4 3 2 1 0 1 2 3 4 5 z 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 10-7 (a) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='02s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 5 0 5 x 5 4 3 2 1 0 1 2 3 4 5 z 3 2 1 0 1 2 3 10-7 (b) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='06s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 5 0 5 x 5 4 3 2 1 0 1 2 3 4 5 z 4 3 2 1 0 1 2 3 4 10-7 (c) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='10s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 5 0 5 x 5 4 3 2 1 0 1 2 3 4 5 z 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 10-7 (d) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='12s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 5 0 5 x 5 4 3 2 1 0 1 2 3 4 5 z 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 10-8 (e) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='16s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 5 0 5 x 5 4 3 2 1 0 1 2 3 4 5 z 4 3 2 1 0 1 2 3 4 10-11 (f) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='20s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Figure 4: 2-D acoustic wave equation: Visualization of numerical errors in the vibrational velocity wavefields v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Intuitively speaking, increasing the layer thickness improves the absorbing effect of PML, at the cost of storing more wavefields and higher computa- tional costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Therefore, it is expected to use absorbing layers as fewer as possible to maintain the accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' To evaluate its effect, we have performed a series of benchmarks under different L and grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The l2-errors at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='2s, as recorded in Table 2, were adopted to measure the accuracy of ADE-PML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The convergence with respect to the mesh size is plotted in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Clearly, the accuracy of ADE- PML can be systematically improved as N increases, and the convergence rate matches the theoretical order 4 when L is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' It is observed that too small L may result in a dramatic reduction in accuracy for sufficiently large N, while its influence becomes negligible when Nx = Nz ≤ 128 because the discretization errors dominate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Therefore, the choice of L should match the size of grid mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' When a coarse grid is used, the thickness might be not too large for the sake of efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' By contrast, 18 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 9 25 20 15 10 5 0 5 (a) t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 9 25 20 15 10 5 0 5 (b) t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='15s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 9 25 20 15 10 5 0 5 (c) t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='20s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Figure 5: 2-D PML: The convergence of l2-errors under different boundary layer thick- nesses L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' It is observed that L = 40-50 might be a good choice to strike a balance in accuracy and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Table 2: 2-D PML: The l2-errors at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='2s under different boundary layer thicknesses L and grid meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The errors induced by PML are negligible when L ≥ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' L Grid N = 322 N = 642 N = 1282 N = 2562 N = 5122 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='405×10−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='089×10−2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='785×10−4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='155×10−5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='453×10−5 20 1.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='787×10−4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='155×10−5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='587×10−6 40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='394×10−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='077×10−2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='786×10−4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='394×10−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='077×10−2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='786×10−4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='156×10−5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='581×10−6 70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='394×10−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='077×10−2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='786×10−4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='154×10−5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='579×10−6 80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='394×10−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='077×10−2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='786×10−4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='155×10−5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='577×10−6 90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='394×10−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='077×10−2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='786×10−4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='155×10−5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='578×10−6 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='394×10−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='077×10−2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='786×10−4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='155×10−5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='580×10−6 when a fine grid mesh is used, one should use a thicker absorbing layer to ensure the accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Based on the results in Figure 5, L = 40 to 50 might be a good choice to strike a balance in accuracy and efficiency, while too large L might not bring in evident improvements as the accuracy is still limited by the numerical solver under the specified grid mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Second, we studied the effect of the reflection coefficient R on the absorb- ing boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The mesh size was fixed to be Nx × Nz = 5122 with the layer thickness L = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The time evolution of the l2-errors are plotted in Figure 6(a), corresponding to the data in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' It is observed that R = 10−6 achieves l2-error about 10−6, whereas it seems to reach the limitation of ac- 19 curacy as the errors vary slightly when R further decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' These findings coincide with the observation made in [32, 35] as the choice of R should match the thickness L of PML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='2 6 5 4 3 2 1 0 (a) Different reflection coefficients R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='2 6 5 4 3 2 1 0 1 2 3 (b) Different cut-off wavenumber kmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Figure 6: 2-D PML: Time evolution of l2-errors under different reflection coefficients R and cut-off wavenumber kmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The mesh size is fixed to be N = 5122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' From numerical perspective, The choice of R should match the thickness L of PML, and cut-off wavenumber kmax = 1 achieves the optimal performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Table 3: 2-D PML: The l2-errors under different reflection coefficients R, with Nx = Nz = 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Here R = 10−6 is suggested to ensure the effectiveness of PML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' R Time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='06s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='08s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='10s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='12s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='14s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='16s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='18s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='20s 10−1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='11×10−6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='73×10−6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='52×10−6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='99×10−5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='15×10−5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='00×10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='38×10−4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='32×10−4 10−2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='11×10−6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='73×10−6 2.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='20×10−6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='67×10−6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='13×10−6 10−6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='11×10−6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='11×10−6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='53×10−6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='16×10−6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='80×10−6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='40×10−6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='92×10−6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='98×10−6 10−7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='11×10−6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='73×10−6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='53×10−6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='13×10−6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='77×10−6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='39×10−6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='58×10−6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='58×10−6 10−8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='11×10−6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='73×10−6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='53×10−6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='12×10−6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='77×10−6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='39×10−6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='51×10−6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='48×10−6 10−9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='11×10−6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='73×10−6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='53×10−6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='12×10−6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='77×10−6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='39×10−6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='51×10−6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='48×10−6 10−10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='11×10−6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='73×10−6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='53×10−6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='12×10−6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='78×10−6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='39×10−6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='53×10−6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='51×10−6 Finally, for the cut-off wavenumber kmax, we also plot the time evolution of the l2-errors under different kmax in Figure 6(b), with data collected in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The mesh size was fixed to be Nx ×Nz = 5122 and the layer thickness was again set as L = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The results show that ADE-PML can work only when kmax = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' In fact, just as pointed out in [34], the sharp variations of the profile of the kx and kz functions might augment the reflection coefficient of waves impinging on the boundary, so that too large kmax is not recommended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 20 Table 4: 2-D PML: The l2-errors under different cut-off wavenumber kmax, with Nx = Nz = 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The ADE-PML may only work when kmax = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' kmax Time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='06s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='08s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='10s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='12s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='14s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='16s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='18s 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='76×10−6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='20×10−6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='67×10−6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='13×10−6 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='11×10−6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='52×10−4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='95×10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='73×10−1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='07×10−1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='61×10−1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='37×10−1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='89×10−1 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='11×10−6 6.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='91×10−1 19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='11×10−6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='11×10−6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='01×10−1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='90×10−1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='21×10−1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='76×10−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='09 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='59×10−1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Elastic wave propagation in 3-D homogenous media To study the wave equation in 3-D homogenous media, we set the ini- tial velocity and stress tensors at equilibrium state and simulated the wave propagation activated by source functions with a Ricker-type wavelet history, f1(x, t) = f2(x, t) = f3(x, t) = A(x)fr(t), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='2) where the amplitude A(x) was a Gaussian profile A(x) = e−(x−x0)2−(y−y0)2−(z−z0)2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='3) and the Ricker wavelet was given by fr(t) = (1 − 2(πfP(t − dr)2))e−(πfP (t−dr))2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='4) where fP was the peak frequency and dr was the temporal delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Here we chose fP = 100Hz, dr = 0 and the centre position x0 = 0, y0 = 0, z0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The model parameters in a homogenous media were set as follows: The group velocities for P-wave and S-wave were cP = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='614km/s and cS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='802km/s, respectively, and the mass density was ρ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='2kg/m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The computational domain was [−40, 40]3 (80km × 80km × 80km).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Four groups of simulations have been performed under N = Nx × Ny × Nz = 1293, 2573, 3853, 5133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The Strang splitting was adopted with time step ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='005s and the final instant was T = 10s (2000 steps in total).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The reference solution was produced by FSM with a fine grid mesh N = 5123 for testing the convergence of LOSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' As a pretreatment in LOSS, the domain was evenly decomposed into 4 × 4 × 4 patches, thereby achieving a perfect balance in overload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Here PMBCs were assembled by truncating the neighborhoods at nnb = 20 [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' A visualization of the domain decomposition in 3-D space is presented in Figure 21 9(a), when the whole computational domain is decomposed into 2 × 2 × 2 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' For the calculations of first-order spatial derivatives, it only requires the local spline expansion in one direction, where PMBCs are assembled at the shared faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' For instance, when one needs the first-order derivatives in x-direction, it only requires the communications in 1 ↔ 2, 3 ↔ 4, 5 ↔ 6 and 7 ↔ 8, thereby greatly reducing the communication cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='8 8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='8 9 10 9 8 7 6 5 4 3 2 1 0 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='8 8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='8 9 4 2 0 2 4 6 8 (a) Convergence with respect to Nz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (left: maximal error ε∞(10), right: l2-error ε2(10)) 40 30 20 10 0 10 20 30 40 4 3 2 1 0 1 2 3 10-5 40 30 20 10 0 10 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='2 (b) v3(0, 0, z) in the homogenuous media (left) and the relative errors (right) at t = 10s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Figure 7: Wave propagation in 3-D homogenous media: A comparison of the vi- brational wavefield v3(0, 0, z) at t = 10s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The convergence of LOSS is verified for smooth data and coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The relative maximal error ε∞(t) and l2-error ε2(t) were again adopted to measure the numerical accuracy and the spatial convergence is plotted in Figure 7(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Clearly, the convergence rate coincides with the theoreti- cal fourth order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' We also investigate the influence of the parameter nnb in assembling PMBCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' As shown in Figure 7(a), the errors induced by trunca- tion in PMBC seem negligible even when nnb = 10, which accords with our observations made in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The snapshots of the vibrational wave-field v3 of the P- and S- wave are given in Figure 8, where three sectional drawings are provided to visualize the wave propagation in the homogenous media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' To further demonstrate the errors of LOSS, we plot the synthetic data v3(0, 0, z) at t = 10s in Figure 22 (a) v3(x, y, z) at t = 4s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (left: FSM, N = 5123, right: LOSS, N = 5133) (b) v3(x, y, z) at t = 6s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (left: FSM, N = 5123, right: LOSS, N = 5133) (c) v3(x, y, z) at t = 8s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (left: FSM, N = 5123, right: LOSS, N = 5133) (d) v3(x, y, z) at t = 10s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (left: FSM, N = 5123, right: LOSS, N = 5133) Figure 8: Wave propagation in 3-D homogenous media: A comparison of snapshots of wavefield v3(x, y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 23 4 2 40 30 0 20 2 2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 4 0 10 6 10 10 0 0 10 8 10 20 20 30 30 10 y 40 40 ×10~540 0 30 20 2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 0 5 10 10 10 0 0 10 10 20 20 10 30 30 y 40 40 ×10-53 2 1 40 0 30 1 20 2 2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 3 0 4 10 10 10 0 0 5 10 10 20 20 6 30 30 y 40 40 ×10~53 2 1 40 0 30 1 20 2 2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 3 0 4 10 10 10 0 0 5 10 10 20 20 6 30 30 y 40 40 ×10~52 1 40 0 30 20 1 2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 2 0 3 10 10 10 0 0 4 10 10 20 20 30 30 5 y 40 40 ×10-52 1 40 0 30 20 1 2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 2 0 3 10 10 10 0 0 4 10 10 20 20 30 30 5 y 40 40 ×10~540 0 30 20 1 2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 0 2 10 10 10 0 0 3 10 10 20 20 30 30 4 y 40 40 ×10~540 0 30 20 1 2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 0 2 10 10 10 0 3 0 10 10 20 20 30 30 4 y 40 40 ×10~57(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Small oscillations are observed around the reference solution when too coarse grid mesh is used (N = 2573).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Nonetheless, they could be suppressed to a large extent when the finer grid mesh was adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Elastic wave propagation in a 3-D double-layer media As a more challenging example, we also studied the wave propagation in a double-layer structure, with the same computational domain [−40, 40]3 and the same source impulse as adopted in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The model parameters in a heterogenous media, iven in Figure 9(b), were set to be: The group velocities for the upper layer 402×[0, 40] were cP = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='614km/s and cS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='802km/s and the mass density was ρ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='2kg/m3, while for the deeper layer 402 ×[−40, 0], cP = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='228km/s, cS = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='604km/s and ρ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5kg/m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (a) Domain decomposition in 3-D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (b) A double-layer heterogeneous structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Figure 9: An illustration of domain decomposition of 3-D space and a double-layer het- erogeneous structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Seven groups of simulations have been performed under N = Nx × Ny × Nz = 1293, 2573, 3853, 5133, 6413, 7693 and 5132 × 1025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The Strang splitting was adopted with time step ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='005s and the final instant was T = 10s (2000 steps in total).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The reference solution was still produced by FSM with a fine grid mesh N = 6403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The snapshots of the vibrational wavefield v3 of the P- and S-wave are given in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Unlike the propagation in the homogenous media, the scattering of wave occurs when it enters a different media, resulting in the reflection of wavepackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' From the synthetic data in Figures 10(b) and 10(c), the superposition of the impulse and reflected wave is clearly observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The convergence of LOSS is still verified in Figure 10(a), albeit with an evident reduction in the order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' This is caused by the sharp variation of the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' As visualized in Figures 10(b) and 10(c), small 24 8 7 40 30 ~ 4 20 ~ 5 10 ~ 6 2 0 3 10 ~ 20 ~ 1 40 30 20 40 2 40 0 20 0 20 20 40 y 4040 30 20 10- cp = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='614km/s, cs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='802km/s,p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='2kg/m3 2 0 10 20 40 30 cp = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='228km/s, cs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='604km/s, p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5kg/m3 20 40 40 0 20 0 20 20 40 y 407 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 0 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 2 1 0 1 2 3 4 5 6 7 (a) Convergence with respect to Nz (left: maximal error ε∞(10), right: l2-error ε2(10)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 40 30 20 10 0 10 20 30 40 6 5 4 3 2 1 0 1 2 3 4 10-5 40 30 20 10 0 10 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='15 (b) v3(0, 0, z) in the heterogenuous media (left) and the relative errors (right) at t = 6s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 40 30 20 10 0 10 20 30 40 4 3 2 1 0 1 2 3 10-5 40 30 20 10 0 10 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='25 (c) v3(0, 0, z) in the heterogenuous media (left) and the relative errors (right) at t = 10s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Figure 10: Wave propagation in 3-D heterogeneous media: A comparison of the vibrational data v3(0, 0, z) at t = 6s and t = 10s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The reduction in the conver- gence order is observed when model parameters have sharp variations, and small oscillations are produced by the spline construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Nevertheless, the errors can be diminished as a finer grid mesh is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' oscillations in the reflected waves are produced by the spline construction under a coarse grid mesh (Nx ≤ 385).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Fortunately, such phenomenon can be alleviated when a finer grid mesh was adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' In particular, we tried to refine the grid mesh in z-direction, where there was a sharp variation in density and group velocities, and found that it succeeded in further reducing the errors (see Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' This manifests that LOSS is still capable to deal 25 with the multi-layer geological model, albeit the grid mesh must be refined near the discontinuity of model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Table 5: The memory requirement for storing nine wavefields (in both serial setting and 4 × 4 × 4 decomposition) and the numerical errors up to T = 10s (with ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='005s, 2000 steps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Grid 1293 2573 5133 6413 5132 × 1025 7693 Memory (Serial) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='14GB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='14GB 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='05GB 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='66GB 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='09GB 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='49GB Memory (MPI) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='17GB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='23GB 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='43GB 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='25GB 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='64GB 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='33GB ε2(t = 10) 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='845 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='078 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='320 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='742 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='334 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='985 ε∞(t = 10) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='785 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='067 Finally, we tested the complexity of LOSS by recording the computational time under N = NxNyNz = 1293, 2573, 5133, 6413, 7693.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' As shown in Figure 11(a), the complexity of LOSS is almost proportional to the mesh size N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Besides, we also calculated the speedup ratio of LOSS in Figure 11(b) by performing the simulations under the fixed grid mesh N = 2413 and time step ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='005s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' For serial realization, it required 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='85 hours to reach the final time T = 10s (2000 steps in total).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' By contrast, when the domain was decomposed into 4 × 4 × 4 patches and 32 cores (64 tasks) were used, it only took 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='85 hour and the speedup ratio was about 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='71%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' In fact, the storage of shared grids is relatively small compared with the total memory require- ment (see Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Thus it is expected that LOSS can achieve even higher speedup ratio when the latency caused by the Hyper-Threading technique is precluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 (a) Computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 5 10 15 20 25 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='9 1 (b) Speedup ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Figure 11: The complexity of LOSS is almost proportional to NxNyNz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' In addition, it achieves a speedup ratio about 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='71% using 32 cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' The platform is AMD Ryzen 7950X (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content='50GHz, 64MB Cache, 16 Cores, 32 Threads) with 64GB Memory (4800Mhz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 26 (a) v3(x, y, z) at t = 4s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (left: FSM, N = 5123, right: LOSS, N = 5132 × 1025) (b) v3(x, y, z) at t = 6s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (left: FSM, N = 5123, right: LOSS, N = 5132 × 1025) (c) v3(x, y, z) at t = 8s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (left: FSM, N = 5123, right: LOSS, N = 5132 × 1025) (d) v3(x, y, z) at t = 10s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' (left: FSM, N = 5123, right: LOSS, N = 5132 × 1025) Figure 12: Wave propagation in 3-D double-layer media: A comparison of snapshots of wavefield v3(x, y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 27 5 40 30 0 20 2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 0 5 10 10 10 0 0 10 10 20 20 30 30 10 y 40 40 ×10~540 30 0 20 2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 0 5 10 10 10 0 0 10 10 20 20 30 30 10 y 40 40 ×10~53 2 40 1 30 0 20 1 2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 2 0 3 10 4 10 10 0 0 5 10 10 20 20 6 30 30 y 40 40 ×10~53 2 1 40 0 30 1 20 2 2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 3 0 4 10 10 10 0 0 5 10 10 20 20 6 30 30 y 40 40 ×10~52 40 1 30 0 20 1 2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 2 0 3 10 10 10 0 0 4 10 10 20 20 5 30 30 y 40 40 ×10~53 2 40 1 30 0 20 1 2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 2 0 3 10 10 10 0 0 4 10 10 20 20 5 30 30 y 40 40 ×10~52 1 40 30 0 20 2 1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 0 2 10 10 10 0 0 3 10 10 20 20 30 30 4 y 40 40 ×10~51 40 0 30 20 1 2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 0 2 10 10 10 0 3 0 10 10 20 20 30 30 4 y 40 40 ×10~55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Conclusions and discussions This paper proposes the distributed local spline simulator (LOSS) for solving the elastic wave propagation, where the wavefields are expanded by patched local cubic B-splines and the first-order spatial derivatives can be calculated accurately with low complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' A perfectly matched boundary condition (PMBC) is introduced by exploiting the exponential decay prop- erty of the wavelet basis in its dual space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' In this manner, the local spline is able to recover the global spline as accurately as possible with only local com- munication costs, thereby greatly facilitating the distributed parallelization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Several typical 2-D and 3-D examples are provided to validate the accuracy, efficiency and parallel scalability of LOSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' For brevity, we only discuss the implementation under a uniform grid mesh, but a nonuniform grid is much more desirable when the model pa- rameters have discontinuities or large variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Actually, the settings of PMBCs can be readily generalized to the non-uniform grid owing to the scaling and flexibility of wavelet basis, and the implementation of LOSS on a structure-driven grid mesh may be a topic of our future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' Acknowledgement This research was supported by the National Natural Science Foundation of China (Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' 12271303).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE4T4oBgHgl3EQfhw0J/content/2301.05127v1.pdf'} +page_content=' X.' metadata={'source': 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D. Nguyen1,∗ J. S. Simon1, J. W. Scott1, Y. C. +Cincia Tsai1, A. M. Zimmerman2, and W. P. Halperin1† +1Department of Physics and Astronomy, Northwestern University and +2Department of Physics, Harvard University +(Dated: January 27, 2023) +Abstract +We perform cluster aggregation simulations to model the structure of anisotropic aerogel. By +biasing the diffusion process, we are able to obtain two distinct types of globally anisotropic aerogel +structures which we call ”nematic”, with long strands along the anisotropy axis, and ”planar”, +with long strands in planes perpendicular to the anisotropy axis. We calculate the auto-correlation +function, the structure factor, and the angular dependence of the free-path distribution for these +samples. +The calculated structure factor from simulated aerogels can be compared with data +from small-angle X-ray scattering (SAXS) of lab-grown aerogel allowing us to classify the spatial +structure of the lab-grown samples. We find that the simulated ”nematic” aerogel has a structure +factor consistent with lab-grown, axially-compressed silica aerogel while the simulated ”planar” +aerogel has a structure factor consistent with lab-grown ”stretched” silica aerogel. Unexpectedly, +compressing previously isotropic silica aerogel leads to the formation of long strands along the +compression axis while stretching silica aerogel leads to formation of planes perpendicular to the +stretching axis. We discuss the implication of this determination on experiments of superfluid 3He +in anisotropic aerogel, in particular the orbital analog of the spin-flop transition. +1 +arXiv:2301.11261v1 [cond-mat.mtrl-sci] 26 Jan 2023 + +INTRODUCTION +Disorder and impurities are important in determining the relative phase stability [1] and +engineering of novel phenomena [2] in a diverse set of superconductors and superfluids. In +the high-Tc cuprate supercondcutors, doping disorder strongly affects the prescence of super- +conductivity [3–5], vortex physics [6], and low-energy surface states [7]. For superconducting +radio frequency cavities used in particle accelerators, surface impurities improve the quality +factor of the cavities [8, 9]. Recently, aerogels have been used to introduce correlated impu- +rities in superfluid 3He to manipulate the phases [10–12] and stabilize new features such as +half-quantum vortices [13]. Different types of aerogel structures and anisotropy will induce +different properties in the superfluid [14–16]. To better understand how aerogel structure +affects these systems, we use diffusion-limited cluster aggregation (DLCA) simulations to +model anisotropic aerogels. +There is extensive literature on using simulations to model globally homogeneous, +isotropic aerogel (HIA) [17–21], similar to those seen in Fig 1a. Here, we present a frame- +work for generating, analyzing, and classifying aerogels with uniaxial anisotropy that have +large scale structure not present in HIA. Finally, we use this classification to propose a +mechanism for the recently observed orbital analog of the spin-flop texture transition of +superfluid 3He [22]. +The DLCA simulations create an aerogel network by a process similar to that described by +Hasmy et al. in Ref. [19] which we summarize in the following section. To obtain anisotropic +aerogel, we modify this procedure by biasing the diffusion process along one axis defined to +be the z-axis. The degree of anisotropy is labeled by a single continuous variable, ϵ = ϵ ˆz, +with ϵ defined to be the ratio of diffusivity along the z-axis to the diffusivity perpendicular +to z. Isotropic aerogel has an anisotropy parameter of ϵ = 1. Samples with ϵ > 1 and +ϵ < 1 have markedly different large scale structures representing two classes of anisotropic +aerogel. These structures have distinct signatures in their correlation functions and structure +factors which can be directly calculated from the aerogel network. The structure factor is a +particularly useful metric as it can be compared with small-angle x-ray scattering (SAXS) +data obtained from aerogel materials [19, 23]. SAXS data for anisotropic aerogels show clear +anisotropy in the scattering pattern but the underlying structure can not be determined +because the scattering data is only proportional to the amplitude of the scattered wave with +2 + +no phase information [24]. Therefore, it is not possible to fully reconstruct the underlying +structure from the SAXS data alone [25]. +On the other hand, starting from simulated +aerogel with a well understood microscopic structure, we can calculate the structure factor +and compare it with the SAXS data. We leverage this connection to classify real silica aerogel +used in superfluid 3He experiments and demonstrate that the structures in the aerogel induce +the orbital-flop transition [22]. +SIMULATED ANISOTROPIC AEROGEL +Aerogel can be accurately simulated using the procedure detailed in Ref. [19]. A random +point field of N particles (ranging from N = 5000 to 200000) is initialized in a periodic box +with volume L3. The particles have a distribution of radii given by a log-normal distribu- +tion with a sample mean of r0 and sample variance σ0. All lengths in the simulation are +normalized to r0 yielding dimensionless distance parameters (such as L/r0 which is used +to characterize finite size effects). The variance was fixed at σ0/r0 = 1/8 because it does +not affect the large scale structure for reasonable values of σ0. The particles are allowed to +diffuse randomly until they collide with another particle. If a collision occurs, the two par- +ticles are joined into an aggregate and thereafter diffuse together. The diffusion coefficient +is controlled by the size (mass) of the aggregate, with larger clusters diffusing slower. When +all particles are joined into a single cluster, the simulation ends yielding an aerogel cluster. +The resulting cluster is a density field denoted ρ(r). For a discrete field of silica spheres, +ρ(r) is simply a list given by ρ(r) = {{ϱ1, r1}, ..., {ϱN, rN}}, where ϱi is the radius and ri +is center of the the ith-particle. Fig. 1 shows a sample density field for isotropic aerogel +showing the complex aerogel structure and various particle sizes. Our work is focused on +high porosity (low density) aerogel with the a filling fraction of ρ0 ∼ 2%, where ρ0 = 4 +3πr3 +0 +N +L3, +corresponding to real silica aerogel with mass density ∼ 45 mg/cm3 [26]. +The full 3-D rendering of ρ(r) obscures the strand and clustering of the aerogel network so +2-D, orthogonal projections are used to better visualize the spatial variation in the density +field. The right-hand panel of Fig. 1 shows the highly-correlated distribution of particle +position, complex strand structure, and characteristic voids in the aerogel network. For the +case of isotropic aerogel, all these properties have no preferred direction in space. In this +work, we show that anisotropy can be introduced by biasing the diffusion step size along +3 + +FIG. 1. +Real and Simulated Isotropic Aerogel. +a Scanning electron microscopy of real, 98% +porous isotropic aerogel shows the complex network of silica particles. b Simulated aeorgel cluster +for isotropic diffusion for a small segment of the sample (∼ 0.5% of the total sample) (middle). +Structural properties such as strand orientation, clustering, and void size are difficult to determine +in the 3-D representation for the full sample. c Projecting the cluster onto orthogonal, 2-D planes +reveals the position of silica spheres to be non-random. Each plane represents a projection of the +aerogel sample along the axis perpendicular to that plane. For isotropic diffusion, the strands of +silica appear to be without a preferred direction. However, characteristic cluster and void sizes are +visibly apparent. +the z-axis. ϵ > 1 indicates faster diffusion (larger step size) along the z-axis while ϵ < 1 +indicates faster diffusion in the XY -plane. +For anisotropic aerogel, these projections reveal clear spatial anisotropy and large scale +structure not found in the isotropic samples. We have numerically created two types of +anisotropic aerogels with uni-axial, anisotropic diffusion which we classify as nematic, with +ϵ < 1, and planar, with ϵ > 1. As seen in the the projected view of ρ in Fig. 2, anisotropic dif- +fusion introduces a preferred direction breaking the full 3-D rotational symmetry of isotropic +aerogel. In the case of ϵ < 1, the strands are preferentially aligned along the anisotropy di- +rection ϵ. This is akin to nematic systems where long molecules have orientational order +along one axis and absence of regular spatial ordering in the perpendicular plane. For ϵ > 1, +the projected view along the X- and Y -axes reveals high-density, planar sheets of aerogel +clustered together with some characteristic thickness. These sheets are separated from each +other by visible gaps of low density regions with fewer particles. We classify samples with +ϵ > 1 as planar aerogels. A more quantitative description of these nematic and planar struc- +4 + +a +bFIG. 2. +Projection of aerogel structure for anisotropic aerogel. +The projections reveal clear +anisotropic strand structure for ϵ ̸= 1 (as compared with isotropic aerogel in Fig. 1c). a For +ϵ = 0.25, the projections onto the XZ- and YZ-planes reveal long structures parallel to the +anisotropy direction, ϵ, corresponding to nematic strands (inset). The projection along Z into +the XY-plane reveals the strands oriented along Z are still correlated in their positions in the +XY-plane. b For ϵ = 4, sheets of aerogel strands form in the XY-plane, perpendicular to ϵ with +gaps between sheets (inset). The projection along Z into the XY-plane reveals a random structure +indicating that the orientation of the strands between distantly separated sheets are uncorrelated. +tures is obtained by calculating the autocorrelation function, the structure factor, and the +distribution of geometric free paths. These three descriptors unambiguously differentiate +between nematic and planar aerogels. +CHARACTERIZATION +Correlation Function +Silica spheres aggregate to form strands which cluster together to form larger struc- +tures that make up the aerogel network. The positions of the spheres are non-uniform and +highly correlated in space. This non-uniformity is encoded in the autocorrelation function, +g(ri, ..., rj), which is the two-point characteristic of ρ obtained by point-wise multiplication +5 + +of ρ evaluated at all pairs of points ri and rj. For a globally homogeneous cluster with N +particles and volume V , g only depends upon the separation, Rij = ri − rj, given by +g(R) = +⟨ρ(ri) ρ(rj)⟩ +N(N − 1)/(2V ) +(1) +where the angled-brackets, ⟨...⟩, represent an ensemble average over all pairs (the homoge- +neous assumption) and the denominator N(N −1)/(2V ) is the mean density of pairs. When +normalized to the density of pairs, the autocorrelation function gives the excess likelihood to +find two particles separated by a vector R, relative to a random uniform Poisson point field +of the same density [19, 27, 28]. The correlation function defined in Eq. (1) is also sometimes +called the ”pair correlation function”, ”pair distribution function”, or ”radial distribution +function” depending upon the application [28–30]. In the limit of large separation, R → ∞, +g(R) → 1, thereby indicating no excess correlation above that of a uniform distribution. +While different samples drawn from the same probability distribution (i.e. simulated +with the same parameters or experimentally grown under the same conditions) will have +a different value for the density field ρ(r) at any point ri, the two samples will have the +same two-point functions because the correlations remain the same. Therefore, g(R) can be +averaged between samples while ρ cannot. In addition, most applications of aerogel are not +interested in the location of the silica spheres themselves but rather the open space between +them, i.e. the negative of the aerogel structure. Characteristic cluster and void sizes can be +determined from the correlation function. Excess correlation (g > 1) indicates clustering at +that separation distance and direction while a deficit in correlation (g < 1) indicates voids. +For isotropic aerogels (ϵ = 1), g(R) further simplifies to g(R), depending only upon +the magnitude of the separation. On the other hand, for anisotropic samples with ϵ ̸= 1, +the angular-dependence of g(R) becomes important. In the case of azimuthally symmetric, +uniaxial anisotropy, g(R) → g(R, θ), where θ is the polar angle with respect to the anisotropy +axis z. The correlation functions determined here have more structure than the correlation +functions proposed in the literature which are simple power-laws with an upper fractal +exponential cutoff [30]. This is not surprising as the aerogel is anisotropic with different +macroscopic structure than isotropic aerogels. As seen in Fig. 3, g(R, θ) for nematic and +planar aerogel have non-uniform θ-dependence, a clear signature of anisotropy. +The R- +dependence of the deficit in correlation gives the characteristic size of voids and while the +θ-dependence of the deficit gives the shape of voids for each sample. In all direction, there +6 + +is a significant nearest-neighbor peak around 2 r0 indicating contact between particles. The +relative height of the nearest-neighbor peaks in different directions indicate whether pairing +along ϵ (cos(θ) = ±1) or in the plane perpendicular (cos(θ) = 0) is more likely. +For +nematic samples (the green curves in panel c and d), there is greater likelihood for the +nearest neighbor to be in the plane perpendicular to ϵ than parallel to ϵ as seen in Fig. 3 a. +However, at intermediate separation, 10 r0 < R < 50 r0, the direction of excess correlation +swaps, indicating particles are more likely to be collimated along ϵ. This is the signature of +the long nematic strands. +The opposite behavior is observed for planar (ϵ > 1) samples (the blue curves in panel c +and d). At small separations, there is preferential pairing along ϵ but for larger separations, a +neighbor is more likely to be found in the plane. Increasing ϵ increases the nearest-neighbor +peak at short separations but also increases the deficit in correlation in the intermediate +range of ∼ 20 r0. +This is interpreted to be the scale of the thickness of the planes of +aerogel strands. A silica sphere located in the planes is less likely to observe a neighbor +above or below it at distances greater than the sheet thickness but less than two sheet +thickness. Correspondingly, as ϵ increases, so does the size of gaps between the sheets for +planar aerogel. In both the ϵ > 1 and ϵ < 1 cases, it is the behavior of the correlation +function at the intermediate length scale of 10 to 50 r0 that is central to understanding the +macroscopic properties of the aerogel. Visually, the correlation function is dominated by the +nearest-neighbor peak at 2 r0, but this only describes the smallest scale correlation. +There are two methods of numerically calculating g(R), the ”direct” method and the +Fourier-Correlation method, each with their own advantages and disadvantages. The direct +method simply applies the definition of the correlation function and loops through every +pair of particles, calculates their separation vector R, and histograms the set of R into +equal volume R and θ bins. The raw bin counts are then normalized by the density of pairs +and the bin volume, +N(N−1) +2V +2πr2dR d(cosθ) for radial bin width dR and theta bin width +d(cosθ). This method can be implemented natively in spherical coordinates but is slow in +time, scaling as O(N 2) for N particles. +Because of finite sample size, particles near the edge of the sample box will have an +artificial deficit in neighbors leading to the tail of the distribution (large R) being incorrect. +This can be corrected by several different methods as described in Ref. [28]. Astronomers +calculating auto-correlation functions of galaxies observe biases for large R [31] and have +7 + +R/r0 +0 +2 +4 +6 +8 +Cos( ) +1.0 +0.5 +0.0 +0.5 +1.0 +Correlation +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +R/r0 +0 +2 +4 +6 +8 +Cos( ) +1.0 +0.5 +0.0 +0.5 +1.0 +g(r) +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +2 +5 +10 +20 +50 +0.1 +0.5 +1 +5 +10 +R/r0 +g(r) +ϵ =8 +ϵ =1 +a +b +c +d +ϵ =0.125 +2 +5 +10 +20 +50 +R/r0 +ϵ =1 +g(r, Cosθ = 0) +perpendicular to ϵ +ϵ =8 +ϵ =0.125 +g(r, Cosθ = 1) +parallel to ϵ +FIG. 3. Anistropic pair correlation of nematic and planar aerogels as a function of separation R/r0 +and cos(θ). a For nematic (ϵ = 0.25) and b for planar (ϵ = 4), where cos(θ) = ±1 indicates pair +correlations parallel to ϵ and cos(θ) = 0 indicates correlations in the perpendicular plane. The +degree of excess and deficit in correlation can be tuned by changing ϵ as seen in the bottom two +panels. c Pair correlations parallel to ϵ for various values of ϵ. Increasing ϵ increases the nearest- +neighbor peak at short separations but also increases the deficit in correlation in the intermediate +range of 10 < R/r0 < 50. d Correlations perpendicular to ϵ, for various ϵ values. Increasing ϵ in +this direction has the opposite effect. The deficit in correlation in d is less than in c. This implies +that the spacing between nematic strands for ϵ < 1 samples is less than the gaps between sheets +in the planar samples. +devised various estimators to correct for this. The central idea is to consider a randomly +distributed sample of similar size and density. This artificial sample will have the same +finite size effects as the simulation of interest. The autocorrelations of the simulation of +interest and of the artificial sample along with the cross-correlation between the artificial +sample and the simulation are combined to remove finite size effects. Different combinations +8 + +of autocorrelations and cross-correlation have been suggested each with their own statistical +bias [28]. We have implemented several of the most widely used estimators and compared +them with a simple procedure of cross-correlating the original aerogel sample ρ(r) with a +copy of itself that has been spatially-shifted in all three directions by the simulation box size, +ρ(r ± L(ˆx, ˆy, ˆz)). This procedure is equivalent to applying the periodic boundary used in +the simulation. We find that the latter method effectively corrects the tail of g(R) consistent +with the other estimators without the need to generate a test sample. The correlations in +Fig. 3 were determined using the direct method with periodic boundary conditions. +The second method uses the Fourier-Correlation (Wiener-Khinchin) theorem and the +efficiency of fast-Fourier transforms (FFT) to speed up the process. The autocorrelation +function can be obtained by first converting ρ(r) into a sparse 3-D matrix, ρijk, whose +indices form a lattice and whose matrix elements represent the density at lattice site (i, j, k). +This matrix ρijk is numerically Fourier transformed into its conjugate field ˆflmn. Then the +cartesian, pair-distribution function, gxyz, is the inverse Fourier transform given by +gxyz = +⟨ρijk ρi′j′k′⟩ +N(N − 1)/(2V ) = +F−1{| ˆflmn|2} +N(N − 1)/(2V ) +(2) +This calculates the circular autocorrelation of ρ(r) which naturally enforces the periodic +boundary used in the simulation so it does not have to be corrected for finite size effects. +Due to the speed and efficiency of FFT algorithms, this method is significantly faster in time +and scales only as O(NLogN). However, this method is memory intensive as the matrix +representation of ρ grows as L3 for L lattice sites in each dimension. For a cubic lattice with +103 sites per dimension, ρijk, stored as a 32-bit float will be ∼4 GB of data. In computational +complexity, the FFT-correlation method scales well in time but poorly in space (memory), +while the direct method is the opposite, efficient in space but slow in time. Importantly, the +FFT method also calculates the three-dimensional, cartesian structure factor, Sxyz. +Structure Factor +From the structure factor, a connection can be made between simulated and real aerogel. +The x-ray scattering intensity, I(q), can be decomposed as I(q) ∝ S(q)F(q). F(q) is the +single-particle form-factor that encodes details about the particle shape which affects the +large-q behavior of I(q). S(q) encodes correlation in position of particles at intermediate +9 + +and large spatial scale (small-q). For small-angle x-ray scattering (SAXS), I(q) is dominated +by the S(q) contribution. Therefore, the structure factor can be used to directly compare +with SAXS data. +Again, there is extensive literature on determining S(q) by calculating g(R) and per- +forming a Fourier transform [19, 30]. This is usually done in spherical coordinates where the +integration kernel simplifies from eiq·R to sin(qr)/(qr) because the aerogel under considera- +tion is isotropic. In the case of an anisotropic ρ(r), it is easier to numerically perform this +Fourier transform in cartesian coordinates, then convert back to spherical coordinates. In +fact, the structure factor is actually calculated first as an intermediate step when employing +the Fourier-correlation theorem to determine gxyz. The cartesian Sxyz is given by [32] +Sxyz = |F{ρxyz}|2 = | ˆflmn|2. +(3) +To compare with SAXS data for x-rays incident perpendicular to ϵ, Sxyz is then converted +to S(q∥, q⊥), where q∥ is the component parallel to ϵ and q⊥ is the component perpedicular. +The calculated structure factor of simulated aerogel in Fig. 4 a, b, c are compared with +the SAXS data from real aerogel Fig. 4 d, e, f. Isotropic aerogel with ϵ = 1 has the expected +isotropic S(q). For anisotropic samples, the structure factor has two features of note. First +is the distinct dipolar angular distribution pattern at short q (corresponding to a length +scale of ∼100 r0) as seen in yellow and orange in panels b and c. Secondly is the ellipsoidal +pattern at intermediate q (corresponding to ∼10 r0) as seen in the purple regions. +The +orientation of these two patterns can be used to unambiguously classify aerogels. Nematic- +like aerogels will have the short-q, dipolar pattern perpendicular to the anisotropy while +planar-like aerogels will have the dipolar pattern parallel to ϵ. This is a general framework +for classifying aerogel and will hold true irrespective of the material or type of aerogel. +The orientation of the anisotropy of the structure factor is a general feature encoding the +difference between nematic and planar correlations. At larger q (in the purple region of +panels b and c), the major (”long”) axis of the ellipsoidal pattern is rotated 90o from the +short-q dipole pattern. Evidently, there are two scales of structure for the aerogel. The large +scale structure is reflected in the dipole scattering pattern and the smaller scale structure +oriented perpendicular to the large structure is reflected in the ellipsoidal pattern of the +SAXS and structure factor. We use the large scale behavior to classify and label the aerogel +samples as being either nematic or planar. +10 + +a +b +c +d +e +f +FIG. 4. Calculated S(q) versus Small-Angle X-ray Scattering Data [33], with the anisotropy axis +vertical. +The top panels display the calculated structure factor S(q). +a isotropic; b nematic +(ϵ = 0.25), and c planar (ϵ = 4). For the nematic simulation, at short q, S(q)) has a dipolar shape +with the ”long axis” perpendicular to ϵ. For the planar case, at short q, S(q) has a dipolar shape +with the ”long axis” along the anisotropy direction. However, at larger q, as seen in the purple +regions, the ”long axis” of the anisotropy pattern is rotated by 90o. The bottom panels display +the Small-angle X-ray scattering (SAXS) for lab-grown d isotropic, e axially compressed (12.7 % +negative strain), and f stretched (13.7 % positive strain) aerogel samples from Ref. [33]. The black +square in the data images is the beam stop. +Fig. 4 d, e, f show the SAXS data for real aerogels. The two types of anisotropic aerogel +analyzed are obtained by either compressing (negative strain) or stretching (positive strain) +isotropic aerogels [26, 33]. It was not previously known how this strain affected the un- +derlying structure of aerogel. Comparing the SAXS data to our calculated S(q), we can +determine if compressing aerogel creates nematic or planar structure. Compressed aerogels, +seen in panel e, has the dipolar scattering pattern at short q with the ”long” dipole axis +perpendicular to the anisotropy axis. For stretched aerogels in panel f, the short q dipole +11 + +pattern is parallel to ϵ while for intermediate q, the ”long” axis is perpendicular to ϵ. In other +words, the direction in which scattering is more intense rotates by 90o as q increases (going +to smaller length scale), which is the same behavior observed in the calculated structure +factors. +Comparing these SAXS patterns to the structure factor of simulated aerogels, we can +identify the structure of experimentally produced aerogels. Axially compressed aerogel has +a scattering intensity consistent with nematic aerogel. On the other hand, experimentally +stretched aerogel is consistent with planar aerogel. +Compressing isotropic aerogel unex- +pectedly leads to the formation of strands along the compression axis. This is contrary to +speculations from Volovik [14]. In his model, stretching aerogel is argued to create long ne- +matic strands parallel to the stretching while compressing aerogel would collapse the strands +into planes. Here we find the opposite behavior. Our identification however is consistent +with experimental results of superfluid 3He imbibed in anisotropic aerogels[22, 34–36] which +we detail in the final section. +Free Path Distribution +The previous two metrics, the correlation function and structure factor, characterize the +aerogel structure itself. For many applications of aerogel, it is the void between the silica +particles that is relevant rather than the aerogel network. An important measure of this +negative space is the distribution of geometric free path through the aerogel (which appears +as a parameter in theoretical calculations of properties of 3He in aerogel [37, 38]). That is to +say, starting at the surface of a random particle, how far can a test ray move before colliding +with the aerogel network? The condition for collision between a ray and a sphere is given +by the discriminant [39]: +disc = ( ˆd · (pf − pi))2 − (|pf − pi|2 − r2 +f) ≥ 0 +(4) +where d = d ˆd is the ray, pi is the origin of the ray, and pf is the center of the final sphere +with a radius rf. If disc ≥ 0 and ˆd · (pf − pi) ≥ 0 (this second condition ensures only +collisions in the forward direction are considered), then the path length, d, is calculated +as d = ( ˆd · (pf − pi)) − +√ +disc. The free path is determined by taking the minimum d +observed along the direction of travel. If no collision is observed within the initial box, +12 + +d/r0 +0 +2 +4 +6 +8 +Cos( ) +1.0 +0.5 +0.0 +0.5 +1.0 +P (d) +0.01 +0.02 +0.03 +d/r0 +0 +2 +4 +6 +8 +Cos( ) +1.0 +0.5 +0.0 +0.5 +1.0 +P (d) +0.01 +0.02 +0.03 +0.04 +c +d +a +100 +101 +102 +10 +4 +10 +3 +10 +2 +10 +1 +cos( )= 1, parallel to +cos( )= 0, perp to +d +0.7 +b +P(d) +d/r0 +FIG. 5. Geometric free paths in the aerogel network. a Two free flights through nematic (ϵ = 0.25) +aerogel starting at the same location at different angles. b Distribution of free path for planar +(ϵ = 8) aerogel parallel (blue curve) and perpendicular (orange curve) to ϵ. There is a power-law +path distribution (∼ d−0.7) for short paths before being exponentially cut off above 100 r0. The +bottom panels show the full P(d, θ) for (ϵ = 0.25) (panel c) and (ϵ = 4) (panel d) with a similar +nearest-neighbor angular dependence to the correlation function, Fig. 3 +periodic boundary conditions are applied. The bounding box plane that the ray intersects is +determined, and then the aerogel sample is shifted in the appropriate direction and collision +detection is applied for the shifted sample. This is repeated until a collision is found. A +13 + +-2000 +0 +Zlro +2000 +4000 +-2000 +-2000 +0 +1000 +Xlro +0 +2000 +1000 +Ylro +2000probability density function, P(d), is obtained by taking a histogram of the catalog of free +paths. A random walk through the aerogel will have a distribution of step sizes given by +P(d). Fig. 5 a shows the diverging path of two random walks through nematic (ϵ = 0.25) +aerogel starting at the same particle but at different angles. These random walks exhibit the +key feature of what are called ”L´evy flights” [40, 41]. The characteristic ”jumps” of a L´evy +flight are observed where the test ray is confined to small regions followed by big jumps to +other regions [41]. +From P(d), we can calculate a mean free path, λ, which has been shown to be inversely +proportional to density, for low density samples [42]. We find that while the distribution +of free path is very different for high porosity aerogel compared with a uniform Poisson +point field of the same density, the mean free path for both systems are similar to within +7% of 60 r0. There are two reasons for this. First, at low density, both a highly correlated +system like aerogel and an uncorrelated uniform distribution will have large lines of sight. +The excess correlation of the aerogel structure, which affects the distribution at short path +lengths, has a smaller effect than density variations. Secondly, aerogel ceases to be a fractal +above what is call the ”upper-fractal cutoff” [30]. If aerogel had no upper fractal cutoff, then +the free path distribution will be scale-free and described by a generalized L´evy distribution +(the namesake of the L´evy flight) with the asymptotic form P(d) ∼ d−α, with 1 < α < 3 +[40, 41]. This power-law distribution is fat-tailed meaning there is significant weight of the +distribution in long free paths with the possibility that the mean is undefined (for α ≤ 2). +However, both SAXS data and theoretical calculations show that aerogel is not fractal at +all lengths [30, 33]. Correspondingly the free path distribution is cut off and the mean is well +defined. These ”truncated L´evy flights” however still retain many properties of L´evy flights +such as super-diffusion [43]. As seen in Fig. 5, the distribution is indeed power-law below +100 r0 with a very weak exponent of α = 0.7 indicating a very flat probability distribution. +At longer length scales above ∼ 100 r0, the distribution is exponentially cut off. The cutoff +is not due to finite size effects of the simulation as it remains constant with increasing L/r0 +from 100 to 350. +For uniaxially anisotropic aerogels, P(d) ⇒ P(d, θ), being a function of both path length, +d, and polar angle, θ. The height of the peak of the distribution has a θ-dependence similar +to what is observed for the correlation function in Fig. 3. Also like the correlation function, +the behavior of P(d, θ) for intermediate distances ∼ 20 r0 is different from short distances. +14 + +For planar aerogel, Fig. 5 b, there are more free paths along ϵ at very short distances but +more free paths perpendicular to ϵ at intermediate distances, consistent with the existence +of large planar gaps in the structure as indicated in Fig. 2 b. +Each angle can be considered an independent probability distribution and distribution +moments can be defined at different angles. Two directions of particular interest are the +mean free path parallel, λ∥ , and mean free path perpendicular, λ⊥, to the anisotropy +direction ϵ. Despite very clear anisotropy, the first moments of P(d, θ) (mean free path +along a certain direction) are similar for each of the two orthogonal directions. For ρ0 ∼ 2%, +λ∥ ∼ λ⊥ ∼ 60 r0 in both the nematic and planar aerogels. For real silica aerogels used in +superfluid 3He experiments, r0 is ∼ 1.5−2 nm, indicating a mean free path of ∼ 90−120 nm +[38], and can be much larger, up to ∼10 nm in general [44, 45]), consistent with experimental +measurements for isotropic aerogel of comparable density [46]. Consequently, the mean is not +a good parameter to characterize the path distribution of high porosity anisotropic aerogel. +In addition, the two length scales in the anisotropic samples are further hidden when only +the mean free path is considered. The blue and orange curves in Fig. 5 b are quite different +and yet they have the same first moment. We conclude that it is insufficient to simply use +the mean free path to encode the effects of anisotropic aerogel in theoretical calculations. +To recap, we find that the correlation function, structure factor, and distribution of +free paths form a set of metrics that can be used to characterize and classify anisotropic +aerogels. From this process, it was determined that axially compressed silica aerogel has +nematic strands while stretched aerogel has planar structure. We use this determination +in the following section to explain a set of superfluid 3He experiments that employ these +aerogels. +ANISOTROPIC AEROGEL AND SUPERFLUID 3HE +Anisotropic aerogels have recently found use in superfluid 3He where it stabilizes novel +order parameter structures such as half-quantum vortices, superfluid polar-phase, and the +orbital angular momentum analog of the spin-flop transition in antiferromagnets [12, 13, 22, +47]. Here we propose a mechanism for the ”orbital-flop” transition based on our identification +of different length scales present in anisotropic silica aerogel. +Superfluid 3He is an unconventional, topological superfluid with quasiparticles forming p- +15 + +wave (L = 1), spin-triplet (S = 1) Cooper pairs creating a manifold of possible phases. In the +chiral ”A-phase”, the Cooper pairs have a net orbital angular momentum, ℓA, with a vector +order parameter. In the isotropic ”B-phase”, the Cooper pairs exist in a superposition of +all three components of spin and orbital angular momentum projections with total angular +momentum J = 0. The relative stability of the phases is strongly affected by aerogel. In the +pure superfluid, both the A and B-phases can exist as stable equilibrium phases depending +upon temperature, pressure, and magnetic field. This phase diagram is drastically altered +in the presence of anisotropic aerogel. For compressed aerogel in zero magnetic field, only +the B-phase is observed for the entire pressure and temperature phase diagram [22, 48]. +On the other hand, for stretched aerogel, the A-phase becomes the equilibrium phase at all +magnetic fields, pressure and temperature [34, 47]. In addition to altering the stability of +phases, anisotropic aerogel has been observed to reorient the orbital degrees of freedom [22]. +Fig. 6 shows phase diagrams of these two systems. +In the presence of symmetry breaking effects such as magnetic fields, boundaries, or +anisotropic disorder, the B-phase becomes distorted in its orbital degrees of freedom giving +rise to a preferred direction denoted ℓB. Recently, sharp transitions have been observed +where the orbital vectors in the two phases spontaneously reorient by 90◦ uniformly across +the entire system as temperature or pressure is varied [22, 35]. +It was determined that +this reorientation is dependent upon the anisotropy of the aerogel and not from competing +orienting effects such as from boundaries as has been observed in isotropic aerogel [49]. +Phase identification of the superfluid, and identification of the direction of the angular +momentum axis can be determined from NMR spectra obtained in a high homogeneity steady +magnetic field, discussed most recently by Zimmerman et al. [49] In the superfluid A-phase +of stretched aerogel, ℓA orients parallel to the anisotropy axis ϵ at high temperature near +the superfluid transition, Tc. NMR experiments [34, 35] show that at a lower temperature +denoted Tx, ℓA spontaneously flops over to being perpendicular to ϵ across the entire sample, +as depicted in panel of Fig. 7 b. In the superfluid B-phase of compressed aerogel, ℓB is +initially perpendicular to ϵ near Tc, the opposite of what is observed in the A-phase of +stretched aerogel. At Tx, ℓB sharply reorients to being parallel to ϵ with a narrow transition +width of ∼ 15 µK [22]. This orbital-flop transition varies with pressure between ∼ 0.67 Tc +at 7.5 bar to 0.88 Tc at 26 bar. The opposite behavior of these two samples is resolved by +considering the underlying structure of the aerogel. +16 + +5 +10 +15 +20 +25 +P (bar) +ℓ ǁ ε +B-phase +ℓ ⟂ ε +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +0 +5 +10 +15 +20 +25 +30 +T (mK) +P (bar) +ℓ ⟂ ε +A-phase +ℓ ǁ ε +FIG. 6. Phase diagram of superfluid 3He in 14 % stretched aerogel (top) and 20 % compressed +aerogel (bottom) extrapolated to zero magnetic field. In stretched aerogel, there is only the A- +phase while there is only the B-phase in compressed aerogel. The orientation of ℓ is reversed in +the two systems. +From both the SAXS data and S(q), it is evident that these anisotropic aerogels have +different structure at long and short length scales. The large scale structure is given by the +dipolar pattern while the small scale structure is given by the ellipsoidal pattern at large +q, Fig. 4. Furthermore, the scattering patterns reveal that large scale structure is oriented +perpendicular to the small scale structure. We propose that this structural crossover in the +aerogel induces the orbital-flop transition. The most important length scale in a superfluid +is the coherence length, ξ, which can be thought of as the size of a Cooper pair (or more +17 + +accurately, the healing length for variations of the order parameter). The coherence length +is largest near the superfluid transition and decreases with temperature. Therefore, at high +temperature near Tc, the superfluid’s orbital degrees of freedom will be sensitive to large +scale disorder. As ξ becomes smaller at lower temperature, the smaller scale structure in +the aerogel dominates. +The analysis presented in this work unambiguously identifies that at long length scales +stretched aerogel has planar structure while compressed aerogel has nematic structure . For +planar aerogels, the surface normal of the large scale structure points along the anisotropy +axis, ˆϵ. Correspondingly in the A-phase of superfluid 3He in planar aerogel we would expect +ℓA∥ˆϵ at high temperatures above Tx, and ℓa ⊥ ˆϵ below Tx [34, 35]. If the preferred orienta- +tion of ℓ is determined solely by aerogel structure, it must be independent of the superfluid +phase. Consequently for a B-phase in nematic aerogel, parallel and perpendicular orbital +orientations are just interchanged as seen in Ref. [22, 35]. Above and below Tx, ℓ preferen- +tially orient perpendicular to the dominant aerogel structure. If this is the mechanism for +the transition, we expect that the coherence length evaluated at Tx, ξ(Tx, P) to be relatively +constant at different pressures. The transition occurs at Tx because that is temperature at +which the superfluid becomes more sensitive to the small scale aerogel structure rather than +the large scale structure. +The coherence length varies with both temperature and pressure with the zero-temperature +coherence length defined to be ξ0(P) = +� +7ζ(3) +12 +�1/2 +ℏ vF (P) +2π kBTc(P), where ζ is the Riemann-zeta +function, vF(P) is the pressure-dependent Fermi velocity, and Tc(P) is the pressure depen- +dent superfluid transition temperature. ξ0(P) varies from 15 to 80 nm between solidification +pressure (34.4 bar) to 0 bar. Most of the pressure dependence of ξ0(P) occurs between 0 +and 6 bar. The experiments in Ref. [22] occur between 7.5 bar and 27 bar where ξ0(P) +varies only from 34 to 18 nm. There are several different definitions for the temperature +dependence of ξ. The most widely used definition is the Ginzburg-Landau (GL) correlation +length given by: +ξGL(T) = ξ0(P)(1 − T/Tc)−1/2. +(5) +ξ diverges near the second order phase transition and decays away with reducing temperature +as (1 − T/Tc)−1/2. The GL coherence length is shown in Fig. 7 for various pressures. +When evaluated at Tx, ξ at the various pressures all collapse into a narrow band of values +around 50 nm, consistent with the model for the orbital-flop transition. The large variation +18 + +T > Tx +T < Tx +l +l +T/Tc +ξ (nm) +stretched +planar +7.5 bar +10 bar +15 bar +26 bar +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +20 +50 +100 +200 +a +compressed +nematic +b +FIG. 7. a, The Ginzburg-Landau coherence length ξ(T, P) for various pressures. The orbital flop +transition Tx in the B-phase of compressed aerogel for each pressure is indicated by the data point +and the vertical dashed line. Tx/Tc varies with pressure but the coherence length evaluated at +various Tx, ξ(Tx, P), all fall into a narrow band around 49 nm. This indicates that the orbital flop +transition occurs when the superfluid coherence length decreases that length scale. b, Orientation +of the orbital angular momentum in stretched aerogel which has been identified as planar aerogel. +Above Tx, the coherence length is large and ℓ is oriented perpendicular to the large scale planar +structure. Below Tx, the coherence length is small and ℓ reorients to being perpendicular to the +small scale structure. +of Tx with pressure more or less converges to a narrow range of length scales. Above Tx, +the ξ is large so ℓ is oriented perpendicular to the large scale structure. At the temperature +when ξ drops below roughly 50 nm, the orbital flop transition occurs and ℓ is reoriented +by the small scale structure. The crossover in scales seen in the correlation function and +structure factor are of order ∼ 20-50 r0. For silica aerogel with r0 ∼ 1.5 nm, this corresponds +to order 30-75 nm compared with ξGL(Tx, P) ∼ 50 nm. A more definitive test of this model +would require going to lower pressure. At low pressure, the coherence length is substantially +larger and we expect Tx to drop in temperature as pressure is lowered. Below 1.5 bar, the +zero-temperature coherence length is greater than 50 nm meaning no crossover transition is +expected. +Other experiments using a different type of planar aerogel also observe a phase diagram +dominated by the A-phase with the orbital angular momentum orienting perpendicular +to the planar sheets [36]. However, an orbital flop transition was not observed in those +experiments because the aerogel has much stronger anisotropy and does not appear to have +the two different length scales. +19 + +The sharpness of the orbital flop transition creates a useful experimental tuning parameter +for probing new physics. +Recently, it was shown that there is a substantial anomalous +thermal hall effect in superfluid 3He in the presence of impurities like aerogel [50]. The +direction of transverse thermal current is strongly dependent upon the orientation of the +orbital angular momentum. Therefore, the orbital-flop can be used as a switch to turn on +or off the transverse current. Because of the sharpness of the transition, the hall current +should drop to zero abruptly as temperature is changed across Tx. This switching will be a +definitive signature of the anomalous thermal hall effect. +CONCLUSION +In summary, we outline a procedure to simulate and characterize anisotropic aerogels +with planar and nematic strands. The anisotropy is induced by biasing the diffusion process +and can be characterized by the autocorrelation function, structure factor, and distribution +of free paths. We make a connection to experimental aerogel by comparing the shape of +the SAXS pattern with the structure factor. Both the calculated structure factor and the +SAXS data exhibit a congruent dipolar shape at small-q and a perpendicular ellipsoidal +pattern at large-q. These two patterns reveal two different length scales of anisotropy in +the aerogels. +From this connection, we are able to classify real aerogel and show that +stretched silica aerogel has large scale planar structure while compressed aerogel has large +scale nematic structure. Finally, we provide a description of the aerogel’s effect on the orbital +angular momentum of superfluid 3He. The orbital angular momentum is oriented by the +large scale structure in the aerogel at high temperature before spontaneously reorienting at +a lower temperature due to the small scale structure. This ”orbital-flop” transition can be +leveraged in future work to observe the anomalous thermal hall effect in superfluid 3He. +This work was supported by the National Science Foundation, grant DMR-2210112. +∗ mannguyen2019@u.northwestern.edu +† w-halperin@northwestern.edu +[1] Y. Imry and S. Ma, Phys. Rev. Lett. 35, 1399 (1975). +[2] M. P. A. Fisher, Phys. Rev. Lett. 62, 1415 (1989). +20 + +[3] S. H. Pan, J. P. O’Neal, R. L. Badzey, C. Chamon, H. Ding, J. R. Engelbrecht, Z. Wang, +H. Eisaki, S. Uchida, A. K. Gupta, K.-W. Ng, E. W. Hudson, K. M. Lang, and J. C. Davis, +Nature 413, 282 (2001). +[4] P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys. 78, 17 (2006). +[5] B. 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Kanda, H. Kawai, N. Kawamura, G. Marshall, H. Ma- +suda, Y. Matsuda, T. Mibe, Y. Miyake, S. Okada, K. Olchanski, A. Olin, H. Onishi, N. Saito, +K. Shimomura, P. Strasser, M. Tabata, D. Tomono, K. Ueno, K. Yokoyama, and S. Yoshida, +Progress of Theoretical and Experimental Physics 2013 (2013). +[47] J. I. A. Li, J. Pollanen, A. M. Zimmerman, C. A. Collett, W. J. Gannon, and W. P. Halperin, +Nature Physics 9, 775 (2013). +[48] A. M. Zimmerman, M. D. Nguyen, J. W. Scott, and W. P. Halperin, Phys. Rev. Lett. 124, +025302 (2020). +[49] N. M. D. H. W. P. Zimmerman, A. M., Journal of Low Temperature Physics 195, 358 (2019). +[50] V. Ngampruetikorn and J. A. Sauls, Phys. Rev. Lett. 124, 157002 (2020). +23 + +APPENDIX +The structure factors in Fig. 4 show S(q) out to q ∼ 0.1 r−1 +0 . While the small-angle X-ray +scattering data is only dependent upon the small q behavior of S(q), we can calculate the +full structure factor out to q = r−1 +0 . As seen in Fig. 8, there are oscillations in the intensity +at large q arising from the interparticle spacing. The differences in the two anisotropies are +still evident at the smallest scale (largest q). +1.00 +0.75 +0.50 +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +0.75 +0.50 +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +S +(1/r0) +0.75 +0.50 +0.25 +0.00 +0.25 +0.50 +0.75 +0.75 +0.50 +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +S +(1/r0) +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +2.2 +S +(1/r0) +S +(1/r0) +S +(1/r0) +FIG. 8. The full structure factor of isotropic (left), nematic ϵ = 0.125 (center), and planar ϵ = 8 +(right) aerogels out to large q. +24 + +2500 +2000 +1500 +1000 +500 +0 +Intensity (a.u.) +0.15 +0.10 +0.05 +Wavenumber (1/r0) +0.20 +0.15 +0.10 +0.05 +Wavenumber (1/r0) + x-axis + y-axis + z-axis +Planar +Nematic +FIG. 9. +Power spectrum of the density for planar ϵ = 8 (left) and nematic ϵ = 0.125 (right) +aerogels. For planar aerogels, there is a sharp peak around wavenumber 0.015 r−1 +0 . This gives a +typical spacing between local maxima in density of about 60 − 70 r0. The size of the gaps between +planes is then roughly half of that at 30 r0. There is not much density variation in the xy-plane. +For nematic aerogel (right), there are peaks in the x- and y-axis density power spectrum which can +be interpreted as the diameter of the nematic bundles. +25 + diff --git a/LtFIT4oBgHgl3EQfbSuz/content/tmp_files/load_file.txt b/LtFIT4oBgHgl3EQfbSuz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..66ba8a748f1b260bfb400e9e5b22a19242d0a6f9 --- /dev/null +++ b/LtFIT4oBgHgl3EQfbSuz/content/tmp_files/load_file.txt @@ -0,0 +1,1030 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf,len=1029 +page_content='Planar and Nematic Aerogels: DLCA and Superfluid 3He M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Nguyen1,∗ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Simon1, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Scott1, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Cincia Tsai1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Zimmerman2, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Halperin1† 1Department of Physics and Astronomy, Northwestern University and 2Department of Physics, Harvard University (Dated: January 27, 2023) Abstract We perform cluster aggregation simulations to model the structure of anisotropic aerogel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' By biasing the diffusion process, we are able to obtain two distinct types of globally anisotropic aerogel structures which we call ”nematic”, with long strands along the anisotropy axis, and ”planar”, with long strands in planes perpendicular to the anisotropy axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' We calculate the auto-correlation function, the structure factor, and the angular dependence of the free-path distribution for these samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The calculated structure factor from simulated aerogels can be compared with data from small-angle X-ray scattering (SAXS) of lab-grown aerogel allowing us to classify the spatial structure of the lab-grown samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' We find that the simulated ”nematic” aerogel has a structure factor consistent with lab-grown, axially-compressed silica aerogel while the simulated ”planar” aerogel has a structure factor consistent with lab-grown ”stretched” silica aerogel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Unexpectedly, compressing previously isotropic silica aerogel leads to the formation of long strands along the compression axis while stretching silica aerogel leads to formation of planes perpendicular to the stretching axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' We discuss the implication of this determination on experiments of superfluid 3He in anisotropic aerogel, in particular the orbital analog of the spin-flop transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='11261v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='mtrl-sci] 26 Jan 2023 INTRODUCTION Disorder and impurities are important in determining the relative phase stability [1] and engineering of novel phenomena [2] in a diverse set of superconductors and superfluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' In the high-Tc cuprate supercondcutors, doping disorder strongly affects the prescence of super- conductivity [3–5], vortex physics [6], and low-energy surface states [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' For superconducting radio frequency cavities used in particle accelerators, surface impurities improve the quality factor of the cavities [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Recently, aerogels have been used to introduce correlated impu- rities in superfluid 3He to manipulate the phases [10–12] and stabilize new features such as half-quantum vortices [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Different types of aerogel structures and anisotropy will induce different properties in the superfluid [14–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' To better understand how aerogel structure affects these systems, we use diffusion-limited cluster aggregation (DLCA) simulations to model anisotropic aerogels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' There is extensive literature on using simulations to model globally homogeneous, isotropic aerogel (HIA) [17–21], similar to those seen in Fig 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Here, we present a frame- work for generating, analyzing, and classifying aerogels with uniaxial anisotropy that have large scale structure not present in HIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Finally, we use this classification to propose a mechanism for the recently observed orbital analog of the spin-flop texture transition of superfluid 3He [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The DLCA simulations create an aerogel network by a process similar to that described by Hasmy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' [19] which we summarize in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' To obtain anisotropic aerogel, we modify this procedure by biasing the diffusion process along one axis defined to be the z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The degree of anisotropy is labeled by a single continuous variable, ϵ = ϵ ˆz, with ϵ defined to be the ratio of diffusivity along the z-axis to the diffusivity perpendicular to z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Isotropic aerogel has an anisotropy parameter of ϵ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Samples with ϵ > 1 and ϵ < 1 have markedly different large scale structures representing two classes of anisotropic aerogel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' These structures have distinct signatures in their correlation functions and structure factors which can be directly calculated from the aerogel network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The structure factor is a particularly useful metric as it can be compared with small-angle x-ray scattering (SAXS) data obtained from aerogel materials [19, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' SAXS data for anisotropic aerogels show clear anisotropy in the scattering pattern but the underlying structure can not be determined because the scattering data is only proportional to the amplitude of the scattered wave with 2 no phase information [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Therefore, it is not possible to fully reconstruct the underlying structure from the SAXS data alone [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' On the other hand, starting from simulated aerogel with a well understood microscopic structure, we can calculate the structure factor and compare it with the SAXS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' We leverage this connection to classify real silica aerogel used in superfluid 3He experiments and demonstrate that the structures in the aerogel induce the orbital-flop transition [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' SIMULATED ANISOTROPIC AEROGEL Aerogel can be accurately simulated using the procedure detailed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' A random point field of N particles (ranging from N = 5000 to 200000) is initialized in a periodic box with volume L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The particles have a distribution of radii given by a log-normal distribu- tion with a sample mean of r0 and sample variance σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' All lengths in the simulation are normalized to r0 yielding dimensionless distance parameters (such as L/r0 which is used to characterize finite size effects).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The variance was fixed at σ0/r0 = 1/8 because it does not affect the large scale structure for reasonable values of σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The particles are allowed to diffuse randomly until they collide with another particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' If a collision occurs, the two par- ticles are joined into an aggregate and thereafter diffuse together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The diffusion coefficient is controlled by the size (mass) of the aggregate, with larger clusters diffusing slower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' When all particles are joined into a single cluster, the simulation ends yielding an aerogel cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The resulting cluster is a density field denoted ρ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' For a discrete field of silica spheres, ρ(r) is simply a list given by ρ(r) = {{ϱ1, r1}, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=', {ϱN, rN}}, where ϱi is the radius and ri is center of the the ith-particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 1 shows a sample density field for isotropic aerogel showing the complex aerogel structure and various particle sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Our work is focused on high porosity (low density) aerogel with the a filling fraction of ρ0 ∼ 2%, where ρ0 = 4 3πr3 0 N L3, corresponding to real silica aerogel with mass density ∼ 45 mg/cm3 [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The full 3-D rendering of ρ(r) obscures the strand and clustering of the aerogel network so 2-D, orthogonal projections are used to better visualize the spatial variation in the density field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The right-hand panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 1 shows the highly-correlated distribution of particle position, complex strand structure, and characteristic voids in the aerogel network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' For the case of isotropic aerogel, all these properties have no preferred direction in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' In this work, we show that anisotropy can be introduced by biasing the diffusion step size along 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Real and Simulated Isotropic Aerogel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' a Scanning electron microscopy of real, 98% porous isotropic aerogel shows the complex network of silica particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' b Simulated aeorgel cluster for isotropic diffusion for a small segment of the sample (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='5% of the total sample) (middle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Structural properties such as strand orientation, clustering, and void size are difficult to determine in the 3-D representation for the full sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' c Projecting the cluster onto orthogonal, 2-D planes reveals the position of silica spheres to be non-random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Each plane represents a projection of the aerogel sample along the axis perpendicular to that plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' For isotropic diffusion, the strands of silica appear to be without a preferred direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' However, characteristic cluster and void sizes are visibly apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' the z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' ϵ > 1 indicates faster diffusion (larger step size) along the z-axis while ϵ < 1 indicates faster diffusion in the XY -plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' For anisotropic aerogel, these projections reveal clear spatial anisotropy and large scale structure not found in the isotropic samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' We have numerically created two types of anisotropic aerogels with uni-axial, anisotropic diffusion which we classify as nematic, with ϵ < 1, and planar, with ϵ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' As seen in the the projected view of ρ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 2, anisotropic dif- fusion introduces a preferred direction breaking the full 3-D rotational symmetry of isotropic aerogel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' In the case of ϵ < 1, the strands are preferentially aligned along the anisotropy di- rection ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' This is akin to nematic systems where long molecules have orientational order along one axis and absence of regular spatial ordering in the perpendicular plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' For ϵ > 1, the projected view along the X- and Y -axes reveals high-density, planar sheets of aerogel clustered together with some characteristic thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' These sheets are separated from each other by visible gaps of low density regions with fewer particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' We classify samples with ϵ > 1 as planar aerogels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' A more quantitative description of these nematic and planar struc- 4 a bFIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Projection of aerogel structure for anisotropic aerogel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The projections reveal clear anisotropic strand structure for ϵ ̸= 1 (as compared with isotropic aerogel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' a For ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='25, the projections onto the XZ- and YZ-planes reveal long structures parallel to the anisotropy direction, ϵ, corresponding to nematic strands (inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The projection along Z into the XY-plane reveals the strands oriented along Z are still correlated in their positions in the XY-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' b For ϵ = 4, sheets of aerogel strands form in the XY-plane, perpendicular to ϵ with gaps between sheets (inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The projection along Z into the XY-plane reveals a random structure indicating that the orientation of the strands between distantly separated sheets are uncorrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' tures is obtained by calculating the autocorrelation function, the structure factor, and the distribution of geometric free paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' These three descriptors unambiguously differentiate between nematic and planar aerogels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' CHARACTERIZATION Correlation Function Silica spheres aggregate to form strands which cluster together to form larger struc- tures that make up the aerogel network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The positions of the spheres are non-uniform and highly correlated in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' This non-uniformity is encoded in the autocorrelation function, g(ri, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=', rj), which is the two-point characteristic of ρ obtained by point-wise multiplication 5 of ρ evaluated at all pairs of points ri and rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' For a globally homogeneous cluster with N particles and volume V , g only depends upon the separation, Rij = ri − rj, given by g(R) = ⟨ρ(ri) ρ(rj)⟩ N(N − 1)/(2V ) (1) where the angled-brackets, ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='⟩, represent an ensemble average over all pairs (the homoge- neous assumption) and the denominator N(N −1)/(2V ) is the mean density of pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' When normalized to the density of pairs, the autocorrelation function gives the excess likelihood to find two particles separated by a vector R, relative to a random uniform Poisson point field of the same density [19, 27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The correlation function defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' (1) is also sometimes called the ”pair correlation function”, ”pair distribution function”, or ”radial distribution function” depending upon the application [28–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' In the limit of large separation, R → ∞, g(R) → 1, thereby indicating no excess correlation above that of a uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' While different samples drawn from the same probability distribution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' simulated with the same parameters or experimentally grown under the same conditions) will have a different value for the density field ρ(r) at any point ri, the two samples will have the same two-point functions because the correlations remain the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Therefore, g(R) can be averaged between samples while ρ cannot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' In addition, most applications of aerogel are not interested in the location of the silica spheres themselves but rather the open space between them, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' the negative of the aerogel structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Characteristic cluster and void sizes can be determined from the correlation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Excess correlation (g > 1) indicates clustering at that separation distance and direction while a deficit in correlation (g < 1) indicates voids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' For isotropic aerogels (ϵ = 1), g(R) further simplifies to g(R), depending only upon the magnitude of the separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' On the other hand, for anisotropic samples with ϵ ̸= 1, the angular-dependence of g(R) becomes important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' In the case of azimuthally symmetric, uniaxial anisotropy, g(R) → g(R, θ), where θ is the polar angle with respect to the anisotropy axis z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The correlation functions determined here have more structure than the correlation functions proposed in the literature which are simple power-laws with an upper fractal exponential cutoff [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' This is not surprising as the aerogel is anisotropic with different macroscopic structure than isotropic aerogels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 3, g(R, θ) for nematic and planar aerogel have non-uniform θ-dependence, a clear signature of anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The R- dependence of the deficit in correlation gives the characteristic size of voids and while the θ-dependence of the deficit gives the shape of voids for each sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' In all direction, there 6 is a significant nearest-neighbor peak around 2 r0 indicating contact between particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The relative height of the nearest-neighbor peaks in different directions indicate whether pairing along ϵ (cos(θ) = ±1) or in the plane perpendicular (cos(θ) = 0) is more likely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' For nematic samples (the green curves in panel c and d), there is greater likelihood for the nearest neighbor to be in the plane perpendicular to ϵ than parallel to ϵ as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 3 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' However, at intermediate separation, 10 r0 < R < 50 r0, the direction of excess correlation swaps, indicating particles are more likely to be collimated along ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' This is the signature of the long nematic strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The opposite behavior is observed for planar (ϵ > 1) samples (the blue curves in panel c and d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' At small separations, there is preferential pairing along ϵ but for larger separations, a neighbor is more likely to be found in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Increasing ϵ increases the nearest-neighbor peak at short separations but also increases the deficit in correlation in the intermediate range of ∼ 20 r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' This is interpreted to be the scale of the thickness of the planes of aerogel strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' A silica sphere located in the planes is less likely to observe a neighbor above or below it at distances greater than the sheet thickness but less than two sheet thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Correspondingly, as ϵ increases, so does the size of gaps between the sheets for planar aerogel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' In both the ϵ > 1 and ϵ < 1 cases, it is the behavior of the correlation function at the intermediate length scale of 10 to 50 r0 that is central to understanding the macroscopic properties of the aerogel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Visually, the correlation function is dominated by the nearest-neighbor peak at 2 r0, but this only describes the smallest scale correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' There are two methods of numerically calculating g(R), the ”direct” method and the Fourier-Correlation method, each with their own advantages and disadvantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The direct method simply applies the definition of the correlation function and loops through every pair of particles, calculates their separation vector R, and histograms the set of R into equal volume R and θ bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The raw bin counts are then normalized by the density of pairs and the bin volume, N(N−1) 2V 2πr2dR d(cosθ) for radial bin width dR and theta bin width d(cosθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' This method can be implemented natively in spherical coordinates but is slow in time, scaling as O(N 2) for N particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Because of finite sample size, particles near the edge of the sample box will have an artificial deficit in neighbors leading to the tail of the distribution (large R) being incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' This can be corrected by several different methods as described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Astronomers calculating auto-correlation functions of galaxies observe biases for large R [31] and have 7 R/r0 0 2 4 6 8 Cos( ) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='0 Correlation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='0 R/r0 0 2 4 6 8 Cos( ) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='0 g(r) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='0 2 5 10 20 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='5 1 5 10 R/r0 g(r) ϵ =8 ϵ =1 a b c d ϵ =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='125 2 5 10 20 50 R/r0 ϵ =1 g(r, Cosθ = 0) perpendicular to ϵ ϵ =8 ϵ =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='125 g(r, Cosθ = 1) parallel to ϵ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Anistropic pair correlation of nematic and planar aerogels as a function of separation R/r0 and cos(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' a For nematic (ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='25) and b for planar (ϵ = 4), where cos(θ) = ±1 indicates pair correlations parallel to ϵ and cos(θ) = 0 indicates correlations in the perpendicular plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The degree of excess and deficit in correlation can be tuned by changing ϵ as seen in the bottom two panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' c Pair correlations parallel to ϵ for various values of ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Increasing ϵ increases the nearest- neighbor peak at short separations but also increases the deficit in correlation in the intermediate range of 10 < R/r0 < 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' d Correlations perpendicular to ϵ, for various ϵ values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Increasing ϵ in this direction has the opposite effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The deficit in correlation in d is less than in c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' This implies that the spacing between nematic strands for ϵ < 1 samples is less than the gaps between sheets in the planar samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' devised various estimators to correct for this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The central idea is to consider a randomly distributed sample of similar size and density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' This artificial sample will have the same finite size effects as the simulation of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The autocorrelations of the simulation of interest and of the artificial sample along with the cross-correlation between the artificial sample and the simulation are combined to remove finite size effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Different combinations 8 of autocorrelations and cross-correlation have been suggested each with their own statistical bias [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' We have implemented several of the most widely used estimators and compared them with a simple procedure of cross-correlating the original aerogel sample ρ(r) with a copy of itself that has been spatially-shifted in all three directions by the simulation box size, ρ(r ± L(ˆx, ˆy, ˆz)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' This procedure is equivalent to applying the periodic boundary used in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' We find that the latter method effectively corrects the tail of g(R) consistent with the other estimators without the need to generate a test sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The correlations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 3 were determined using the direct method with periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The second method uses the Fourier-Correlation (Wiener-Khinchin) theorem and the efficiency of fast-Fourier transforms (FFT) to speed up the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The autocorrelation function can be obtained by first converting ρ(r) into a sparse 3-D matrix, ρijk, whose indices form a lattice and whose matrix elements represent the density at lattice site (i, j, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' This matrix ρijk is numerically Fourier transformed into its conjugate field ˆflmn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Then the cartesian, pair-distribution function, gxyz, is the inverse Fourier transform given by gxyz = ⟨ρijk ρi′j′k′⟩ N(N − 1)/(2V ) = F−1{| ˆflmn|2} N(N − 1)/(2V ) (2) This calculates the circular autocorrelation of ρ(r) which naturally enforces the periodic boundary used in the simulation so it does not have to be corrected for finite size effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Due to the speed and efficiency of FFT algorithms, this method is significantly faster in time and scales only as O(NLogN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' However, this method is memory intensive as the matrix representation of ρ grows as L3 for L lattice sites in each dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' For a cubic lattice with 103 sites per dimension, ρijk, stored as a 32-bit float will be ∼4 GB of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' In computational complexity, the FFT-correlation method scales well in time but poorly in space (memory), while the direct method is the opposite, efficient in space but slow in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Importantly, the FFT method also calculates the three-dimensional, cartesian structure factor, Sxyz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Structure Factor From the structure factor, a connection can be made between simulated and real aerogel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The x-ray scattering intensity, I(q), can be decomposed as I(q) ∝ S(q)F(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' F(q) is the single-particle form-factor that encodes details about the particle shape which affects the large-q behavior of I(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' S(q) encodes correlation in position of particles at intermediate 9 and large spatial scale (small-q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' For small-angle x-ray scattering (SAXS), I(q) is dominated by the S(q) contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Therefore, the structure factor can be used to directly compare with SAXS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Again, there is extensive literature on determining S(q) by calculating g(R) and per- forming a Fourier transform [19, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' This is usually done in spherical coordinates where the integration kernel simplifies from eiq·R to sin(qr)/(qr) because the aerogel under considera- tion is isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' In the case of an anisotropic ρ(r), it is easier to numerically perform this Fourier transform in cartesian coordinates, then convert back to spherical coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' In fact, the structure factor is actually calculated first as an intermediate step when employing the Fourier-correlation theorem to determine gxyz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The cartesian Sxyz is given by [32] Sxyz = |F{ρxyz}|2 = | ˆflmn|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' (3) To compare with SAXS data for x-rays incident perpendicular to ϵ, Sxyz is then converted to S(q∥, q⊥), where q∥ is the component parallel to ϵ and q⊥ is the component perpedicular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The calculated structure factor of simulated aerogel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 4 a, b, c are compared with the SAXS data from real aerogel Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 4 d, e, f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Isotropic aerogel with ϵ = 1 has the expected isotropic S(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' For anisotropic samples, the structure factor has two features of note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' First is the distinct dipolar angular distribution pattern at short q (corresponding to a length scale of ∼100 r0) as seen in yellow and orange in panels b and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Secondly is the ellipsoidal pattern at intermediate q (corresponding to ∼10 r0) as seen in the purple regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The orientation of these two patterns can be used to unambiguously classify aerogels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Nematic- like aerogels will have the short-q, dipolar pattern perpendicular to the anisotropy while planar-like aerogels will have the dipolar pattern parallel to ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' This is a general framework for classifying aerogel and will hold true irrespective of the material or type of aerogel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The orientation of the anisotropy of the structure factor is a general feature encoding the difference between nematic and planar correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' At larger q (in the purple region of panels b and c), the major (”long”) axis of the ellipsoidal pattern is rotated 90o from the short-q dipole pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Evidently, there are two scales of structure for the aerogel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The large scale structure is reflected in the dipole scattering pattern and the smaller scale structure oriented perpendicular to the large structure is reflected in the ellipsoidal pattern of the SAXS and structure factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' We use the large scale behavior to classify and label the aerogel samples as being either nematic or planar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 10 a b c d e f FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Calculated S(q) versus Small-Angle X-ray Scattering Data [33], with the anisotropy axis vertical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The top panels display the calculated structure factor S(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' a isotropic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' b nematic (ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='25), and c planar (ϵ = 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' For the nematic simulation, at short q, S(q)) has a dipolar shape with the ”long axis” perpendicular to ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' For the planar case, at short q, S(q) has a dipolar shape with the ”long axis” along the anisotropy direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' However, at larger q, as seen in the purple regions, the ”long axis” of the anisotropy pattern is rotated by 90o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The bottom panels display the Small-angle X-ray scattering (SAXS) for lab-grown d isotropic, e axially compressed (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='7 % negative strain), and f stretched (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='7 % positive strain) aerogel samples from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The black square in the data images is the beam stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 4 d, e, f show the SAXS data for real aerogels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The two types of anisotropic aerogel analyzed are obtained by either compressing (negative strain) or stretching (positive strain) isotropic aerogels [26, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' It was not previously known how this strain affected the un- derlying structure of aerogel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Comparing the SAXS data to our calculated S(q), we can determine if compressing aerogel creates nematic or planar structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Compressed aerogels, seen in panel e, has the dipolar scattering pattern at short q with the ”long” dipole axis perpendicular to the anisotropy axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' For stretched aerogels in panel f, the short q dipole 11 pattern is parallel to ϵ while for intermediate q, the ”long” axis is perpendicular to ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' In other words, the direction in which scattering is more intense rotates by 90o as q increases (going to smaller length scale), which is the same behavior observed in the calculated structure factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Comparing these SAXS patterns to the structure factor of simulated aerogels, we can identify the structure of experimentally produced aerogels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Axially compressed aerogel has a scattering intensity consistent with nematic aerogel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' On the other hand, experimentally stretched aerogel is consistent with planar aerogel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Compressing isotropic aerogel unex- pectedly leads to the formation of strands along the compression axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' This is contrary to speculations from Volovik [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' In his model, stretching aerogel is argued to create long ne- matic strands parallel to the stretching while compressing aerogel would collapse the strands into planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Here we find the opposite behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Our identification however is consistent with experimental results of superfluid 3He imbibed in anisotropic aerogels[22, 34–36] which we detail in the final section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Free Path Distribution The previous two metrics, the correlation function and structure factor, characterize the aerogel structure itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' For many applications of aerogel, it is the void between the silica particles that is relevant rather than the aerogel network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' An important measure of this negative space is the distribution of geometric free path through the aerogel (which appears as a parameter in theoretical calculations of properties of 3He in aerogel [37, 38]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' That is to say, starting at the surface of a random particle, how far can a test ray move before colliding with the aerogel network?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The condition for collision between a ray and a sphere is given by the discriminant [39]: disc = ( ˆd · (pf − pi))2 − (|pf − pi|2 − r2 f) ≥ 0 (4) where d = d ˆd is the ray, pi is the origin of the ray, and pf is the center of the final sphere with a radius rf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' If disc ≥ 0 and ˆd · (pf − pi) ≥ 0 (this second condition ensures only collisions in the forward direction are considered), then the path length, d, is calculated as d = ( ˆd · (pf − pi)) − √ disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The free path is determined by taking the minimum d observed along the direction of travel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' If no collision is observed within the initial box, 12 d/r0 0 2 4 6 8 Cos( ) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='0 P (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='03 d/r0 0 2 4 6 8 Cos( ) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='0 P (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='04 c d a 100 101 102 10 4 10 3 10 2 10 1 cos( )= 1, parallel to cos( )= 0, perp to d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='7 b P(d) d/r0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Geometric free paths in the aerogel network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' a Two free flights through nematic (ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='25) aerogel starting at the same location at different angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' b Distribution of free path for planar (ϵ = 8) aerogel parallel (blue curve) and perpendicular (orange curve) to ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' There is a power-law path distribution (∼ d−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='7) for short paths before being exponentially cut off above 100 r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The bottom panels show the full P(d, θ) for (ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='25) (panel c) and (ϵ = 4) (panel d) with a similar nearest-neighbor angular dependence to the correlation function, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 3 periodic boundary conditions are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The bounding box plane that the ray intersects is determined, and then the aerogel sample is shifted in the appropriate direction and collision detection is applied for the shifted sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' This is repeated until a collision is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' A 13 2000 0 Zlro 2000 4000 2000 2000 0 1000 Xlro 0 2000 1000 Ylro 2000probability density function, P(d), is obtained by taking a histogram of the catalog of free paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' A random walk through the aerogel will have a distribution of step sizes given by P(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 5 a shows the diverging path of two random walks through nematic (ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='25) aerogel starting at the same particle but at different angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' These random walks exhibit the key feature of what are called ”L´evy flights” [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The characteristic ”jumps” of a L´evy flight are observed where the test ray is confined to small regions followed by big jumps to other regions [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' From P(d), we can calculate a mean free path, λ, which has been shown to be inversely proportional to density, for low density samples [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' We find that while the distribution of free path is very different for high porosity aerogel compared with a uniform Poisson point field of the same density, the mean free path for both systems are similar to within 7% of 60 r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' There are two reasons for this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' First, at low density, both a highly correlated system like aerogel and an uncorrelated uniform distribution will have large lines of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The excess correlation of the aerogel structure, which affects the distribution at short path lengths, has a smaller effect than density variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Secondly, aerogel ceases to be a fractal above what is call the ”upper-fractal cutoff” [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' If aerogel had no upper fractal cutoff, then the free path distribution will be scale-free and described by a generalized L´evy distribution (the namesake of the L´evy flight) with the asymptotic form P(d) ∼ d−α, with 1 < α < 3 [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' This power-law distribution is fat-tailed meaning there is significant weight of the distribution in long free paths with the possibility that the mean is undefined (for α ≤ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' However, both SAXS data and theoretical calculations show that aerogel is not fractal at all lengths [30, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Correspondingly the free path distribution is cut off and the mean is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' These ”truncated L´evy flights” however still retain many properties of L´evy flights such as super-diffusion [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 5, the distribution is indeed power-law below 100 r0 with a very weak exponent of α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='7 indicating a very flat probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' At longer length scales above ∼ 100 r0, the distribution is exponentially cut off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The cutoff is not due to finite size effects of the simulation as it remains constant with increasing L/r0 from 100 to 350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' For uniaxially anisotropic aerogels, P(d) ⇒ P(d, θ), being a function of both path length, d, and polar angle, θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The height of the peak of the distribution has a θ-dependence similar to what is observed for the correlation function in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Also like the correlation function, the behavior of P(d, θ) for intermediate distances ∼ 20 r0 is different from short distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 14 For planar aerogel, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 5 b, there are more free paths along ϵ at very short distances but more free paths perpendicular to ϵ at intermediate distances, consistent with the existence of large planar gaps in the structure as indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 2 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Each angle can be considered an independent probability distribution and distribution moments can be defined at different angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Two directions of particular interest are the mean free path parallel, λ∥ , and mean free path perpendicular, λ⊥, to the anisotropy direction ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Despite very clear anisotropy, the first moments of P(d, θ) (mean free path along a certain direction) are similar for each of the two orthogonal directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' For ρ0 ∼ 2%, λ∥ ∼ λ⊥ ∼ 60 r0 in both the nematic and planar aerogels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' For real silica aerogels used in superfluid 3He experiments, r0 is ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='5−2 nm, indicating a mean free path of ∼ 90−120 nm [38], and can be much larger, up to ∼10 nm in general [44, 45]), consistent with experimental measurements for isotropic aerogel of comparable density [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Consequently, the mean is not a good parameter to characterize the path distribution of high porosity anisotropic aerogel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' In addition, the two length scales in the anisotropic samples are further hidden when only the mean free path is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The blue and orange curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 5 b are quite different and yet they have the same first moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' We conclude that it is insufficient to simply use the mean free path to encode the effects of anisotropic aerogel in theoretical calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' To recap, we find that the correlation function, structure factor, and distribution of free paths form a set of metrics that can be used to characterize and classify anisotropic aerogels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' From this process, it was determined that axially compressed silica aerogel has nematic strands while stretched aerogel has planar structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' We use this determination in the following section to explain a set of superfluid 3He experiments that employ these aerogels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' ANISOTROPIC AEROGEL AND SUPERFLUID 3HE Anisotropic aerogels have recently found use in superfluid 3He where it stabilizes novel order parameter structures such as half-quantum vortices, superfluid polar-phase, and the orbital angular momentum analog of the spin-flop transition in antiferromagnets [12, 13, 22, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Here we propose a mechanism for the ”orbital-flop” transition based on our identification of different length scales present in anisotropic silica aerogel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Superfluid 3He is an unconventional, topological superfluid with quasiparticles forming p- 15 wave (L = 1), spin-triplet (S = 1) Cooper pairs creating a manifold of possible phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' In the chiral ”A-phase”, the Cooper pairs have a net orbital angular momentum, ℓA, with a vector order parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' In the isotropic ”B-phase”, the Cooper pairs exist in a superposition of all three components of spin and orbital angular momentum projections with total angular momentum J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The relative stability of the phases is strongly affected by aerogel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' In the pure superfluid, both the A and B-phases can exist as stable equilibrium phases depending upon temperature, pressure, and magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' This phase diagram is drastically altered in the presence of anisotropic aerogel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' For compressed aerogel in zero magnetic field, only the B-phase is observed for the entire pressure and temperature phase diagram [22, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' On the other hand, for stretched aerogel, the A-phase becomes the equilibrium phase at all magnetic fields, pressure and temperature [34, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' In addition to altering the stability of phases, anisotropic aerogel has been observed to reorient the orbital degrees of freedom [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 6 shows phase diagrams of these two systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' In the presence of symmetry breaking effects such as magnetic fields, boundaries, or anisotropic disorder, the B-phase becomes distorted in its orbital degrees of freedom giving rise to a preferred direction denoted ℓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Recently, sharp transitions have been observed where the orbital vectors in the two phases spontaneously reorient by 90◦ uniformly across the entire system as temperature or pressure is varied [22, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' It was determined that this reorientation is dependent upon the anisotropy of the aerogel and not from competing orienting effects such as from boundaries as has been observed in isotropic aerogel [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Phase identification of the superfluid, and identification of the direction of the angular momentum axis can be determined from NMR spectra obtained in a high homogeneity steady magnetic field, discussed most recently by Zimmerman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' [49] In the superfluid A-phase of stretched aerogel, ℓA orients parallel to the anisotropy axis ϵ at high temperature near the superfluid transition, Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' NMR experiments [34, 35] show that at a lower temperature denoted Tx, ℓA spontaneously flops over to being perpendicular to ϵ across the entire sample, as depicted in panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 7 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' In the superfluid B-phase of compressed aerogel, ℓB is initially perpendicular to ϵ near Tc, the opposite of what is observed in the A-phase of stretched aerogel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' At Tx, ℓB sharply reorients to being parallel to ϵ with a narrow transition width of ∼ 15 µK [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' This orbital-flop transition varies with pressure between ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='67 Tc at 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='5 bar to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='88 Tc at 26 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The opposite behavior of these two samples is resolved by considering the underlying structure of the aerogel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 16 5 10 15 20 25 P (bar) ℓ ǁ ε B-phase ℓ ⟂ ε 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='5 0 5 10 15 20 25 30 T (mK) P (bar) ℓ ⟂ ε A-phase ℓ ǁ ε FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Phase diagram of superfluid 3He in 14 % stretched aerogel (top) and 20 % compressed aerogel (bottom) extrapolated to zero magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' In stretched aerogel, there is only the A- phase while there is only the B-phase in compressed aerogel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The orientation of ℓ is reversed in the two systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' From both the SAXS data and S(q), it is evident that these anisotropic aerogels have different structure at long and short length scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The large scale structure is given by the dipolar pattern while the small scale structure is given by the ellipsoidal pattern at large q, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Furthermore, the scattering patterns reveal that large scale structure is oriented perpendicular to the small scale structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' We propose that this structural crossover in the aerogel induces the orbital-flop transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The most important length scale in a superfluid is the coherence length, ξ, which can be thought of as the size of a Cooper pair (or more 17 accurately, the healing length for variations of the order parameter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The coherence length is largest near the superfluid transition and decreases with temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Therefore, at high temperature near Tc, the superfluid’s orbital degrees of freedom will be sensitive to large scale disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' As ξ becomes smaller at lower temperature, the smaller scale structure in the aerogel dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The analysis presented in this work unambiguously identifies that at long length scales stretched aerogel has planar structure while compressed aerogel has nematic structure .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' For planar aerogels, the surface normal of the large scale structure points along the anisotropy axis, ˆϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Correspondingly in the A-phase of superfluid 3He in planar aerogel we would expect ℓA∥ˆϵ at high temperatures above Tx, and ℓa ⊥ ˆϵ below Tx [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' If the preferred orienta- tion of ℓ is determined solely by aerogel structure, it must be independent of the superfluid phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Consequently for a B-phase in nematic aerogel, parallel and perpendicular orbital orientations are just interchanged as seen in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' [22, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Above and below Tx, ℓ preferen- tially orient perpendicular to the dominant aerogel structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' If this is the mechanism for the transition, we expect that the coherence length evaluated at Tx, ξ(Tx, P) to be relatively constant at different pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The transition occurs at Tx because that is temperature at which the superfluid becomes more sensitive to the small scale aerogel structure rather than the large scale structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The coherence length varies with both temperature and pressure with the zero-temperature coherence length defined to be ξ0(P) = � 7ζ(3) 12 �1/2 ℏ vF (P) 2π kBTc(P), where ζ is the Riemann-zeta function, vF(P) is the pressure-dependent Fermi velocity, and Tc(P) is the pressure depen- dent superfluid transition temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' ξ0(P) varies from 15 to 80 nm between solidification pressure (34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='4 bar) to 0 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Most of the pressure dependence of ξ0(P) occurs between 0 and 6 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The experiments in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' [22] occur between 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='5 bar and 27 bar where ξ0(P) varies only from 34 to 18 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' There are several different definitions for the temperature dependence of ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The most widely used definition is the Ginzburg-Landau (GL) correlation length given by: ξGL(T) = ξ0(P)(1 − T/Tc)−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' (5) ξ diverges near the second order phase transition and decays away with reducing temperature as (1 − T/Tc)−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The GL coherence length is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 7 for various pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' When evaluated at Tx, ξ at the various pressures all collapse into a narrow band of values around 50 nm, consistent with the model for the orbital-flop transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The large variation 18 T > Tx T < Tx l l T/Tc ξ (nm) stretched planar 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='5 bar 10 bar 15 bar 26 bar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='0 20 50 100 200 a compressed nematic b FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' a, The Ginzburg-Landau coherence length ξ(T, P) for various pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The orbital flop transition Tx in the B-phase of compressed aerogel for each pressure is indicated by the data point and the vertical dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Tx/Tc varies with pressure but the coherence length evaluated at various Tx, ξ(Tx, P), all fall into a narrow band around 49 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' This indicates that the orbital flop transition occurs when the superfluid coherence length decreases that length scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' b, Orientation of the orbital angular momentum in stretched aerogel which has been identified as planar aerogel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Above Tx, the coherence length is large and ℓ is oriented perpendicular to the large scale planar structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Below Tx, the coherence length is small and ℓ reorients to being perpendicular to the small scale structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' of Tx with pressure more or less converges to a narrow range of length scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Above Tx, the ξ is large so ℓ is oriented perpendicular to the large scale structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' At the temperature when ξ drops below roughly 50 nm, the orbital flop transition occurs and ℓ is reoriented by the small scale structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The crossover in scales seen in the correlation function and structure factor are of order ∼ 20-50 r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' For silica aerogel with r0 ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='5 nm, this corresponds to order 30-75 nm compared with ξGL(Tx, P) ∼ 50 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' A more definitive test of this model would require going to lower pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' At low pressure, the coherence length is substantially larger and we expect Tx to drop in temperature as pressure is lowered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Below 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='5 bar, the zero-temperature coherence length is greater than 50 nm meaning no crossover transition is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Other experiments using a different type of planar aerogel also observe a phase diagram dominated by the A-phase with the orbital angular momentum orienting perpendicular to the planar sheets [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' However, an orbital flop transition was not observed in those experiments because the aerogel has much stronger anisotropy and does not appear to have the two different length scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 19 The sharpness of the orbital flop transition creates a useful experimental tuning parameter for probing new physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Recently, it was shown that there is a substantial anomalous thermal hall effect in superfluid 3He in the presence of impurities like aerogel [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The direction of transverse thermal current is strongly dependent upon the orientation of the orbital angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Therefore, the orbital-flop can be used as a switch to turn on or off the transverse current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Because of the sharpness of the transition, the hall current should drop to zero abruptly as temperature is changed across Tx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' This switching will be a definitive signature of the anomalous thermal hall effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' CONCLUSION In summary, we outline a procedure to simulate and characterize anisotropic aerogels with planar and nematic strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The anisotropy is induced by biasing the diffusion process and can be characterized by the autocorrelation function, structure factor, and distribution of free paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' We make a connection to experimental aerogel by comparing the shape of the SAXS pattern with the structure factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Both the calculated structure factor and the SAXS data exhibit a congruent dipolar shape at small-q and a perpendicular ellipsoidal pattern at large-q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' These two patterns reveal two different length scales of anisotropy in the aerogels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' From this connection, we are able to classify real aerogel and show that stretched silica aerogel has large scale planar structure while compressed aerogel has large scale nematic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Finally, we provide a description of the aerogel’s effect on the orbital angular momentum of superfluid 3He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The orbital angular momentum is oriented by the large scale structure in the aerogel at high temperature before spontaneously reorienting at a lower temperature due to the small scale structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' This ”orbital-flop” transition can be leveraged in future work to observe the anomalous thermal hall effect in superfluid 3He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' This work was supported by the National Science Foundation, grant DMR-2210112.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' of Low Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 150, 453 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' [15] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Askhadullin, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 23 APPENDIX The structure factors in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 4 show S(q) out to q ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='1 r−1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' While the small-angle X-ray scattering data is only dependent upon the small q behavior of S(q), we can calculate the full structure factor out to q = r−1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 8, there are oscillations in the intensity at large q arising from the interparticle spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The differences in the two anisotropies are still evident at the smallest scale (largest q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='50 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='2 S (1/r0) S (1/r0) S (1/r0) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The full structure factor of isotropic (left), nematic ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='125 (center), and planar ϵ = 8 (right) aerogels out to large q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 24 2500 2000 1500 1000 500 0 Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='05 Wavenumber (1/r0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='05 Wavenumber (1/r0) x-axis y-axis z-axis Planar Nematic FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' Power spectrum of the density for planar ϵ = 8 (left) and nematic ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='125 (right) aerogels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' For planar aerogels, there is a sharp peak around wavenumber 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content='015 r−1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' This gives a typical spacing between local maxima in density of about 60 − 70 r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' The size of the gaps between planes is then roughly half of that at 30 r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' There is not much density variation in the xy-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFIT4oBgHgl3EQfbSuz/content/2301.11261v1.pdf'} +page_content=' For nematic aerogel (right), there are peaks in the x- and y-axis density power spectrum which can be interpreted as the diameter 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+1 +Robust Domain Adaptive Object Detection with +Unified Multi-Granularity Alignment +Libo Zhang, Wenzhang Zhou, Heng Fan, Tiejian Luo, and Haibin Ling +Abstract—Domain adaptive detection aims to improve the generalization of detectors on target domain. To reduce discrepancy in feature +distributions between two domains, recent approaches achieve domain adaption through feature alignment in different granularities via +adversarial learning. However, they neglect the relationship between multiple granularities and different features in alignment, degrading +detection. Addressing this, we introduce a unified multi-granularity alignment (MGA)-based detection framework for domain-invariant +feature learning. The key is to encode the dependencies across different granularities including pixel-, instance-, and category-levels +simultaneously to align two domains. Specifically, based on pixel-level features, we first develop an omni-scale gated fusion (OSGF) +module to aggregate discriminative representations of instances with scale-aware convolutions, leading to robust multi-scale detection. +Besides, we introduce multi-granularity discriminators to identify where, either source or target domains, different granularities of samples +come from. Note that, MGA not only leverages instance discriminability in different categories but also exploits category consistency +between two domains for detection. Furthermore, we present an adaptive exponential moving average (AEMA) strategy that explores +model assessments for model update to improve pseudo labels and alleviate local misalignment problem, boosting detection robustness. +Extensive experiments on multiple domain adaption scenarios validate the superiority of MGA over other approaches on FCOS and +Faster R-CNN detectors. Code will be released at https://github.com/tiankongzhang/MGA. +Index Terms—Domain Adaptive Object Detection, Multi-Granularity Alignment, Omni-scale Gated Fusion, Model Assessment, +Adaptive Exponential Moving Average. +! +1 +INTRODUCTION +O +BJECT detection has been one of the most fundamental +problems in computer vision with a long list of ap- +plications such as visual surveillance, self-driving, robotics, +etc. Owing to the powerful representation by deep learning +(e.g., [2], [3], [4]), object detection has witnessed consider- +able advancement in recent years with numerous excellent +frameworks (e.g., [5], [6], [7], [8], [9], [10], [11], [12]). These +modern detectors are usually trained and evaluated on a +large-scale annotated dataset (e.g., [13]). Despite the great +achievement, they may suffer from poor generalization +when applied to images from a new target domain. To +remedy this, a simple and straightforward solution is to +build a benchmark for the new target domain and re-train +the detector. Nevertheless, benchmark creation is both time- +consuming and costly. In addition, the new target domain +could arbitrary and it is almost impossible to develop bench- +marks for all new target domains. +In order to deal with the above issues, researchers have +explored the unsupervised domain adaption (UDA) detection, +with the goal of transferring knowledge learned from an +annotated source domain to an unlabeled target domain. +One popular trend is to leverage adversarial learning [14] +• +Libo Zhang is with the State Key Laboratory of Computer Science, +Institute of Software Chinese Academy of Sciences, China. E-mail: +libo@iscas.ac.cn. +• +Wenzhang Zhou and Tiejian Luo are with the University of Chinese +Academy of Sciences, China. E-mail: zhouwenzhang19@mails.ucas.ac.cn, +tjluo@ucas.ac.cn. +• +Heng Fan is with the Department of Computer Science and Engineering, +University of North Texas, USA. E-mail: heng.fan@unt.edu. +• +Haibin Ling is with the Department of Computer Science, Stony Brook +University, USA. E-mail: hling@cs.stonybrook.edu. +• +Libo Zhang and Wenzhang Zhou make equal contributions to this work. +• +A preliminary version [1] of this work has appeared in CVPR 2022. +Student Detector +(i.e., Adapted Detector) +Pixels +Categories +Instances +Domains +Source +or +Target? +Omni-scale +Gated Fusion +Teacher Detector +(i.e., Original Detector) +Pixel-level +Discriminator +Instance-level +Discriminator +Category-level +Discriminator +𝛿 +Refined +Pseudo +Labels +Pseudo +Labels +Update +Model +Assessment +AEMA +Fig. 1. Illustration of the proposed Multi-Granularity Alignment (MGA) +framework for domain adaptive object detection. Specifically, MGA en- +codes the dependencies across multiple granularities simultaneously, +including pixel-, instance-, and category-levels. In addition, a dynamic +update mechanism guided by update factor δ (as detailed later) through +model assessment during training is used to improve the quality of +pseudo labels and meanwhile mitigate the local misalignment problem, +further enhancing the detection robustness. Best viewed in color and by +zooming in for all figures throughout this paper. +to narrow the discrepancy between domains. Specifically, a +domain discriminator is introduced to distinguish which do- +main, the source or the target, the image comes from. Then, +the detector learns the domain-invariant feature representa- +tion by confusing the discriminator [15]. Despite achieving +promising results, previous domain adaption approaches +may suffer from target scale variations in cluttered regions +due to the fixed kernel design in ConvNets [2], [3], which re- +sults in difficulty in learning discriminative representations +for objects of different scales and thus degrades detection +performance. For example, the features of small targets +may contain too much background noise because of too +large receptive field in the convolutional layer. Meanwhile, +arXiv:2301.00371v1 [cs.CV] 1 Jan 2023 + +SUBMITTED TO IEEE JOURNAL +2 +the features of large objects may lack global structural +information owing to too small receptive field. In addition, +the intrinsic relation of feature distributions between two +domains are neglected. +To address the aforementioned problems, various feature +alignment strategies have been introduced in the adversarial +learning manner [14], [16] for better target domain adap- +tion. These alignment approaches can be summarized into +three categories based on different granularity perspectives, +consisting of pixel-, instance-, and category-level. Pixel-level +alignment [17], [18], [19] aims at aligning lower-level pixel +feature distribution of objects and background regions. Nev- +ertheless, there may exist a large gap between the pixel- +level features for objects of different scales within the same +category, resulting in limited detection performance. Differ- +ent from pixel-level alignment, instance-level alignment [20], +[21], [22], [23], [24] first pools the feature maps of detection +proposals and then leverages the pooled proposal features +for the domain discriminator training. Despite avoiding +the gap in pixel-level alignment, this strategy suffers from +feature distortion for objects of different scales and aspect +ratios caused by the pooling operation, which may lead to +inaccurate feature representation and degenerated results. +In addition to the aforementioned two types of alignments, +recent approaches have attempted to utilize category-level +alignment [25], [26], [27], [28], [29], [30] for UDA detection. +In specific, taking into consideration the intrinsic relation +of feature distributions in two domains, the category-level +alignment leverages the categorical discriminability to han- +dle hard aligned instances. However, this alignment mech- +anism focuses only on the global consistency of feature +distribution between two domains while ignores other local +consistency constrains. +Contribution. Although each type of alignment strategy +brings in improvement, they are limited in several aspects +as discussed above. In order to address these issues and +make full use of the advantages of these three alignments, +we propose a novel unified Multi-Granularity Alignment +(MGA) framework for UDA detection, as illustrated in +Figure 1. +Instead of performing simple combination of different +alignment methods, MGA simultaneously encodes the de- +pendencies across different granularities, consisting of pixel- +, instance-, and category-levels, for domain alignment. More +specifically, we first introduce an omni-scale gated fusion +(OSGF) module in MGA. The OSGF module is able to adapt +to instances of different scales by automatically choosing +the most plausible convolutions from the low- and high- +resolution streams for feature extraction. Concretely, we first +predict coarse detections based on pixel-level backbone fea- +ture maps. Then, using these coarse detections as guidance, +we design a set of parallel convolutions in OSGF and adopt +a gate mechanism to aggregate the discriminative features +of instances (i.e., the coarse detections) with similar scales +and aspect ratios. By doing so, the following detection head +can more accurately predict multi-scale objects. +Besides the OSGF module, we present a new category- +level discriminator. Different from previous approaches, our +category-level discriminator takes into account not only the +instance discriminability in different classes but also the cat- +egory consistency between two domains, leading to better +detection. In order to supervise the category-level discrim- +inator, pseudo labels are assigned to important instances +with high confidence based on object detection results. +Considering that the quality of pseudo labels is crucial +for learning a good category-level discriminator, we propose +a simple yet effective adaptive exponential moving average +(AEMA) strategy to train the teacher detector (i.e., the orig- +inal detector). As shown in Figure 1, during the training +phase, we assess both the teacher detector and the student +detector (i.e., the final adaptive detector) on the source do- +main. Based on their model assessments, a dynamic update +factor δ (as described later) is learned and utilized as a +guidance to adjust the coefficient parameter in exponential +moving average (EMA) update in an adaptive manner. The +resulted AEMA helps better train the teacher detector to +produce high-quality pseudo labels and meanwhile allevi- +ate the local misalignment caused by low-quality pseudo +labels, significantly enhancing the detection robustness. We +will elaborate on the details of our AEMA later. +By developing the multi-granularity discriminators, our +MGA exploits and integrates rich complementary informa- +tion from different levels, and hence achieves better UDA +detection. Besides, the proposed AEMA further enhances +the robustness with high-quality pseudo labels. +To validate the effectiveness of our approach, we carry +out extensive experiments on multiple domain-shift scenar- +ios using various benchmarks including Cityscapes [31], +FoggyCityscapes [32], Sim10k [33], KITTI [34], PASCAL +VOC [35], Clipart [36] and Watercolor [36]. We evaluate +the proposed method on the top of two popular detection +frameworks, the anchor-free FCOS [8] and the anchor-based +Faster R-CNN [7], with VGG-16 [4] and ResNet-101 [2] +backbones. Experiment results demonstrate that our MGA +together with the dynamic model update significantly im- +prove the baseline detectors with superior results over other +state-of-the-arts. +To sum up, we make the following key contributions: +• +We propose a novel unified multi-granularity align- +ment (MGA) framework that encodes dependencies +across different pixel-, instance-, and category-levels +for UDA detection. Notably, our MGA framework is +general and applicable to different object detectors. +• +We present an omni-scale gate fusion (OSGF) module +to extract discriminative feature representation for +instances with different scales and aspect ratios. +• +we propose a new category-level discriminator by +exploiting both the instance discriminability in dif- +ferent classes and the category consistency between +two domains, leading to better detection. +• +We introduce a simple yet effective dynamic adap- +tive exponential moving average (AEMA) strategy +to improve the quality of pseudo labels and mean- +while mitigate the local misalignment issue in UDA +detection, which significantly boosts the robustness +of detector. +• +On extensive experiments on multiple domain adap- +tion scenarios, the proposed approach outperforms +other state-of-the-art UDA detectors on the top of +two detection frameworks, evidencing its effective- + +SUBMITTED TO IEEE JOURNAL +3 +ness and generality. +This paper builds upon our preliminary conference ver- +sion [1] and significantly extends it in different aspects. +(1) We propose an effective assessment-based AEMA for +model update of the teacher detector. This way, we are able +to obtain pseudo labels with better quality, which largely +boosts the detection robustness. Meanwhile, it is beneficial +in alleviating the local misalignment issue caused by low- +quality pseudo labels, further improving the performance. +(2) We modify the structure of the OSGF module by sharing +the convolutional layer (as described later). Note that, this +modification is not trivial. It not only brings in improvement +on the detection results but also decreases the number +of parameters. (3) We incorporate more experiments and +comparisons with in-depth analysis and ablation studies +to further show the effectiveness of our approach. (4) We +supplement thorough visual analysis of our detector, which +allows the readers to better understand our method. +The rest of this paper is organized as follows. Section 2 +discusses approaches related to this paper. Our approach is +elaborated in Section 3. In Section 4, we demonstrate the +experimental results, including comparisons with state-of- +the-arts, ablation studies and visual analysis, followed by +conclusion in Section 5. +2 +RELATED WORK +In this section, we review approaches relevant to this paper +from four aspects, including object detection, UDA detec- +tion, alignment strategy for UDA detection and exponential +moving average. +2.1 +Object Detection +Object detection is a fundamental topic in computer vision +and has been extensively studied for decades. In general, +existing modern detectors can be categorized into either +anchor-based or anchor-free. Anchor-based detectors usu- +ally contain a set of anchor boxes with different scales +and aspect ratios, which are applied to generate object +proposals for further processing (in two-stage frameworks) +or final detections (in one-stage frameworks). One of the +most popular anchor-based detectors is Faster R-CNN [7]. +It introduces a novel region proposal network (RPN) to +produce object proposals based on anchors and then applies +another network to further process the proposals for detec- +tion. The approaches of SSD [10] and YOLOv2 [37] present +one-stage anchor-based detectors that strikes a good balance +between accuracy and speed. Later, more excellent anchor- +based detectors [12], [38], [39], [40], [41], [42] are proposed +for improvements. Different from anchor-based approaches, +anchor-free detectors remove the manual design of anchor +boxes and directly predict the class and coordinates of +objects. YOLO [11] directly predicts the object class and po- +sition from grid cells. CornerNet [9] proposes to predict the +object bounding boxes as keypoint detection. The work of +CenterNet [43] improves CornetNet by considering an extra +center point. FCOS [8] introduces the fully convolutional +networks to predict object box of each pixel in feature maps. +Recently, DETR [44] applies Transformer [45] to develop an +anchor-free detector and exhibits impressive performance. +In this paper, we utilize the proposed MGA upon the +popular anchor-based Faster R-CNN [7] and anchor-free +FCOS [8] to verify its effectiveness. But please note that, +our MGA is general and flexible and can be used in more +frameworks for UDA detection. +2.2 +Unsupervised Domain Adaption (UDA) Detection +The task of unsupervised domain adaption (UDA) detection +focuses on improving the generality of object detectors +learned from labeled source images on unlabeled target +images. Because of its great practicability, UDA object de- +tection has attracted extensive attention in recent years. One +popular framework is to leverage adversarial learning to +achieve UDA detection [46], [47]. These approaches intro- +duce a discriminator to identify which domain the features +of pixels, regions or images come from. Then, the goal is to +confuse the discriminator to learn domain-invariant features +for detection. In addition, many other researchers propose to +apply graph methods for UDA detection [48], [49], [50], [51]. +These graph-based approaches propose to construct a graph +based on regions or instances in an image and leverages +the intra-class and inter-class relation of intra-domain and +inter-domain for detection. Self-training strategy has also +been explored in UDA detection [52], [53], [54], [55]. The +main idea of these methods is to generate high-quality or +class-balanced pseudo-labels, which can be utilized to train +the detection model on the unlabeled target domain. The +approaches of [19], [56], [57] leverage the idea of style trans- +fer for UDA detection. In specific, these methods reduce +the discrepancy of data distribution between two domains +by translating the images of source domain to target style, +which improves the generalization ability of the detector. +Besides the aforementioned methods, recent approaches +propose to apply mean-teacher for UDA detection [58], [59], +[60]. These models adopt a teacher-student training frame- +work to maintain the consistency of the teacher and student +detector networks for boosting the generality of detection. +Different from the above approaches, in this paper we tackle +the UDA detection problem from a different perspective by +unifying alignments of multiple granularities. +2.3 +Alignment for UDA Detection +Alignment of feature distribution between source and target +domains has demonstrated effectiveness for UDA detection. +Accordingly to the features involved, recent alignment- +based approaches can be categorized into three types in- +cluding pixel-, instance-, and category-level alignments. +Pixel-level Alignment. Pixel-level alignment focuses on +aligning the pixel feature distributions of objects and back- +ground regions between two domains for UDA detection. +The work of [18] takes into account every pixel for do- +main adaption and introduces a center-aware pixel-level +alignment by paying more attention to foreground pixels +for UDA detection. The approach of [17] designs the multi- +domain-invariant representation learning to encourage un- +biased semantic representation through adversarial learn- +ing. The method of [19] introduces an intermediate domain +for progressive adaption and utilize adversarial learning for +pixel-level feature alignment. + +SUBMITTED TO IEEE JOURNAL +4 +Teacher FCOS Detector +FPN +Backbone +Omni-scale +Gated Fusion +guidance +Object +Detection +Target +Student FCOS Detector +FPN +Backbone +guidance +Object +Detection +Source +Target +F +Assessment-based AEMA +Memory +… +Labels Images +… +C0 +… +… +C1 +… +… +CK +… +store +Domain Shift +Simulation +AEMA +𝛿 +Loss 𝐿!"# +𝒮 +Loss 𝐿!"# +𝒯 +Coarse +Detections +Parameters 𝜃𝒮 +Parameters 𝜃𝒯 +Update 𝜃𝒯 +Cat.-level +Discriminator +Ins.-level +Discriminator +Pix.-level +Discriminator +GRL +GRL +Source or +Target +Source +Path +Gradient Reversal Layer +GRL +Target +Path +AEMA Adaptative Exponential +Moving Average +Coarse +Detections +Teacher Faster R-CNN Detector +Backbone +Omni-scale +Gated Fusion +guidance +Object +Detection +Target +Student Faster R-CNN +Detector +Backbone +guidance +RoI Head +Source +Target +F1 +Source Labels / +Target Pseudo Labels +Assessment-based AEMA +Memory +… +Labels Images +… +C0 +… +… +C1 +… +… +CK +… +store +Domain Shift +Simulation +Model +Assessment +AEMA +𝛿 +Loss 𝐿!"# +𝒮 +Loss 𝐿!"# +𝒯 +RPN +Parameters 𝜃𝒮 +Parameters 𝜃𝒯 +Update 𝜃𝒯 +Cat.-level +Discriminator +Ins.-level +Discriminator +Pix.-level +Discriminator +GRL +GRL +GRL +Source Path +Gradient Reversal Layer +GRL +Target Path +AEMA Adaptative Exponential +Moving Average +RPN +F2 +IoU +Source Labels / +Target Pseudo Labels +IoU +Intersection over Union +(a) Applying MGA on top of anchor-free FCOS for UDA detection +(b) Applying MGA on top of anchor-based Faster R-CNN for UDA detection +Model +Assessment +Omni-scale +Gated Fusion +Omni-scale +Gated Fusion +GRL +Source or +Target +Source or +Target +Source or +Target +Source or +Target +Source or +Target +Fig. 2. Framework of our MGA on the top of popular anchor-free FCOS [8] (see left image (a)) and anchor-based Faster R-CNN [61] (see right +image (b)) for UDA detection with assessment-based AEMA. Note that for Faster R-CNN, the region proposal network (RPN) and the RoI head are +used for coarse detection and final detection, respectively. F in (a), F1 and F2 in (b) represent the features from the feature pyramid network in +FCOS and backbone in Faster R-CNN. +Instance-level Alignment. Instance-level alignment usually +leverages features of regions or instances to train a domain +discriminator. The approach of [20] explores the relation of +different instances using mean-teacher based on Faster R- +CNN [7] for UDA detection. The work of [21] introduces +a hierarchical framework to align both instance and do- +main features for detection. The method of [22] proposes to +adapt Faster R-CNN on target domain images by aligning +features on instance- and image-levels, exhibiting promis- +ing performance. The approach of [23] introduces attention +mechanisms for better alignment in UDA detection. Instead +of using all instances or regions for alignment, the work +of [24] proposes to mine the discriminative ones and focuses +on aligning them across two different domains for adaption +detection. +Category-level Alignment. Category-level alignment con- +siders the intrinsic relation of feature distributions in source +and target domains and exploits the categorical discrim- +inability for alignment. The approaches of [25], [26], [27], +[62] learn a category-specific discriminator for each category +and focus on classification between two domains using +pseudo labels. Despite effectiveness, it is difficult for these +approaches to learn discriminative category-wise represen- +tation among multiple discriminators. The method of [28] +retains one discriminator to distinguish different categories +within one domain, whereas it neglects the consistency of +feature subspaces in the same category across two domains. +Besides, the work of [29] develops a categorical regulariza- +tion method that focuses on important regions and instances +to reduce the domain discrepancy. The method of [30] seeks +for category-level domain alignment by enhancing intra- +class compactness and inter-class separability. +Our Alignment. Despite sharing similar spirit in applying +alignment for UDA object detection, our approach (i.e., +Multi-Granularity Alignment (MGA)) is significantly differ- +ent from others. Specifically, MGA is a unified framework +that effectively encodes the dependencies across differ- +ent granularities, including pixel-, instance-, and category- +levels, for domain adaption detection, while other methods +do not consider this important dependency relation in align- +ment. In addition, we specially design the omni-scale gated +fusion (OSGF) module and present a new category-level +discriminator in MGA to improve the discriminative ability, +as detailed later. +2.4 +Exponential Moving Average +Exponential moving average (EMA) is a simple but effective +strategy for updating the model parameters and commonly +utilized in distillation technology [63], [64], [65] and mean- +teacher [66], [67]. In these approaches, the EMA process is +usually controlled by a constant weight coefficient to update +the teacher network. Despite effectiveness, this mechanism +may hurt the performance of UDA detection due to the +low-quality pseudo labels generated by the teacher detector. +Therefore, unlike previous studies, we design an adaptive +EMA (AEMA) by exploring the assessments of the teacher +and student detector networks during training. AEMA is +able to adjust the weight coefficient of EMA in an adap- +tive manner, leading to higher-quality pseudo labels for +improvements. +3 +MULTI-GRANULARITY ALIGNMENT (MGA) +3.1 +Overview +In this paper, we propose a novel unified multi-granularity +alignment (MGA) framework for domain adaption detec- +tion. The overall architecture is illustrated in Figure 2. + +SUBMITTED TO IEEE JOURNAL +5 +As displayed in Figure 2, given images from the source +domain s and the target domain t, we first extract the base +pixel-level feature representation from the backbone. Then, +these features are merged in the omni-scale gated fusion +(OSGF) module to produce discriminative representations +of multi-scale instances. Based on the fused feature repre- +sentations, more accurate candidate objects can be predicted +by the object detection head. Meanwhile, we introduce the +multi-granularity discriminators to distinguish the feature +distributions between two domains from different perspec- +tives, including pixel-level, instance-level and category- +level. Moreover, in order to improve the quality of pseudo +labels and mitigate the misalignment issue caused by noisy +pseudo labels during training, we propose a simple but +effective assessment-based adaptive EMA (AEMA) strategy +to refine the pseudo labels, further enhancing the robustness +of our MGA for domain adaption detection. +It is worth noting that, our MGA is a general framework +and can be easily applied in various detectors (e.g., anchor- +free FCOS [8] and anchor-based Faster R-CNN [7]) with +different backbones (e.g., VGG-16 [4] and ResNet-101 [2]). +Without loss of generality, we first apply the proposed MGA +in FCOS [8] for UDA object detection as in Figure 2 (a), and +then explain how it can be used in Faster R-CNN [7] as in +Figure 2 (b). For FCOS [8], we extract feature maps from +the last three stages of the backbone and combine them into +multi-level feature maps F k, where k ∈ {3, 4, 5, 6, 7}, using +FPN representation [68]. +3.2 +Omni-Scale Gated Object Detection +In most previous studies on domain adaption detection, the +main goal is to designate discriminators at a specific level +and some attentive regions. Nevertheless, the use of point +representation at pixel-level in anchor-free models [18], [69] +may cause difficulties in learning robust and discriminative +feature in cluttered background, while the pooling operation +(e.g., RoIAlign [39]) in anchor-based models [15], [70] may +distort the features of the instances with different scales and +aspect ratios. +In order to handle this problem, we introduce an omni- +scale gated fusion (OSGF) module for object detection, +which enables the adaption of the feature learning to object +with various scales and aspect ratios. Specifically in OSGF, +with the scale guidance from coarse detections, we can +choose the most plausible convolutions with different ker- +nels to extract compact features of instances in terms of ob- +ject scales, which can significantly boost the discriminative +capacity of the features. Our OSGF module is designed for +general purpose and thus can be easily applied in different +detectors. +3.2.1 +Scale Guidance by Coarse Detection +In order to select the most plausible convolutions for feature +extraction, it is necessary to obtain the scale information +of the objects. To this end, we introduce a coarse detection +step to provide the scale guidance. In specific, followed by +the multi-level feature maps F k (k ∈ {3, 4, 5, 6, 7} denotes +the level index) from the backbone (see Figure 2 (a)), we +can predict the candidate object boxes ˜bk through a series +of convolutional layers. Drawing inspiration from [71], we +utilize the cross-entropy Intersection over Union (IoU) loss +[72] to regress the bounding boxes of objects in foreground +pixels as follows, +Lgui = − +� +k +� +(i,j) +ln(IoU(˜bk +i,j, bk +i,j)), +(1) +where IoU(·, ·) represents the function to calculate the IoU +score between predicted box ˜bk and ground-truth box bk. +For each pixel (i, j) in the feature map, the corresponding +box bk +i,j can be defined as a 4-dimensional vector as follows, +bk +i,j = (xti,j, xbi,j, xli,j, xri,j) +(2) +where xti,j, xbi,j, xli,j, and xri,j respectively represent the +distances between current location and the top, bottom, +left and right bounds of ground-truth box. Therefore, the +normalized object scale (i.e., width wk and height hk) at each +level can be computed as follows, +� +wk +i,j = (˜xri,j + ˜xli,j)/stridek, +hk +i,j = (˜xbi,j + ˜xti,j)/stridek, +(3) +where stridek denotes how many steps we move in each +round of convolution operation1. As in FCOS [8], the feature +maps at each level are utilized to individually detect the +objects of different scales in the range {[−1, 64], [64, 128], +[128, 256], [256, 512]}, [512, +∞]}. Therefore, the majority of +object scales is less than 8, i.e., wk ≤ 8, hk ≤ 8. For notation +simplicity, we omit the superscript k and write F for F k and +˜b for ˜bk in the following sections. +3.2.2 +Omni-scale Gated Fusion (OSGF) +With the scale guidance as in Section 3.2.1, we present an +omni-scale gated fusion module (OSGF), which is composed +of both low-resolution and high-resolution feature streams, to +adapt to objects of various scales and aspect ratios. Specif- +ically, as illustrated in Figure 3, the low-resolution stream +consists of three parallel convolutional layers with different +kernels ω ∈ {3×3, 3×5, 5×3}, which is applied for feature +extraction of relatively small objects (wk ≤ 5, hk ≤ 5). +Meanwhile, in the high-resolution feature stream, we use +another set of parallel convolutional layers with kernels ω +to handle large objects (wk > 5, hk > 5). The different +from the low-resolution feature branch is that, we utilize an +extra upsampling operation after each convolutional layer +in the high-resolution stream to upscale the feature maps. It +is worth noticing that, the structure of OSGF in this paper +is different from that in the conference publication [1]. In +specific, the major modifications in this paper include: (i) +removal of the 3 × 3 convolutional layer before the two +streams, (ii) incorporation of simpler averaging pooling and +1 × 1 convolutional layers in the low-resolution stream, (iii) +replacement of the 3 × 3 convolutional layers with shared +averaging pooling and 1 × 1 convolutional layers, and (iv) +change of the 1×1 convolutional layer to 3×3 convolutional +layer in the residual connection. By doing so, the overall +number of parameters are significantly reduced because of +less convolutional layers used. In addition, we observe that +the detection performance has been improved by designing +1. We have {(k, stride)|(3, 8), (4, 16), (5, 32), (6, 64), (7, 128)}. + +SUBMITTED TO IEEE JOURNAL +6 +Conv +3×5_1 +Conv +5×3_1 +Conv +3×3_1 +Conv +3×3_1 +Conv +3×3_1 +Conv +3×3_1 +AVG +1×1_2 +Conv +1×1_1 +Conv +3×5_1 +Conv +5×3_1 +Conv +3×3_1 +Conv +3×3_1 +Conv +3×3_1 +Conv +3×3_1 +AVG +1×1_2 +Conv +1×1_1 +upsample +upsample +upsample +G3×5_1 +G5×3_1 +G3×3_1 +G3×5_2 +G5×3_2 +G3×3_2 +add +Conv +3×3_1 +merged +feature +maps +backbone +feature +maps +coarse +detections +guidance +ℎ!, 𝑤! +low-resolution stream +high-resolution stream +Conv +3×5_1 +AVG +1×1_2 +Conv +1×1_1 +Conv +3×5_1 +Conv +5×3_1 +Conv +3×3_1 +AVG +1×1_2 +Conv +1×1_1 +upsample +upsample +upsample +backbone +feature +maps +low-resolution stream +high-resolution stream +C +Conv +5×3_1 +Conv +3×3_1 +concat. +RoI +Align +split +Conv +3×3_1 +Conv +3×3_1 +Conv +3×3_1 +G3×5_1 +G5×3_1 +G3×3_1 +Conv +3×3_1 +Conv +3×3_1 +Conv +3×3_1 +G5×3_2 +G3×3_2 +Conv +3×3_1 +merged +feature +maps +𝑘 candidate +boxes +guidance +ℎ, 𝑤 +(a) Omni-scale Gate Fusion (OSGF) for anchor-free FCOS +(b) Omni-scale Gate Fusion (OSGF) for anchor-based Faster R-CNN +add +G3×5_2 +RoI +Align +Fig. 3. Illustration of the proposed omni-scale gated fusion (OSGF) module for anchor-free FCOS [8] (see left image (a)) and anchor-based Faster +R-CNN [61] (see right image (b)). The parameters of the modules with the same color are shared. +the shared convolutional layer and increasing the kernel +size in the convolutional layer of residual connection, as +evidenced by our experiments. +After the two branches of low- and high-resolution fea- +tures, we introduce a gate mask G to weight each convolu- +tional layer based on the predicted coarse boxes ˜b as follows, +Gω = +exp(τ(oω − ˆo)/(ˆo + ϵ)) +� +ω exp(τ(oω − ˆo)/(ˆo + ϵ)), +(4) +where τ represents the temperature factor, oω = IoU(˜b, ω) +denotes the overlap between the predicted box and the +convolution kernel ω, and ˆo is the maximal overlap among +them. Finally, we merge the pixel-level features to exploit +the scale-wise representation of instances as follows, +M = +� +ω +Fω ⊙ Gω + F3×3, +(5) +where ⊙ denotes the element-wise product, and Fω denotes +the feature maps after the convolutional layer with kernel +ω. +3.2.3 +Object Detection +After obtaining the merged feature maps M from the OSGF +module, we can predict the categories and bounding boxes +of objects. In FCOS [8], the object detection heads contain +three branches for classification, centerness and regression, +respectively. The classification and centerness branches are +optimized by the focal loss [73] Lcls and cross-entropy loss +[8] Lctr, respectively. The regression branch is optimized by +the IoU loss [72] Lreg. Thus, the final loss function Ldet for +the object detection is defined as +Ldet = Lcls + Lctr + Lreg. +(6) +Please refer to [8] for more details regarding the loss func- +tions. It is worthy to notice that, in the UDA detection, we +implement two detectors, including a teacher detector and +a student detector (see Figure 2 (a)). These two detectors +share the same architecture but independent parameters. +We denote the loss functions for the teacher and the student +detectors as LT +det and LS +det, respectively. +3.3 +Multi-Granularity Discriminators +As discussed earlier, we propose the multi-granularity dis- +criminators to distinguish whether the sample belongs to +the source domain or the target domain from various per- +spectives, consisting of pixels, instances and categories. The +discrepancy between two domains is reduced using Gradi- +ent Reversal Layer (GRL) [14] that transfers reverse gradient +when optimizing the object detection network. The discrim- +inator contains four stacked convolution-groupnorm-relu +layers and an extra 3 × 3 convolutional layer. Below we will +elaborate on our multi-granularity discriminators. +3.3.1 +Pixel- and Instance-level Discriminators +The pixel- and instance-level discriminators are leveraged +to respectively perform pixel-level and instance-level align- +ments of feature maps between two domains. As demon- +strated in Figure 2 (a), given the input multi-level features +F and the merged feature M, the pixel-level and instance- +level discriminators Dpix and Dins are learned through the +loss functions Lpix and Lins. Similar to previous work [18], +we adopt the same loss function. Then, Lpix and Lins are +defined as follows +Lpix = − +� +(i,j) +ypix +i,j log Dpix(F s(i, j)) ++ (1 − ypix +i,j ) log(1 − Dpix(F t(i, j))), +(7) +Lins = − +� +(i,j) +yins +i,j log Dins(M s(i, j)) ++ (1 − yins +i,j) log(1 − Dins(M t(i, j))), +(8) +where F(i, j) is the feature at pixel (i, j) in F, and M(i, j) +the feature at instance (i, j) in M. We have the domain label +ypix +i,j = 1 if pixel at (i, j) in F is from source domain and 0 +otherwise. Likewise, yins +i,j = 1 if the instance at (i, j) in M +belongs to source domain and 0 otherwise. +3.3.2 +Category-level Discriminator +In order to keep the semantic consistency between dif- +ferent domain distributions, a category-level discrimina- +tor is applied. Previous methods design either category- +specific discriminators for each category (e.g., [25], [26], + +SUBMITTED TO IEEE JOURNAL +7 +(a) category-specific discriminator +𝐷! +%&' +𝐷%&' +𝐷" +%&' +𝐷&'% +"#$ +… +𝑠! +𝑡! +𝑠" +𝑡" +𝑠#$" +𝑡#$" +𝑠! +𝑠" +… +𝑠#$" +𝑡! +𝑡" +… +𝑡#$" +source +alignment +target +alignment +(b) domain-consistent discriminator +𝐷%&' +𝑠! +𝑠" +… +𝑠#$" +𝑡! +𝑡" +… +𝑡#$" +𝑐 = 0 +𝑐 = 1 +𝑐 = 𝐶 − 1 +𝑠! +𝑠" +… +𝑠#$" +𝑡! +𝑡" +… +𝑡#$" +0 +2 +1 +𝐶 − 1 +𝑐: +𝑐: +𝑐: +𝑐: +𝑐: +… +instance +discriminability ℒ/01 +category +consistency ℒ102 +(c) category- and domain-consistent discriminator (ours) +Fig. 4. Illustration of different category-level discriminators D, where sc +and tc denote the c-th category (c = 0, 1, · · · , C − 1) in source domain +and target domain respectively. (a) Category-specific discriminators for +each category [25], [26], [27]. (b) Domain-consistent discriminator to +distinguish different categories within one domain [28]. (c) Our category- +and domain-consistent discriminator to consider both instance discrim- +inability in different categories and category consistency between two +domains. +[27], see Figure 4 (a)) or a domain-consistent discriminator +to distinguish categories within one domain (e.g., [28], see +Figure 4 (b)). By contrast, our approach considers jointly +instance discriminability in different categories and category +consistency between two domains and introduces a novel +category- and domain-consistent discriminator (see Figure 4 +(c)). Specifically, in our discriminator, we predict the cate- +gory and domain labels of pixel (i, j) in each image based +on feature map ˆ +M ∈ RH×W ×2C, where ˆ +M ∈ RH×W ×2C is +the output by feeding M to the category-level discriminator, +H and W are the height and width respectively, and 2C +represents the total number of categories for source and +target domains. +Since there is no ground-truth to supervise the category- +level discriminator, we assign pseudo labels to important +samples with high confidence from object detection (see +Sec. 3.2). In practice, given a batch of input images, we +can output the category probability map P using the object +detection heads, and obtain the set S of pseudo labels by +utilizeing the probability threshold τprob and non-maximum +suppression (NMS) threshold τnms. Then, the instances in +different categories are classified by Eq. (9), while the same +category in two domains is aligned by Eq. (11), as follows: +• +In order to keep instance discriminability in different +categories, we separate the category distribution by +using the following loss function, +Ldis = − 1 +|S| +� +(i,j)∈S +C−1 +� +c=0 +ˆydis +i,j,c log(pdis +i,j,c). +(9) +By normalizing confidence over the domain channel, +pdis +i,j,c represents the probability of the c-th category +of the pixel (i, j), i.e., +pdis +i,j,c = +exp ( ˆ +Mi,j,2c + ˆ +Mi,j,2c+1) +�C−1 +c=0 exp ( ˆ +Mi,j,2c + ˆ +Mi,j,2c+1) +, +(10) +where ˆ +Mi,j,2c and ˆ +Mi,j,2c+1 represent the confidence +of the c-th category in source and target domains, +respectively (see again Figure 4(c)). ˆydis ∈ RH×W ×C +is the pseudo category label. We have ˆydis +i,j,c = 1 if the +instance at (i, j) in ˆ +M is an important one of the c-th +category and ˆydis +i,j,c = 0 otherwise. +• +Category consistency between two domains. After +classifying instances of different categories, we need +to further identify which domain the instance be- +longs to. With GRL [14], we write the loss function +as follows, +Lsim = − 1 +|S| +� +(i,j)∈S +2C−1 +� +m=0 +ˆysim +i,j,m log(psim +i,j,m), +(11) +where ysim ∈ RH×W ×2C is the pseudo domain label. +Similarly, ˆysim +i,j,m = 1 if the instance at (i, j) in ˆ +M is +an important one of the ⌊ m +2 ⌋-th category in specific +domain and ˆysim +i,j,m = 0 otherwise. The domain prob- +ability psim is obtained as follows, +psim +i,j,m = +� +� +� +exp( ˆ +Mi,j,m) +exp( ˆ +Mi,j,m−1)+exp( ˆ +Mi,j,m), +if m is odd +exp( ˆ +Mi,j,m) +exp( ˆ +Mi,j,m)+exp( ˆ +Mi,j,m+1). +if m is even +(12) +With the above analysis, we define the final loss function +Lcat for the category-level discriminator Dcat as follows, +Lcat = λdisLdis + λsimLsim, +(13) +where Ldis and Lsim are loss functions for instance discrim- +inability and category consistency as in Eq. (9) and Eq. (11), +and λdis and λsim are the balancing factors. +3.4 +Adaptive Exponential Moving Average (AEMA) +As mentioned in Section 3.3.2, the pseudo labels, which are +generated by the teacher detector (see Figure 2 (a)), are +required for supervising the learning of the category-level +discriminator. During the training procedure, the teacher de- +tector is usually updated using exponential moving average +(EMA) as follows, +θη +T = (1 − γ) · θη−1 +T ++ γ · θη−1 +S +(14) +where θη +T represents the weights of the teacher detector at +iteration η, θη−1 +S +denotes the weights of the student detector +at iteration η − 1, and α is a constant coefficient. + +SUBMITTED TO IEEE JOURNAL +8 +(a) EMA +(b) AEMA +(c) GT +(d) Comparison of pseudo label quality +Fig. 5. Comparison of the pseudo label quality between EMA and AEMA. +Image (a) displays the pseudo label generated by EMA, image (b) the +pseudo label by our AEMA, and image (c) the GT pseudo label. In image +(d), we demonstrate the mAP scores of the generated pseudo labels +of different strategies, and we can observe that AEMA produces better +pseudo labels. +Despite simplicity, the EMA approach may lead to some +low-quality pseudo labels (see Figure 5 (a)), because it does +not consider the feedback from the two detectors, degrading +the final detection performance. To address this problem for +improving quality of pseudo labels, we propose an adaptive +EMA (AEMA). Specifically, unlike EMA, AEMA considers +the intermediate assessments of both teacher and student +detectors during update by evaluating their performance +on the source domain. Using the assessments as a guidance, +an update factor δ (as described later) is learned to adjust +the coefficient α in Eq. (14). More concretely, as in Figure 2, +we maintain a memory bank, which is used for generating +the assessments. The memory is dynamically updated by +storing the images and labels of source domain into it. In +order to accurately assess the generalization ability of the +detector, we introduce a domain shift simulation (DSS, see +Figure. 2 (a)) module, and apply it on the memory bank +to generate discrepancy of data distribution on the source +domain. Specifically, given the sampling data of images xm +and labels ym of all categories from the memory bank, +we randomly adjust the mean xu and variance σ2 of xm +to generate the variant data distribution from the source +domain, as follows +xm = �σ xm − xu +√ +σ2 + ϵ + �xu, +(15) +Where the mean �xu and the standard deviation �σ for the +new variant data distribution are obtained by using the uni- +form distribution under xu and σ, respectively, as follows, +�xu = U(au, bu)xu, +(16) +�σ = U(aσ, bσ)σ, +(17) +Here, U(a, b) represents the uniform distribution between +a and b and is predefined. Afterwards, the detection losses +LT +det and LS +det, obtained by evaluating the teacher and stu- +dent detectors on the above input date, are employed as the +assessment results to derive the update factor δ, as follows, +δ = +� +eτ1·(0.5−ρ), ρ < 0.5 +eτ2·(0.5−ρ), ρ ≥ 0.5 +ρ = +LS +det +LS +det + LT +det +(18) +where τ1 and τ2 are two constant values. +Finally, the weights of the teacher detection model can +be updated by our AEMA with δ as follows, +θη +T = (1 − γ · δ) · θη−1 +T ++ γ · δ · θη−1 +S +(19) +By using AEMA, we can take into account the assess- +ments of two detectors to guide the update the teacher +detection, resulting in better pseudo labels, as shown in +Figure 5 (b). Furthermore, we show the statistic comparison +of the pseudo labels obtained by teacher detector with EMA +and our proposed AEMA in term of accuracy in Figure 5 (d). +As demonstrated in Figure 5 (d), we can see that, the quality +of the pseudo labels is clearly improved. We will further +analyze the effectiveness of our AEMA in later experimental +section. +3.5 +Overall Loss Function and Optimization +As discussed above, the omni-scale gated object detection +network is supervised by Lgui and Ldet. Meanwhile, the +multi-granularity discriminators are optimized in different +granularities, including pixel-level Lpix, instance-level Lins +and category-level Lcat. In summary, the overall loss func- +tion is defined as +L = (Lgui + Ldet +� +�� +� +object detection +) + α (Lpix + Lins + Lcat) +� +�� +� +multi-granularity discriminators +(20) +where α is the balancing factor between object detection and +multi-granularity discriminators. +The training process of our proposed method is divided +into two stages. In stage 1 (S1), we train teacher detector by +using SGD optimizer and random sampling with Eq. (6) on +source domain. Next, in stage 2 (S2), the student detector is +optimized by using SDG optimizer and Eq. (20) on source +with labels and target domains with pseudo labels, and the +teacher detector is updated by adaptive exponential moving +average (AEMA). +4 +EXPERIMENTS +Extension of our framework. Our MGA framework is +designed for general purpose and applicable to both one- +and two-stage detection models. To verify this, in addition +to the representative one-stage FCOS [75], we further extend +our MGA to the popular two-stage Faster-RCNN [7] that +consists of Region Proposal Network (RPN) and RCNN +with classification and regression branches. As shown in +Figure 2 (b), we employ RPN as our coarse detection mod- +ule, whose loss function is replaced by the original RPN +loss, i.e., Lgui = Lrpn; and we use RCNN as object detection +module with the loss defined as Ldet = Lcls+Lreg. For omni- +scale gate fusion, we first obtain the top K proposals by +using RPN based on the backbone feature layer with stride +16. Then, we further extract the pixel-level features with +low-resolution and high-resolution streams, and generate +instance features of 7 × 7 under the pixel-level feature map +and original input feature maps by using the ROIAlign op- +eration. Finally, the instance features are merged according +to the RPN outputs and Eq. (5) after using a convolution of +3 × 3, as shown in Figure 3 (b). + +Quality of Pseudo Label +0.5 +mAPAEMA +mAPEMA +0.4 +0.3 +mAP +0.2 +0.1 +0.0 +0.5 +0.55 +0.6 +0.65 +0.7 +0.75 +0.8 +0.85 +0.9 +0.95 +loUcar:100% +car:carcar:100% +car:1car:100person:679% +car:86% +山 +car:55% +car:73%r:76.ccar:61%.m +car:51%6perperson:68% +car:89% +Car:60% +car:62% +car:49%SUBMITTED TO IEEE JOURNAL +9 +TABLE 1 +Results of our approach and comparison to state-of-the-arts on weather adaptation from Cityscapes to FoggyCityscapes. The best two results are +highlighted in red and blue fonts, respectively. Note that, MGA-DA [1] is the method from our conference version. +Method +Detector +Backbone +person +rider +car +truck +bus +train +mbike +bicycle +mAP +Baseline +Faster-RCNN +VGG-16 +17.8 +23.6 +27.1 +11.9 +23.8 +9.1 +14.4 +22.8 +18.8 +DAF [22] +Faster-RCNN +VGG-16 +25.0 +31.0 +40.5 +22.1 +35.3 +20.2 +20.0 +27.1 +27.6 +SC-DA [74] +Faster-RCNN +VGG-16 +33.5 +38.0 +48.5 +26.5 +39.0 +23.3 +28.0 +33.6 +33.8 +MAF [75] +Faster-RCNN +VGG-16 +28.2 +39.5 +43.9 +23.8 +39.9 +33.3 +29.2 +33.9 +34.0 +SW-DA [15] +Faster-RCNN +VGG-16 +29.9 +42.3 +43.5 +24.5 +36.2 +32.6 +30.0 +35.3 +34.3 +DAM [17] +Faster-RCNN +VGG-16 +30.8 +40.5 +44.3 +27.2 +38.4 +34.5 +28.4 +32.2 +34.6 +MOTR [76] +Faster-RCNN +ResNet-50 +30.6 +41.4 +44.0 +21.9 +38.6 +40.6 +28.3 +35.6 +35.1 +CST [77] +Faster-RCNN +VGG-16 +32.7 +44.4 +50.1 +21.7 +45.6 +25.4 +30.1 +36.8 +35.9 +PD [78] +Faster-RCNN +VGG-16 +33.1 +43.4 +49.6 +22.0 +45.8 +32.0 +29.6 +37.1 +36.6 +CDN [79] +Faster-RCNN +VGG-16 +35.8 +45.7 +50.9 +30.1 +42.5 +29.8 +30.8 +36.5 +36.6 +SFOD-Masoic-Defoggy [80] +Faster-RCNN +VGG-16 +34.1 +44.4 +51.9 +30.4 +41.8 +25.7 +30.3 +37.2 +37.0 +ATF [70] +Faster-RCNN +VGG-16 +34.6 +46.5 +49.2 +23.5 +43.1 +29.2 +33.2 +39.0 +37.3 +SW-Faster-ICR-CCR [29] +Faster-RCNN +VGG-16 +32.9 +43.8 +49.2 +27.2 +45.1 +36.4 +30.3 +34.6 +37.4 +SCL [81] +Faster-RCNN +VGG-16 +31.6 +44.0 +44.8 +30.4 +41.8 +40.7 +33.6 +36.2 +37.9 +CFFA [82] +Faster-RCNN +VGG-16 +43.2 +37.4 +52.1 +34.7 +34.0 +46.9 +29.9 +30.8 +38.6 +GPA [30] +Faster-RCNN +ResNet-50 +32.9 +46.7 +54.1 +24.7 +45.7 +41.1 +32.4 +38.7 +39.5 +SAPNet [23] +Faster-RCNN +VGG-16 +40.8 +46.7 +59.8 +24.3 +46.8 +37.5 +30.4 +40.7 +40.9 +DSS [83] +Faster-RCNN +ResNet-50 +42.9 +51.2 +53.6 +33.6 +49.2 +18.9 +36.2 +41.8 +40.9 +D-adapt [62] +Faster-RCNN +VGG-16 +44.9 +54.2 +61.7 +25.6 +36.3 +24.7 +37.3 +46.1 +41.3 +UMT [84] +Faster-RCNN +VGG-16 +56.5 +37.3 +48.6 +30.4 +33.0 +46.7 +46.8 +34.1 +41.7 +MeGA-CDA [46] +Faster-RCNN +VGG-16 +37.7 +49.0 +52.4 +25.4 +49.2 +46.9 +34.5 +39.0 +41.8 +CDG [53] +Faster-RCNN +VGG-16 +38.0 +47.4 +53.1 +34.2 +47.5 +41.1 +38.3 +38.9 +42.3 +TIA [47] +Faster-RCNN +VGG-16 +52.1 +38.1 +49.7 +37.7 +34.8 +46.3 +48.6 +31.1 +42.3 +SDA [85] +Faster-RCNN +VGG-16 +38.3 +47.2 +58.8 +34.9 +57.7 +48.3 +35.7 +42.0 +45.2 +TDD [59] +Faster-RCNN +VGG-16 +39.6 +47.5 +55.7 +33.8 +47.6 +42.1 +37.0 +41.4 +43.1 +SIGMA [50] +Faster-RCNN +VGG-16 +46.9 +48.4 +63.7 +27.1 +50.7 +35.9 +34.7 +41.4 +43.5 +MGA-DA [1] +Faster-RCNN +VGG-16 +43.9 +49.6 +60.6 +29.6 +50.7 +39.0 +38.3 +42.8 +44.3 +Baseline (ours) +Faster-RCNN +VGG-16 +39.4 +46.8 +48.2 +23.2 +36.0 +16.8 +35.2 +43.2 +36.1 +MGA (ours) +Faster-RCNN +VGG-16 +47.0 +54.6 +64.8 +28.5 +52.1 +41.5 +40.9 +49.5 +47.4 +oracle +Faster-RCNN +VGG-16 +48.2 +53.3 +68.5 +31.7 +55.3 +33.1 +41.9 +49.3 +47.7 +SST-AL [69] +FCOS +- +45.1 +47.4 +59.4 +24.5 +50.0 +25.7 +26.0 +38.7 +39.6 +CFA [18] +FCOS +VGG-16 +41.9 +38.7 +56.7 +22.6 +41.5 +26.8 +24.6 +35.5 +36.0 +SCAN [86] +FCOS +VGG-16 +41.7 +43.9 +57.3 +28.7 +48.6 +48.7 +31.0 +37.3 +42.1 +MGA-DA [1] +FCOS +VGG-16 +45.7 +47.5 +60.6 +31.0 +52.9 +44.5 +29.0 +38.0 +43.6 +CFA [18] +FCOS +ResNet-101 +41.5 +43.6 +57.1 +29.4 +44.9 +39.7 +29.0 +36.1 +40.2 +MGA-DA [1] +FCOS +ResNet-101 +43.1 +47.3 +61.5 +30.2 +53.2 +50.3 +27.9 +36.9 +43.8 +Baseline (ours) +FCOS +VGG-16 +31.4 +31.0 +42.7 +14.2 +26.9 +2.3 +17.6 +31.6 +24.7 +Baseline (ours) +FCOS +ResNet-101 +35.2 +37.2 +43.5 +17.3 +31.8 +7.2 +27.0 +34.5 +29.2 +MGA (ours) +FCOS +VGG-16 +47.9 +50.1 +64.9 +34.8 +58.0 +45.6 +38.3 +43.7 +47.9 +MGA (ours) +FCOS +ResNet-101 +47.2 +48.1 +63.7 +37.5 +54.6 +50.8 +28.8 +44.2 +46.9 +oracle +FCOS +VGG-16 +51.7 +48.9 +69.5 +39.1 +52.8 +56.0 +31.8 +40.0 +48.7 +oracle +FCOS +ResNet-101 +46.3 +46.0 +66.8 +38.8 +57.3 +52.2 +36.4 +36.9 +47.6 +Implementation. In this work, we implement our method +based on different detectors (i.e., Faster-RCNN and FCOS) +and backbones (i.e., VGG-16 and ResNet-101) using PyTorch +[87] to show generality of our approach. Both VGG-16 and +ResNet-101 are pre-trained on ImageNet [3]. We utilize a +unified optimization framework by using training process +of two stages and warm-up followed the previous works +[18] and [50] for different detectors. Similar to [23], we apply +the Adam optimizer with an initial learning rate of 3e-4, a +momentum of 0.9 and weight decay of 1e-4 in Faster-RCNN +framework. For FCOS framework, we use SGD optimizer +with an initial learning rate of 5e-3, a momentum of 0.9 and +weight decay of 1e-4, being consistent with CFA [18]. γ is 0.1 +in Eq. (14). The parameters au and bu are set respectively to +0.4 and 0.5 in Eq. (16), and the aδ and bδ respectively to 0.8 +and 0.9 in Eq. (17). The thresholds τprob and τnms for obtain- +ing S are empirically set to 0.42 and 0.5. All our experiments +are conducted on the machine with an Intel(R) Xeon(R) CPU +and 4 Tesla V100 GPUs. Our code will be made publicly +available at https://github.com/tiankongzhang/MGA. +4.1 +Datasets +To verify the proposed method, we conduct extensive exper- +iments on different adaption settings, as described below. +Weather adaptation. For weather adaptation, we explore +generalization of the detector on Cityscapes [31] and Fog- +gyCityscapes [32]. Cityscapes [31] is a popular street scene +dataset with normal weather, which comprises 2,975 train- +ing images and 500 validation images. FoggyCityscapes [32] +is synthesized on Cityscapes with different levels of fog (i.e., +0.005, 0.01 and 0.02). For fair comparison, we choose the +level of 0.02 for experiment as in other methods in Table 1. +In weather adaptation, we use Cityscapes [31] as the source +domain and FoggyCityscapes [32] as the target domain. +Cross-Camera adaptation. In Cross-Camera adaptation, we +evaluate our algorithm on KITTI [34] and Cityscapes. KITTI +[34] is a popular traffic scene dataset containing 7,481 train- +ing images. In this adaption experiment, KITTI is the source +domain and Cityscapes is the target domain. Following +previous works [50], [81], we only report the results on the +category of car. + +SUBMITTED TO IEEE JOURNAL +10 +TABLE 2 +Results of our approach and comparison to state-of-the-arts on real-to-artistic adaptation from PASCAL VOC to Clipart. The best two results are +highlighted in red and blue fonts, respectively. Note, there are no oracle results for Clipart because all images in Clipart are used for evaluation. +Method +Detector +Backbone +acro +bicycle +bird +boat +bottle +bus +car +cat +chair +cow +Baseline +Faster-RCNN +ResNet-101 +35.6 +52.5 +24.3 +23.0 +20.0 +43.9 +32.8 +10.7 +30.6 +11.7 +SW-DA [15] +Faster-RCNN +ResNet-101 +26.2 +48.5 +32.6 +33.7 +38.5 +54.3 +37.1 +18.6 +34.8 +58.3 +SCL [81] +Faster-RCNN +ResNet-101 +44.7 +50.0 +33.6 +27.4 +42.2 +55.6 +38.3 +19.2 +37.9 +69.0 +ATF [70] +Faster-RCNN +ResNet-101 +41.9 +67.0 +27.4 +36.4 +41.0 +48.5 +42.0 +13.1 +39.2 +75.1 +PD [78] +Faster-RCNN +ResNet-101 +41.5 +52.7 +34.5 +28.1 +43.7 +58.5 +41.8 +15.3 +40.1 +54.4 +SAPNet [23] +Faster-RCNN +ResNet-101 +27.4 +70.8 +32.0 +27.9 +42.4 +63.5 +47.5 +14.3 +48.2 +46.1 +UMT [58] +Faster-RCNN +ResNet-101 +39.1 +59.1 +32.4 +35.0 +45.1 +61.9 +48.4 +7.5 +46.0 +67.6 +SFOD-ODS [51] +Faster-RCNN +ResNet-101 +43.1 +61.4 +40.1 +36.8 +48.2 +45.8 +48.3 +20.4 +44.8 +53.3 +D-adapt [62] +Faster-RCNN +ResNet-101 +56.4 +63.2 +42.3 +40.9 +45.3 +77.0 +48.7 +25.4 +44.3 +58.4 +MGA-DA [1] +Faster-RCNN +ResNet-101 +35.5 +64.6 +27.8 +34.5 +41.6 +66.4 +49.8 +26.8 +43.6 +56.7 +Baseline (ours) +Faster-RCNN +ResNet-101 +30.5 +35.3 +24.8 +23.5 +34.8 +65.7 +32.6 +9.0 +35.1 +26.4 +MGA (ours) +Faster-RCNN +ResNet-101 +38.7 +77.2 +39.0 +35.4 +53.8 +78.1 +47.5 +17.5 +38.2 +49.9 +table +dog +horse +bike +person +plant +sheep +sofa +train +tv +mAP +Baseline +Faster-RCNN +ResNet-101 +13.8 +6.0 +36.8 +45.9 +48.7 +41.9 +16.5 +7.3 +22.9 +32.0 +27.8 +SW-DA [15] +Faster-RCNN +ResNet-101 +17.0 +12.5 +33.8 +65.5 +61.6 +52.0 +9.3 +24.9 +54.1 +49.1 +38.1 +SCL [81] +Faster-RCNN +ResNet-101 +30.1 +26.3 +34.4 +67.3 +61.0 +47.9 +21.4 +26.3 +50.1 +47.3 +41.5 +ATF [70] +Faster-RCNN +ResNet-101 +33.4 +7.9 +41.2 +56.2 +61.4 +50.6 +42.0 +25.0 +53.1 +39.1 +42.1 +PD [78] +Faster-RCNN +ResNet-101 +26.7 +28.5 +37.7 +75.4 +63.7 +48.7 +16.5 +30.8 +54.5 +48.7 +42.1 +SAPNet [23] +Faster-RCNN +ResNet-101 +31.8 +17.9 +43.8 +68.0 +68.1 +49.0 +18.7 +20.4 +55.8 +51.3 +42.2 +UMT [58] +Faster-RCNN +ResNet-101 +21.4 +29.5 +48.2 +75.9 +70.5 +56.7 +25.9 +28.9 +39.4 +43.6 +44.1 +SFOD-ODS [51] +Faster-RCNN +ResNet-101 +32.5 +26.1 +40.6 +86.3 +68.5 +48.9 +25.4 +33.2 +44.0 +56.5 +45.2 +D-adapt [62] +Faster-RCNN +ResNet-101 +31.4 +24.5 +47.1 +75.3 +69.3 +43.5 +27.9 +34.1 +60.7 +64.0 +49.0 +MGA-DA [1] +Faster-RCNN +ResNet-101 +24.3 +20.9 +43.2 +84.3 +74.2 +41.1 +17.4 +27.6 +56.5 +57.6 +44.8 +Baseline (ours) +Faster-RCNN +ResNet-101 +24.2 +12.2 +31.2 +55.5 +40.4 +52.2 +5.7 +18.4 +45.0 +38.4 +32.0 +MGA (ours) +Faster-RCNN +ResNet-101 +20.0 +18.0 +44.2 +83.5 +74.6 +57.7 +26.7 +26.0 +55.4 +58.3 +47.0 +TABLE 3 +Results of our approach and comparison to state-of-the-arts on real-to-artistic adaptation from PASCAL VOC to Watercolor. The best two results +are highlighted in red and blue fonts, respectively. +Method +Detector +Backbone +bike +bird +car +cat +dog +person +mAP +Baseline +Faster-RCNN +ResNet-101 +68.8 +46.8 +37.2 +32.7 +21.3 +60.7 +44.6 +SW-DA [15] +Faster-RCNN +ResNet-101 +82.3 +55.9 +46.5 +32.7 +35.5 +66.7 +53.3 +SCL [81] +Faster-RCNN +ResNet-101 +82.2 +55.1 +51.8 +39.6 +38.4 +64.0 +55.2 +ATF [70] +Faster-RCNN +ResNet-101 +78.8 +59.9 +47.9 +41.0 +34.8 +66.9 +54.9 +PD [78] +Faster-RCNN +ResNet-101 +95.8 +54.3 +48.3 +42.4 +35.1 +65.8 +56.9 +SAPNet [23] +Faster-RCNN +ResNet-101 +81.1 +51.1 +53.6 +34.3 +39.8 +71.3 +55.2 +UMT [58] +Faster-RCNN +ResNet-101 +88.2 +55.3 +51.7 +39.8 +43.6 +69.9 +58.1 +SFOD-ODS [51] +Faster-RCNN +ResNet-101 +95.2 +53.1 +46.9 +37.2 +47.6 +69.3 +58.2 +AT [60] +Faster-RCNN +ResNet-101 +93.6 +56.1 +58.9 +37.3 +39.6 +73.8 +59.9 +MGA-DA [1] +Faster-RCNN +ResNet-101 +87.6 +49.9 +56.9 +37.4 +44.6 +72.5 +58.1 +Baseline (ours) +Faster-RCNN +ResNet-101 +76.0 +46.7 +52.0 +27.7 +33.3 +54.9 +48.4 +MGA (ours) +Faster-RCNN +ResNet-101 +85.3 +59.7 +59.7 +43.3 +46.6 +77.7 +62.1 +oracle +Faster-RCNN +ResNet-101 +67.1 +53.4 +43.9 +46.3 +50.5 +79.8 +56.8 +Synthetic-to-Real adaptation. For Synthetic-to-Real adapta- +tion, we utilize SIM10k [33] and Cityscapes for experiments. +SIM10k [33] is a synthetic scene dataset from the game video +Grand Theft Auto V (GTA5). It contains 10k training images, +and we conduct comparisons on the car class, similar to [81]. +In this adaptation experiment, we utilize the SIM10k as the +source domain and Cityscapes as the target domain. +Real-to-Artistic adaptation. In Real-to-Artistic adaptation, +we verify our method on PASCAL VOC [35], Clipart [36] +and Watercolor [36] datasets. PASCAL VOC [35] is a real- +scene dataset including two sub-datasets (i.e., PASCAL VOC +2007 and PASCAL VOC 2012). PASCAL VOC 2007 consists +of 2,501 images for training and 2,510 images for validation, +and PASCAL VOC 2012 contains 5,717 images for training +and 5,823 mages for validation. Clipart [36] is a carton +dataset with 1k images and has the same categories as +PASCAL VOC. Watercolor [36] is a watercolor style dataset +containing 1,000 training images and 1,000 testing images, +and it shares 6 classes with PASCAL VOC. In this setting, +we use PASCAL VOC as the source domain and Clipart or +Watercolor as the target domain. +4.2 +State-of-the-Art Comparison +In this section, we demonstrate our results and comparison +with state-of-the-art methods using different base detectors +(i.e., Faster-RCNN [7] and FCOS [8]) and backbones (i.e., +VGG-16 [4] and ReseNet-101 [2]) on different adaptation +scenarios. In all comparison table, “Baseline (ours)” means +that the baseline detector is equipped with our OSGF and +trained using data augmentation as in our method but +without adaption strategy, and “oracle” indicates that the +baseline detector is trained and tested on the target domain +without any adaptation strategy. +Weather adaptation. In Table 1, we report the results from +Cityscapes to FoggyCityscapes. As displayed in Table 1, for +Faster-RCNN detector, our method achieves the best mAP + +SUBMITTED TO IEEE JOURNAL +11 +Baseline +MGA-DA +MGA +GT +Fig. 6. Qualitative results and comparison (from left column to right column: weather adaptation from Cityscapes to FoggyCityscapes, real-to- +artistic adaptation from PASCAL VOC to Clipart and Watercolor, cross-camera adaption from Kitti to Cityscapes, and synthetic-to-real adaption +from SIM10k to Cityscapes). We can see that MGA achieves superior results than MGA-DA in our conference version and the baseline method. +TABLE 4 +Results of our approach and comparison to state-of-the-arts on +cross-camera/synthetic-to-real adaptation detection results from +Kitti/SIM10k to Cityscapes. The best two results are highlighted in red +and blue fonts, respectively. +Method +Detector +Backbone +APcar +Baseline Faster-RCNN +VGG-16 +30.2/30.1 +DAF [22] Faster-RCNN +VGG-16 +38.5/39.0 +MAF [75] Faster-RCNN +VGG-16 +41.0/41.1 +ATF [70] Faster-RCNN +VGG-16 +42.1/42.8 +SC-DA [74] Faster-RCNN +VGG-16 +42.5/43.0 +UMT [84] Faster-RCNN +VGG-16 +-/43.1 +SFOD-Mosaic [80] Faster-RCNN +VGG-16 +44.6/43.1 +CST [77] Faster-RCNN +VGG-16 +43.6/44.5 +MeGA-CDA [46] Faster-RCNN +VGG-16 +43.0/44.8 +SAPNet [23] Faster-RCNN +VGG-16 +43.4/44.9 +CDN [79] Faster-RCNN +VGG-16 +44.9/49.3 +TIA [47] Faster-RCNN +VGG-16 +44.0/ – +DSS [83] Faster-RCNN +ResNet-50 +42.7/44.5 +SSD [88] Faster-RCNN +ResNet-50 +47.6/49.3 +SIGMA [50] Faster-RCNN +VGG-16 +45.8/53.7 +TDD [59] Faster-RCNN +VGG-16 +47.4/53.4 +MGA-DA [1] Faster-RCNN +VGG-16 +45.2/49.8 +Baseline (ours) Faster-RCNN +VGG-16 +43.7/44.1 +MGA (ours) Faster-RCNN +VGG-16 +54.3/55.5 +oracle Faster-RCNN +VGG-16 +68.1 +SST-AL [69] +FCOS +- +45.6/51.8 +CFA [86] +FCOS +VGG-16 +43.2/49.0 +SCAN [86] +FCOS +VGG-16 +45.8/52.6 +CFA [18] +FCOS +ResNet-101 45.0/51.2 +MGA-DA [1] +FCOS +VGG-16 +48.5/54.6 +MGA-DA [1] +FCOS +ResNet-101 46.5/54.1 +Baseline (ours) +FCOS +VGG-16 +43.1/43.0 +Baseline (ours) +FCOS +ResNet-101 41.3/43.7 +MGA (ours) +FCOS +VGG-16 +49.9/55.8 +MGA (ours) +FCOS +ResNet-101 47.6/55.4 +oracle +FCOS +VGG-16 +73.4 +oracle +FCOS +ResNet-101 +71.8 +of 47.4% and outperforms the second best SDA [85] with +45.2% mAP by 2.4%. Compared to the baseline (ours) of +36.1%, MGA obtains 11.3% performance gains. For FCOS +detector, we also obtain the best mAP scores of 47.9% +with VGG-16 and 46.9% with ResNet-101. Compared with +the approaches of SCAN [86] with VGG-16 and CFA [18] +with ResNet-101, our MGA respectively shows the 5.8% +and 6.7% gains. In addition, our method observes obvious +improvements over the baseline (ours) with 23.2% using +VGG-16 and 17.7% gains using ResNet-101, which verifies +the effectiveness of our method. Furthermore, compared +with MGA-DA [1] with 44.3% mAP score for Fatser-RCNN +and 43.6% and 43.8 % mAP scores for FCOS on VGG-16 and +ResNet-101 backbone in our conference version, our MGA +shows 3.1%, 4.3% and 3.1% gains, showing the effectiveness +of our new contributions. +Real-to-Artistic adaptation. Table 2 and 3 show the results +and comparison in real-to-artistic adaptation. As in Table +2, from PASCAL VOC to Clipart, our method achieves +the second mAP score of 47.0%, and D-adapt [62] obtains +the best performance of 49.0%. Compared to our baseline, +we achieve 15.0% performance gain. In Table 3, our MGA +performs the best with 62.1% mAP score and surpasses +the second best AT [60] with 59.9% by 2.2%. Besides, our +method outperforms the oracle, indicating that our MGA +makes full use of the information between source and target +domains for robust UDA detection. +Cross-Camera adaptation. Table 4 displays the results and +comparison from Kitti to Cityscapes. As shown in Table +4, on Faster-RCNN detector, our method shows the best +result of 54.3% APcar with VGG-16. In contrast to the +baseline (ours), we obtain 10.6% gain. Using FCOS detector, +our MGA outperforms SCAN [86] by 4.1% with VGG-16 +and CFA [18] by 2.6% with ResNet-101. Compared to the +baseline (ours), MGA obtains 6.8% and 6.3% performance + +person:87%car:74% +car:88% +car:7/1% +car:56% +car:46% +car:65%car:85% +car:64% +:65 +car:62%car:86% +car:58% +car:66% +car:82% +car:62%car:10o +car:10 +ar:100% +2ar:100% +ar1oo +car:100c0% +ar:100% +100%0person:100% +person:98% +pers +person:91% +person:99%person:96% +person:9 +person:95% +person:88% +person +person:9 +person:97% +person:97% +person:99person:99%:89% +person:99% +person: +person:92% +person:96%rson:9 +1080 +diningtabie:90% +A +persorbottle:97% +person:99% +9 +person:9 +bottle:99%person:100% +eison:n +person:100% +person: +diningtable:100% +pon:lc +persorbottle:100% +s0n:100% +person:1 +bottle:100%oer +person:100% +person:100% +person:97%nerson:92% +person:100% +person:98% +person:97% +person:99% +person:97%person:99% +person:99% +person:100% +person:86% +person:100% +person:100%person:89% +person:75% +bicycle:48% +car:80% +car-55eperson:100% +person:100% +person:100% +person:100% +person:100% +person:100%person:79% +person:60%pe +065% +car:76%car:63% +car:45%39% +ar:48% +car:71%car:48% +Car4B3348%person:100% +bicycle:100% +car:1005ar100% +car:1occar.100%% +car:100% +ar100% +ar100% +car:100% +car:100%r:10 +person:100%scar:74% +65%car:79% +car:77% +car:64% +880%car:85% +car:81%car:100% +ar:oo% +car:100% +100 +car:100%100% +car:10 +ar100%car:79% +8% +car:47%SUBMITTED TO IEEE JOURNAL +12 +gains with VGG-16 and ResNet-101, respectively, showing +its advantages. +Synthetic-to-Real adaptation. Table 4 shows the result from +Sim10k to Cityscapes. On Faster-RCNN detector, our MGA +achieves the best result of 55.5% APcar. In contrast to the +baseline (ours), it obtains a 11.4% gain. On FCOS detector, +our method shows the best APcar of 55.8% with VGG-16 +and 55.4% with ResNet-101. In comparison with SCAN +[86] with VGG-16 and CFA [18] with ResNet-101, we ob- +tains 3.2% and 4.2% gains with VGG-16 and ResNet-101, +respectively. Compared to baselines (ours), it demonstrates +gains of 12.8% and 11.7% with VGG-16 and ResNet-101, +respectively. +Besides quantitative results, we demonstrate qualitative +results our method and comparison to other approaches in +Figure 6. From Figure 6, we can observe that MGA achieves +superior detection results than MGA-DA in our conference +version and the baseline method. +4.3 +Ablation Study +To further analyze our approach, we conduct ablation ex- +periments on different components. The results are reported +under the weather adaptation. +Effectiveness of different components. In order to further +validate the effectiveness of different components including +omni-scale gated fusion (OSGF), multi-granularity discrim- +inators (MGD) and adaptive exponential moving average +(AEMA) in MGA, we demonstrate the results by gradually +adding them to the baseline, which is FCOS with VGG16. +Table 5 shows the results. As shown in Table 5, with our +OSGF, the performance of the baseline is improved from +22.0% to 24.7% with a gain of 2.7%, showing the effec- +tiveness of OSGF in discriminative representation learning. +When employing the proposed MGD for adaption, we +achieve significant improvement by boosting the result from +24.7% to 45.3% with a 20.6%, which clearly evidences the +effectiveness of our method. Further, when adopting AEMA +for better pseudo labels, the final result is improved from +45.3% to 47.9%. +Comparison of category-level discriminators. As displayed +in the Table 6, we compare our category-level discriminator +Dcat and other related class-level discriminators, including +Dcen [86], Dgrp [26] and Dcls [28] using FCOS with VGG16. +The “baseline” indicates that the detector is learned under +all strategies but the class-level discriminator is removed +from MGA module. Dcen focuses on reducing the differences +on the center-aware distributions of source and target do- +mains, which is composed of features of the central positions +of objects. Dgrp aligns the feature distributions by building a +sole domain discriminator for each category. Dcls simultane- +ously takes domain and class information and expands the +binary domain labels by inserting the binary class labels. +From Table 6, we obverse that the performance of de- +tector is improved from 44.7% to 46.2%/46.0%/45.2% with +1.5%/1.3%/0.5% gains by Dcen/Dgrp/Dcls over the base- +line. In contrast, our Dcat improves the baseline detector +performance from 44.7% to 47.9%, which shows 3.2% per- +formance gains over the baseline and outperforms other +three category-level discriminators by 1.7%, 1.9% and 2.7%, +respectively, evidencing the effectiveness of our approach. +TABLE 5 +Effectiveness of different components. +OSGF +MGD +AEMA +mAP +22.0 +✓ +24.7 +✓ +✓ +45.3 +✓ +✓ +✓ +47.9 +TABLE 6 +Comparison between different category-level discriminators. +discriminator +baseline +Dcen +Dgrp +Dcls +Dcat (ours) +mAP +44.7 +46.2 +46.0 +45.2 +47.9 +TABLE 7 +Comparison between different fusion methods for OSGF. +S1 +S2 +w/o +OSGF +Conv +fusion +Average +fusion +Gated +fusion (ours) +✓ +22.0 +23.2 +22.8 +24.7 +✓ +✓ +45.6 +46.2 +45.1 +47.9 +TABLE 8 +Comparison of different OSGF in this work (i.e., OSGF (ours)) and +conference version (i.e., OSGF (conference)) [1]. The backbone of the +detectors in this table is VGG16. +method +detector +# Params (M) +S1 +S1&S2 +OSGF (conference) [1] +FCOS +17 +23.4 +46.8 +OSGF (ours) +FCOS +14 +24.7 +47.9 +OSGF (conference) [1] Faster-RCNN +55 +33.8 +46.1 +OSGF (ours) Faster-RCNN +50 +36.1 +47.4 +Analysis on gated fusion. In this paper, we introduce a +novel gated fusion approach for improved discriminative +feature representation. In order to further analyze the gated +fusion, we compare it with other commonly used fusion +strategies including Conv fusion and Average fusion in +Table 7 using FCOS with VGG16. In Table 7, “w/o OSGF” +means that the OSGF module is removed from our method. +“Conv fusion” indicates the fusion weights in OSGF are +obtained by using 1 × 1 convolution, and “Average fu- +sion” means averaging features of convolution kernels in +OSGF. From Table 7, we can observe that, in S1, all three +fusion methods can improve the performance. In specific, +the OSGF with our gated fusion strategy improves the +mAP score from 22.0% to 24.7% with 2.7% performance +gains, outperforming those using Conv and Average fusion +methods with 23.2% and 22.8% mAP scores. Likewise, in S2, +OSGF with our gated fusion achieves the best performance, +evidencing the effectiveness of gated mechanism for repre- +sentation learning. +In addition, from Table 8, we can observe that the im- +proved OSGF in this work contains less parameters and +meanwhile achieves better performance compared to that +in the preliminary conference version [1], which indicates +the advantages of our improved OSGF. +Comparison of EMA and AEMA. To show the effectiveness +of our proposed AEMA, we compare it with EMA using +FCOS with VGG16 as shown in Table 9. From Table 9, we +obverse that EMA brings in a 1.2% performance gain by +improving the result from 45.3% to 46.5%. However, when +using the proposed AEMA, the performance is improved to + +SUBMITTED TO IEEE JOURNAL +13 +TABLE 9 +Comparison between EMA and AEMA for detection performance. +EMA +AEMA +mAP +45.3 +✓ +46.5 +✓ +47.9 +TABLE 10 +Comparison of computational complexity. The backbone of the +detectors in this table is VGG16. +method +detector +# Total Params (M) FPS +CFA [18] +FCOS +177 +17.5 +MGA (ours) +FCOS +429 +9.1 +SCL [81] Faster-RCNN +580 +11.8 +SAPNet [23] Faster-RCNN +556 +25.2 +MGA (ours) Faster-RCNN +535 +11.0 +47.9% with 2.6% gains, which demonstrates the superiority +of our AEMA in learning better pseudo labels for detection. +Computational complexity. In Table 10, we show the com- +parison of computational complexity between our method +and other state-of-the-art approaches. As shown in Tab. +10, on FCOS, our MGA performs better with reasonable +increased complexity over its primary contender CFA [18]; +while our method has less parameters on the top of Faster- +RCNN than two recent methods SCL [81] and SAPNet [23]. +5 +CONCLUSION +In this paper, we propose a novel unified multi-granularity +alignment (MGA) framework, which encodes dependencies +among different pixel-, instance-, and category-levels to +achieve alignment of feature distributions for unsupervised +domain adaptive detection. Notably, we design the omni- +scale gated fusion module with different scales and aspect +rations to extract discriminative instance-level feature rep- +resentation. In order to improve the quality of pseudo labels +and mitigate the local misalignment problem in our MGA +framework, we further propose a simple but effective dy- +namic adaptive exponential moving average strategy. 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Schiele, “Seeking similarities over differences: Similarity-based +domain alignment for adaptive object detection,” in ICCV, 2021. + diff --git a/PdAyT4oBgHgl3EQfhPgA/content/tmp_files/load_file.txt b/PdAyT4oBgHgl3EQfhPgA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..143965240dbf23932e78659f2cc4837c144fd829 --- /dev/null +++ b/PdAyT4oBgHgl3EQfhPgA/content/tmp_files/load_file.txt @@ -0,0 +1,2240 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf,len=2239 +page_content='SUBMITTED TO IEEE JOURNAL 1 Robust Domain Adaptive Object Detection with Unified Multi-Granularity Alignment Libo Zhang, Wenzhang Zhou, Heng Fan, Tiejian Luo, and Haibin Ling Abstract—Domain adaptive detection aims to improve the generalization of detectors on target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' To reduce discrepancy in feature distributions between two domains, recent approaches achieve domain adaption through feature alignment in different granularities via adversarial learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' However, they neglect the relationship between multiple granularities and different features in alignment, degrading detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Addressing this, we introduce a unified multi-granularity alignment (MGA)-based detection framework for domain-invariant feature learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' The key is to encode the dependencies across different granularities including pixel-, instance-, and category-levels simultaneously to align two domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Specifically, based on pixel-level features, we first develop an omni-scale gated fusion (OSGF) module to aggregate discriminative representations of instances with scale-aware convolutions, leading to robust multi-scale detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Besides, we introduce multi-granularity discriminators to identify where, either source or target domains, different granularities of samples come from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Note that, MGA not only leverages instance discriminability in different categories but also exploits category consistency between two domains for detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Furthermore, we present an adaptive exponential moving average (AEMA) strategy that explores model assessments for model update to improve pseudo labels and alleviate local misalignment problem, boosting detection robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Extensive experiments on multiple domain adaption scenarios validate the superiority of MGA over other approaches on FCOS and Faster R-CNN detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Code will be released at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='com/tiankongzhang/MGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Index Terms—Domain Adaptive Object Detection, Multi-Granularity Alignment, Omni-scale Gated Fusion, Model Assessment, Adaptive Exponential Moving Average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' 1 INTRODUCTION O BJECT detection has been one of the most fundamental problems in computer vision with a long list of ap- plications such as visual surveillance, self-driving, robotics, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Owing to the powerful representation by deep learning (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=', [2], [3], [4]), object detection has witnessed consider- able advancement in recent years with numerous excellent frameworks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=', [5], [6], [7], [8], [9], [10], [11], [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' These modern detectors are usually trained and evaluated on a large-scale annotated dataset (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=', [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Despite the great achievement, they may suffer from poor generalization when applied to images from a new target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' To remedy this, a simple and straightforward solution is to build a benchmark for the new target domain and re-train the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Nevertheless, benchmark creation is both time- consuming and costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In addition, the new target domain could arbitrary and it is almost impossible to develop bench- marks for all new target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In order to deal with the above issues, researchers have explored the unsupervised domain adaption (UDA) detection, with the goal of transferring knowledge learned from an annotated source domain to an unlabeled target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' One popular trend is to leverage adversarial learning [14] Libo Zhang is with the State Key Laboratory of Computer Science, Institute of Software Chinese Academy of Sciences, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' E-mail: libo@iscas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Wenzhang Zhou and Tiejian Luo are with the University of Chinese Academy of Sciences, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' E-mail: zhouwenzhang19@mails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='ucas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='cn, tjluo@ucas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Heng Fan is with the Department of Computer Science and Engineering, University of North Texas, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' E-mail: heng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='fan@unt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Haibin Ling is with the Department of Computer Science, Stony Brook University, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' E-mail: hling@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='stonybrook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Libo Zhang and Wenzhang Zhou make equal contributions to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' A preliminary version [1] of this work has appeared in CVPR 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Student Detector (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=', Adapted Detector) Pixels Categories Instances Domains Source or Target?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Omni-scale Gated Fusion Teacher Detector (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=', Original Detector) Pixel-level Discriminator Instance-level Discriminator Category-level Discriminator 𝛿 Refined Pseudo Labels Pseudo Labels Update Model Assessment AEMA Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Illustration of the proposed Multi-Granularity Alignment (MGA) framework for domain adaptive object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Specifically, MGA en- codes the dependencies across multiple granularities simultaneously, including pixel-, instance-, and category-levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In addition, a dynamic update mechanism guided by update factor δ (as detailed later) through model assessment during training is used to improve the quality of pseudo labels and meanwhile mitigate the local misalignment problem, further enhancing the detection robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Best viewed in color and by zooming in for all figures throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' to narrow the discrepancy between domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Specifically, a domain discriminator is introduced to distinguish which do- main, the source or the target, the image comes from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Then, the detector learns the domain-invariant feature representa- tion by confusing the discriminator [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Despite achieving promising results, previous domain adaption approaches may suffer from target scale variations in cluttered regions due to the fixed kernel design in ConvNets [2], [3], which re- sults in difficulty in learning discriminative representations for objects of different scales and thus degrades detection performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' For example, the features of small targets may contain too much background noise because of too large receptive field in the convolutional layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Meanwhile, arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='00371v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='CV] 1 Jan 2023 SUBMITTED TO IEEE JOURNAL 2 the features of large objects may lack global structural information owing to too small receptive field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In addition, the intrinsic relation of feature distributions between two domains are neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' To address the aforementioned problems, various feature alignment strategies have been introduced in the adversarial learning manner [14], [16] for better target domain adap- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' These alignment approaches can be summarized into three categories based on different granularity perspectives, consisting of pixel-, instance-, and category-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Pixel-level alignment [17], [18], [19] aims at aligning lower-level pixel feature distribution of objects and background regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Nev- ertheless, there may exist a large gap between the pixel- level features for objects of different scales within the same category, resulting in limited detection performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Differ- ent from pixel-level alignment, instance-level alignment [20], [21], [22], [23], [24] first pools the feature maps of detection proposals and then leverages the pooled proposal features for the domain discriminator training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Despite avoiding the gap in pixel-level alignment, this strategy suffers from feature distortion for objects of different scales and aspect ratios caused by the pooling operation, which may lead to inaccurate feature representation and degenerated results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In addition to the aforementioned two types of alignments, recent approaches have attempted to utilize category-level alignment [25], [26], [27], [28], [29], [30] for UDA detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In specific, taking into consideration the intrinsic relation of feature distributions in two domains, the category-level alignment leverages the categorical discriminability to han- dle hard aligned instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' However, this alignment mech- anism focuses only on the global consistency of feature distribution between two domains while ignores other local consistency constrains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Although each type of alignment strategy brings in improvement, they are limited in several aspects as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In order to address these issues and make full use of the advantages of these three alignments, we propose a novel unified Multi-Granularity Alignment (MGA) framework for UDA detection, as illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Instead of performing simple combination of different alignment methods, MGA simultaneously encodes the de- pendencies across different granularities, consisting of pixel- , instance-, and category-levels, for domain alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' More specifically, we first introduce an omni-scale gated fusion (OSGF) module in MGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' The OSGF module is able to adapt to instances of different scales by automatically choosing the most plausible convolutions from the low- and high- resolution streams for feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Concretely, we first predict coarse detections based on pixel-level backbone fea- ture maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Then, using these coarse detections as guidance, we design a set of parallel convolutions in OSGF and adopt a gate mechanism to aggregate the discriminative features of instances (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=', the coarse detections) with similar scales and aspect ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' By doing so, the following detection head can more accurately predict multi-scale objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Besides the OSGF module, we present a new category- level discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Different from previous approaches, our category-level discriminator takes into account not only the instance discriminability in different classes but also the cat- egory consistency between two domains, leading to better detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In order to supervise the category-level discrim- inator, pseudo labels are assigned to important instances with high confidence based on object detection results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Considering that the quality of pseudo labels is crucial for learning a good category-level discriminator, we propose a simple yet effective adaptive exponential moving average (AEMA) strategy to train the teacher detector (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=', the orig- inal detector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' As shown in Figure 1, during the training phase, we assess both the teacher detector and the student detector (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=', the final adaptive detector) on the source do- main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Based on their model assessments, a dynamic update factor δ (as described later) is learned and utilized as a guidance to adjust the coefficient parameter in exponential moving average (EMA) update in an adaptive manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' The resulted AEMA helps better train the teacher detector to produce high-quality pseudo labels and meanwhile allevi- ate the local misalignment caused by low-quality pseudo labels, significantly enhancing the detection robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' We will elaborate on the details of our AEMA later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' By developing the multi-granularity discriminators, our MGA exploits and integrates rich complementary informa- tion from different levels, and hence achieves better UDA detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Besides, the proposed AEMA further enhances the robustness with high-quality pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' To validate the effectiveness of our approach, we carry out extensive experiments on multiple domain-shift scenar- ios using various benchmarks including Cityscapes [31], FoggyCityscapes [32], Sim10k [33], KITTI [34], PASCAL VOC [35], Clipart [36] and Watercolor [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' We evaluate the proposed method on the top of two popular detection frameworks, the anchor-free FCOS [8] and the anchor-based Faster R-CNN [7], with VGG-16 [4] and ResNet-101 [2] backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Experiment results demonstrate that our MGA together with the dynamic model update significantly im- prove the baseline detectors with superior results over other state-of-the-arts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' To sum up, we make the following key contributions: We propose a novel unified multi-granularity align- ment (MGA) framework that encodes dependencies across different pixel-, instance-, and category-levels for UDA detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Notably, our MGA framework is general and applicable to different object detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' We present an omni-scale gate fusion (OSGF) module to extract discriminative feature representation for instances with different scales and aspect ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' we propose a new category-level discriminator by exploiting both the instance discriminability in dif- ferent classes and the category consistency between two domains, leading to better detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' We introduce a simple yet effective dynamic adap- tive exponential moving average (AEMA) strategy to improve the quality of pseudo labels and mean- while mitigate the local misalignment issue in UDA detection, which significantly boosts the robustness of detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' On extensive experiments on multiple domain adap- tion scenarios, the proposed approach outperforms other state-of-the-art UDA detectors on the top of two detection frameworks, evidencing its effective- SUBMITTED TO IEEE JOURNAL 3 ness and generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' This paper builds upon our preliminary conference ver- sion [1] and significantly extends it in different aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' (1) We propose an effective assessment-based AEMA for model update of the teacher detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' This way, we are able to obtain pseudo labels with better quality, which largely boosts the detection robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Meanwhile, it is beneficial in alleviating the local misalignment issue caused by low- quality pseudo labels, further improving the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' (2) We modify the structure of the OSGF module by sharing the convolutional layer (as described later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Note that, this modification is not trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' It not only brings in improvement on the detection results but also decreases the number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' (3) We incorporate more experiments and comparisons with in-depth analysis and ablation studies to further show the effectiveness of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' (4) We supplement thorough visual analysis of our detector, which allows the readers to better understand our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Section 2 discusses approaches related to this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Our approach is elaborated in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In Section 4, we demonstrate the experimental results, including comparisons with state-of- the-arts, ablation studies and visual analysis, followed by conclusion in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' 2 RELATED WORK In this section, we review approaches relevant to this paper from four aspects, including object detection, UDA detec- tion, alignment strategy for UDA detection and exponential moving average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='1 Object Detection Object detection is a fundamental topic in computer vision and has been extensively studied for decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In general, existing modern detectors can be categorized into either anchor-based or anchor-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Anchor-based detectors usu- ally contain a set of anchor boxes with different scales and aspect ratios, which are applied to generate object proposals for further processing (in two-stage frameworks) or final detections (in one-stage frameworks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' One of the most popular anchor-based detectors is Faster R-CNN [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' It introduces a novel region proposal network (RPN) to produce object proposals based on anchors and then applies another network to further process the proposals for detec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' The approaches of SSD [10] and YOLOv2 [37] present one-stage anchor-based detectors that strikes a good balance between accuracy and speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Later, more excellent anchor- based detectors [12], [38], [39], [40], [41], [42] are proposed for improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Different from anchor-based approaches, anchor-free detectors remove the manual design of anchor boxes and directly predict the class and coordinates of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' YOLO [11] directly predicts the object class and po- sition from grid cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' CornerNet [9] proposes to predict the object bounding boxes as keypoint detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' The work of CenterNet [43] improves CornetNet by considering an extra center point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' FCOS [8] introduces the fully convolutional networks to predict object box of each pixel in feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Recently, DETR [44] applies Transformer [45] to develop an anchor-free detector and exhibits impressive performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In this paper, we utilize the proposed MGA upon the popular anchor-based Faster R-CNN [7] and anchor-free FCOS [8] to verify its effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' But please note that, our MGA is general and flexible and can be used in more frameworks for UDA detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='2 Unsupervised Domain Adaption (UDA) Detection The task of unsupervised domain adaption (UDA) detection focuses on improving the generality of object detectors learned from labeled source images on unlabeled target images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Because of its great practicability, UDA object de- tection has attracted extensive attention in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' One popular framework is to leverage adversarial learning to achieve UDA detection [46], [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' These approaches intro- duce a discriminator to identify which domain the features of pixels, regions or images come from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Then, the goal is to confuse the discriminator to learn domain-invariant features for detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In addition, many other researchers propose to apply graph methods for UDA detection [48], [49], [50], [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' These graph-based approaches propose to construct a graph based on regions or instances in an image and leverages the intra-class and inter-class relation of intra-domain and inter-domain for detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Self-training strategy has also been explored in UDA detection [52], [53], [54], [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' The main idea of these methods is to generate high-quality or class-balanced pseudo-labels, which can be utilized to train the detection model on the unlabeled target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' The approaches of [19], [56], [57] leverage the idea of style trans- fer for UDA detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In specific, these methods reduce the discrepancy of data distribution between two domains by translating the images of source domain to target style, which improves the generalization ability of the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Besides the aforementioned methods, recent approaches propose to apply mean-teacher for UDA detection [58], [59], [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' These models adopt a teacher-student training frame- work to maintain the consistency of the teacher and student detector networks for boosting the generality of detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Different from the above approaches, in this paper we tackle the UDA detection problem from a different perspective by unifying alignments of multiple granularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='3 Alignment for UDA Detection Alignment of feature distribution between source and target domains has demonstrated effectiveness for UDA detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Accordingly to the features involved, recent alignment- based approaches can be categorized into three types in- cluding pixel-, instance-, and category-level alignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Pixel-level Alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Pixel-level alignment focuses on aligning the pixel feature distributions of objects and back- ground regions between two domains for UDA detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' The work of [18] takes into account every pixel for do- main adaption and introduces a center-aware pixel-level alignment by paying more attention to foreground pixels for UDA detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' The approach of [17] designs the multi- domain-invariant representation learning to encourage un- biased semantic representation through adversarial learn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' The method of [19] introduces an intermediate domain for progressive adaption and utilize adversarial learning for pixel-level feature alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' SUBMITTED TO IEEE JOURNAL 4 Teacher FCOS Detector FPN Backbone Omni-scale Gated Fusion guidance Object Detection Target Student FCOS Detector FPN Backbone guidance Object Detection Source Target F Assessment-based AEMA Memory … Labels Images … C0 … … C1 … … CK … store Domain Shift Simulation AEMA 𝛿 Loss 𝐿!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' "# 𝒮 Loss 𝐿!' metadata={'source': 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+page_content='-level Discriminator Ins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='-level Discriminator Pix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='-level ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='Discriminator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='GRL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='GRL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='GRL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='Source Path ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='Gradient Reversal Layer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='GRL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='Target Path ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='AEMA Adaptative Exponential ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='Moving Average ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='RPN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='F2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='IoU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='Source Labels / ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='Target Pseudo Labels ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='IoU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='Intersection over Union ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='(a) Applying MGA on top of anchor-free FCOS for UDA detection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='(b) Applying MGA on top of anchor-based Faster R-CNN for UDA detection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='Assessment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='Omni-scale ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='Gated Fusion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='Omni-scale ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='Gated Fusion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='GRL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='Source or ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='Target ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='Source or ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='Target ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='Source or ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='Target ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='Source or ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='Target ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='Source or ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='Target ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Framework of our MGA on the top of popular anchor-free FCOS [8] (see left image (a)) and anchor-based Faster R-CNN [61] (see right image (b)) for UDA detection with assessment-based AEMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Note that for Faster R-CNN, the region proposal network (RPN) and the RoI head are used for coarse detection and final detection, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' F in (a), F1 and F2 in (b) represent the features from the feature pyramid network in FCOS and backbone in Faster R-CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Instance-level Alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Instance-level alignment usually leverages features of regions or instances to train a domain discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' The approach of [20] explores the relation of different instances using mean-teacher based on Faster R- CNN [7] for UDA detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' The work of [21] introduces a hierarchical framework to align both instance and do- main features for detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' The method of [22] proposes to adapt Faster R-CNN on target domain images by aligning features on instance- and image-levels, exhibiting promis- ing performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' The approach of [23] introduces attention mechanisms for better alignment in UDA detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Instead of using all instances or regions for alignment, the work of [24] proposes to mine the discriminative ones and focuses on aligning them across two different domains for adaption detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Category-level Alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Category-level alignment con- siders the intrinsic relation of feature distributions in source and target domains and exploits the categorical discrim- inability for alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' The approaches of [25], [26], [27], [62] learn a category-specific discriminator for each category and focus on classification between two domains using pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Despite effectiveness, it is difficult for these approaches to learn discriminative category-wise represen- tation among multiple discriminators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' The method of [28] retains one discriminator to distinguish different categories within one domain, whereas it neglects the consistency of feature subspaces in the same category across two domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Besides, the work of [29] develops a categorical regulariza- tion method that focuses on important regions and instances to reduce the domain discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' The method of [30] seeks for category-level domain alignment by enhancing intra- class compactness and inter-class separability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Our Alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Despite sharing similar spirit in applying alignment for UDA object detection, our approach (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=', Multi-Granularity Alignment (MGA)) is significantly differ- ent from others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Specifically, MGA is a unified framework that effectively encodes the dependencies across differ- ent granularities, including pixel-, instance-, and category- levels, for domain adaption detection, while other methods do not consider this important dependency relation in align- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In addition, we specially design the omni-scale gated fusion (OSGF) module and present a new category-level discriminator in MGA to improve the discriminative ability, as detailed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='4 Exponential Moving Average Exponential moving average (EMA) is a simple but effective strategy for updating the model parameters and commonly utilized in distillation technology [63], [64], [65] and mean- teacher [66], [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In these approaches, the EMA process is usually controlled by a constant weight coefficient to update the teacher network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Despite effectiveness, this mechanism may hurt the performance of UDA detection due to the low-quality pseudo labels generated by the teacher detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Therefore, unlike previous studies, we design an adaptive EMA (AEMA) by exploring the assessments of the teacher and student detector networks during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' AEMA is able to adjust the weight coefficient of EMA in an adap- tive manner, leading to higher-quality pseudo labels for improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' 3 MULTI-GRANULARITY ALIGNMENT (MGA) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='1 Overview In this paper, we propose a novel unified multi-granularity alignment (MGA) framework for domain adaption detec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' The overall architecture is illustrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' SUBMITTED TO IEEE JOURNAL 5 As displayed in Figure 2, given images from the source domain s and the target domain t, we first extract the base pixel-level feature representation from the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Then, these features are merged in the omni-scale gated fusion (OSGF) module to produce discriminative representations of multi-scale instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Based on the fused feature repre- sentations, more accurate candidate objects can be predicted by the object detection head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Meanwhile, we introduce the multi-granularity discriminators to distinguish the feature distributions between two domains from different perspec- tives, including pixel-level, instance-level and category- level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Moreover, in order to improve the quality of pseudo labels and mitigate the misalignment issue caused by noisy pseudo labels during training, we propose a simple but effective assessment-based adaptive EMA (AEMA) strategy to refine the pseudo labels, further enhancing the robustness of our MGA for domain adaption detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' It is worth noting that, our MGA is a general framework and can be easily applied in various detectors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=', anchor- free FCOS [8] and anchor-based Faster R-CNN [7]) with different backbones (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=', VGG-16 [4] and ResNet-101 [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Without loss of generality, we first apply the proposed MGA in FCOS [8] for UDA object detection as in Figure 2 (a), and then explain how it can be used in Faster R-CNN [7] as in Figure 2 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' For FCOS [8], we extract feature maps from the last three stages of the backbone and combine them into multi-level feature maps F k, where k ∈ {3, 4, 5, 6, 7}, using FPN representation [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='2 Omni-Scale Gated Object Detection In most previous studies on domain adaption detection, the main goal is to designate discriminators at a specific level and some attentive regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Nevertheless, the use of point representation at pixel-level in anchor-free models [18], [69] may cause difficulties in learning robust and discriminative feature in cluttered background, while the pooling operation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=', RoIAlign [39]) in anchor-based models [15], [70] may distort the features of the instances with different scales and aspect ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In order to handle this problem, we introduce an omni- scale gated fusion (OSGF) module for object detection, which enables the adaption of the feature learning to object with various scales and aspect ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Specifically in OSGF, with the scale guidance from coarse detections, we can choose the most plausible convolutions with different ker- nels to extract compact features of instances in terms of ob- ject scales, which can significantly boost the discriminative capacity of the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Our OSGF module is designed for general purpose and thus can be easily applied in different detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='1 Scale Guidance by Coarse Detection In order to select the most plausible convolutions for feature extraction, it is necessary to obtain the scale information of the objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' To this end, we introduce a coarse detection step to provide the scale guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In specific, followed by the multi-level feature maps F k (k ∈ {3, 4, 5, 6, 7} denotes the level index) from the backbone (see Figure 2 (a)), we can predict the candidate object boxes ˜bk through a series of convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Drawing inspiration from [71], we utilize the cross-entropy Intersection over Union (IoU) loss [72] to regress the bounding boxes of objects in foreground pixels as follows, Lgui = − � k � (i,j) ln(IoU(˜bk i,j, bk i,j)), (1) where IoU(·, ·) represents the function to calculate the IoU score between predicted box ˜bk and ground-truth box bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' For each pixel (i, j) in the feature map, the corresponding box bk i,j can be defined as a 4-dimensional vector as follows, bk i,j = (xti,j, xbi,j, xli,j, xri,j) (2) where xti,j, xbi,j, xli,j, and xri,j respectively represent the distances between current location and the top, bottom, left and right bounds of ground-truth box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Therefore, the normalized object scale (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=', width wk and height hk) at each level can be computed as follows, � wk i,j = (˜xri,j + ˜xli,j)/stridek, hk i,j = (˜xbi,j + ˜xti,j)/stridek, (3) where stridek denotes how many steps we move in each round of convolution operation1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' As in FCOS [8], the feature maps at each level are utilized to individually detect the objects of different scales in the range {[−1, 64], [64, 128], [128, 256], [256, 512]}, [512, +∞]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Therefore, the majority of object scales is less than 8, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=', wk ≤ 8, hk ≤ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' For notation simplicity, we omit the superscript k and write F for F k and ˜b for ˜bk in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='2 Omni-scale Gated Fusion (OSGF) With the scale guidance as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='1, we present an omni-scale gated fusion module (OSGF), which is composed of both low-resolution and high-resolution feature streams, to adapt to objects of various scales and aspect ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Specif- ically, as illustrated in Figure 3, the low-resolution stream consists of three parallel convolutional layers with different kernels ω ∈ {3×3, 3×5, 5×3}, which is applied for feature extraction of relatively small objects (wk ≤ 5, hk ≤ 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Meanwhile, in the high-resolution feature stream, we use another set of parallel convolutional layers with kernels ω to handle large objects (wk > 5, hk > 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' The different from the low-resolution feature branch is that, we utilize an extra upsampling operation after each convolutional layer in the high-resolution stream to upscale the feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' It is worth noticing that, the structure of OSGF in this paper is different from that in the conference publication [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In specific, the major modifications in this paper include: (i) removal of the 3 × 3 convolutional layer before the two streams, (ii) incorporation of simpler averaging pooling and 1 × 1 convolutional layers in the low-resolution stream, (iii) replacement of the 3 × 3 convolutional layers with shared averaging pooling and 1 × 1 convolutional layers, and (iv) change of the 1×1 convolutional layer to 3×3 convolutional layer in the residual connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' By doing so, the overall number of parameters are significantly reduced because of less convolutional layers used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In addition, we observe that the detection performance has been improved by designing 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' We have {(k, stride)|(3, 8), (4, 16), (5, 32), (6, 64), (7, 128)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' SUBMITTED TO IEEE JOURNAL 6 Conv 3×5_1 Conv 5×3_1 Conv 3×3_1 Conv 3×3_1 Conv 3×3_1 Conv 3×3_1 AVG 1×1_2 Conv 1×1_1 Conv 3×5_1 Conv 5×3_1 Conv 3×3_1 Conv 3×3_1 Conv 3×3_1 Conv 3×3_1 AVG 1×1_2 Conv 1×1_1 upsample upsample upsample G3×5_1 G5×3_1 G3×3_1 G3×5_2 G5×3_2 G3×3_2 add Conv 3×3_1 merged feature maps backbone feature maps coarse detections guidance ℎ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=', 𝑤!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' low-resolution stream high-resolution stream Conv 3×5_1 AVG 1×1_2 Conv 1×1_1 Conv 3×5_1 Conv 5×3_1 Conv 3×3_1 AVG 1×1_2 Conv 1×1_1 upsample upsample upsample backbone feature maps low-resolution stream high-resolution stream C Conv 5×3_1 Conv 3×3_1 concat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' RoI Align split Conv 3×3_1 Conv 3×3_1 Conv 3×3_1 G3×5_1 G5×3_1 G3×3_1 Conv 3×3_1 Conv 3×3_1 Conv 3×3_1 G5×3_2 G3×3_2 Conv 3×3_1 merged feature maps 𝑘 candidate boxes guidance ℎ, 𝑤 (a) Omni-scale Gate Fusion (OSGF) for anchor-free FCOS (b) Omni-scale Gate Fusion (OSGF) for anchor-based Faster R-CNN add G3×5_2 RoI Align Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Illustration of the proposed omni-scale gated fusion (OSGF) module for anchor-free FCOS [8] (see left image (a)) and anchor-based Faster R-CNN [61] (see right image (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' The parameters of the modules with the same color are shared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' the shared convolutional layer and increasing the kernel size in the convolutional layer of residual connection, as evidenced by our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' After the two branches of low- and high-resolution fea- tures, we introduce a gate mask G to weight each convolu- tional layer based on the predicted coarse boxes ˜b as follows, Gω = exp(τ(oω − ˆo)/(ˆo + ϵ)) � ω exp(τ(oω − ˆo)/(ˆo + ϵ)), (4) where τ represents the temperature factor, oω = IoU(˜b, ω) denotes the overlap between the predicted box and the convolution kernel ω, and ˆo is the maximal overlap among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Finally, we merge the pixel-level features to exploit the scale-wise representation of instances as follows, M = � ω Fω ⊙ Gω + F3×3, (5) where ⊙ denotes the element-wise product, and Fω denotes the feature maps after the convolutional layer with kernel ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='3 Object Detection After obtaining the merged feature maps M from the OSGF module, we can predict the categories and bounding boxes of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In FCOS [8], the object detection heads contain three branches for classification, centerness and regression, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' The classification and centerness branches are optimized by the focal loss [73] Lcls and cross-entropy loss [8] Lctr, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' The regression branch is optimized by the IoU loss [72] Lreg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Thus, the final loss function Ldet for the object detection is defined as Ldet = Lcls + Lctr + Lreg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' (6) Please refer to [8] for more details regarding the loss func- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' It is worthy to notice that, in the UDA detection, we implement two detectors, including a teacher detector and a student detector (see Figure 2 (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' These two detectors share the same architecture but independent parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' We denote the loss functions for the teacher and the student detectors as LT det and LS det, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='3 Multi-Granularity Discriminators As discussed earlier, we propose the multi-granularity dis- criminators to distinguish whether the sample belongs to the source domain or the target domain from various per- spectives, consisting of pixels, instances and categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' The discrepancy between two domains is reduced using Gradi- ent Reversal Layer (GRL) [14] that transfers reverse gradient when optimizing the object detection network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' The discrim- inator contains four stacked convolution-groupnorm-relu layers and an extra 3 × 3 convolutional layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Below we will elaborate on our multi-granularity discriminators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='1 Pixel- and Instance-level Discriminators The pixel- and instance-level discriminators are leveraged to respectively perform pixel-level and instance-level align- ments of feature maps between two domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' As demon- strated in Figure 2 (a), given the input multi-level features F and the merged feature M, the pixel-level and instance- level discriminators Dpix and Dins are learned through the loss functions Lpix and Lins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Similar to previous work [18], we adopt the same loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Then, Lpix and Lins are defined as follows Lpix = − � (i,j) ypix i,j log Dpix(F s(i, j)) + (1 − ypix i,j ) log(1 − Dpix(F t(i, j))), (7) Lins = − � (i,j) yins i,j log Dins(M s(i, j)) + (1 − yins i,j) log(1 − Dins(M t(i, j))), (8) where F(i, j) is the feature at pixel (i, j) in F, and M(i, j) the feature at instance (i, j) in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' We have the domain label ypix i,j = 1 if pixel at (i, j) in F is from source domain and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Likewise, yins i,j = 1 if the instance at (i, j) in M belongs to source domain and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='2 Category-level Discriminator In order to keep the semantic consistency between dif- ferent domain distributions, a category-level discrimina- tor is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Previous methods design either category- specific discriminators for each category (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=', [25], [26], SUBMITTED TO IEEE JOURNAL 7 (a) category-specific discriminator 𝐷!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' %&\' 𝐷%&\' 𝐷" %&\' 𝐷&\'% "#$ … 𝑠!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' 𝑡!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' 𝑠" 𝑡" 𝑠#$" 𝑡#$" 𝑠!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' 𝑠" … 𝑠#$" 𝑡!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' 𝑡" … 𝑡#$" source alignment target alignment (b) domain-consistent discriminator 𝐷%&\' 𝑠!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' 𝑠" … 𝑠#$" 𝑡!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' 𝑡" … 𝑡#$" 𝑐 = 0 𝑐 = 1 𝑐 = 𝐶 − 1 𝑠!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' 𝑠" … 𝑠#$" 𝑡!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' 𝑡" … 𝑡#$" 0 2 1 𝐶 − 1 𝑐: 𝑐: 𝑐: 𝑐: 𝑐: … instance discriminability ℒ/01 category consistency ℒ102 (c) category- and domain-consistent discriminator (ours) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Illustration of different category-level discriminators D, where sc and tc denote the c-th category (c = 0, 1, · · · , C − 1) in source domain and target domain respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' (a) Category-specific discriminators for each category [25], [26], [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' (b) Domain-consistent discriminator to distinguish different categories within one domain [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' (c) Our category- and domain-consistent discriminator to consider both instance discrim- inability in different categories and category consistency between two domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' [27], see Figure 4 (a)) or a domain-consistent discriminator to distinguish categories within one domain (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=', [28], see Figure 4 (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' By contrast, our approach considers jointly instance discriminability in different categories and category consistency between two domains and introduces a novel category- and domain-consistent discriminator (see Figure 4 (c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Specifically, in our discriminator, we predict the cate- gory and domain labels of pixel (i, j) in each image based on feature map ˆ M ∈ RH×W ×2C, where ˆ M ∈ RH×W ×2C is the output by feeding M to the category-level discriminator, H and W are the height and width respectively, and 2C represents the total number of categories for source and target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Since there is no ground-truth to supervise the category- level discriminator, we assign pseudo labels to important samples with high confidence from object detection (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In practice, given a batch of input images, we can output the category probability map P using the object detection heads, and obtain the set S of pseudo labels by utilizeing the probability threshold τprob and non-maximum suppression (NMS) threshold τnms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Then, the instances in different categories are classified by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' (9), while the same category in two domains is aligned by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' (11), as follows: In order to keep instance discriminability in different categories, we separate the category distribution by using the following loss function, Ldis = − 1 |S| � (i,j)∈S C−1 � c=0 ˆydis i,j,c log(pdis i,j,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' (9) By normalizing confidence over the domain channel, pdis i,j,c represents the probability of the c-th category of the pixel (i, j), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=', pdis i,j,c = exp ( ˆ Mi,j,2c + ˆ Mi,j,2c+1) �C−1 c=0 exp ( ˆ Mi,j,2c + ˆ Mi,j,2c+1) , (10) where ˆ Mi,j,2c and ˆ Mi,j,2c+1 represent the confidence of the c-th category in source and target domains, respectively (see again Figure 4(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' ˆydis ∈ RH×W ×C is the pseudo category label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' We have ˆydis i,j,c = 1 if the instance at (i, j) in ˆ M is an important one of the c-th category and ˆydis i,j,c = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Category consistency between two domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' After classifying instances of different categories, we need to further identify which domain the instance be- longs to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' With GRL [14], we write the loss function as follows, Lsim = − 1 |S| � (i,j)∈S 2C−1 � m=0 ˆysim i,j,m log(psim i,j,m), (11) where ysim ∈ RH×W ×2C is the pseudo domain label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Similarly, ˆysim i,j,m = 1 if the instance at (i, j) in ˆ M is an important one of the ⌊ m 2 ⌋-th category in specific domain and ˆysim i,j,m = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' The domain prob- ability psim is obtained as follows, psim i,j,m = � � � exp( ˆ Mi,j,m) exp( ˆ Mi,j,m−1)+exp( ˆ Mi,j,m), if m is odd exp( ˆ Mi,j,m) exp( ˆ Mi,j,m)+exp( ˆ Mi,j,m+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' if m is even (12) With the above analysis, we define the final loss function Lcat for the category-level discriminator Dcat as follows, Lcat = λdisLdis + λsimLsim, (13) where Ldis and Lsim are loss functions for instance discrim- inability and category consistency as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' (9) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' (11), and λdis and λsim are the balancing factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='4 Adaptive Exponential Moving Average (AEMA) As mentioned in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='2, the pseudo labels, which are generated by the teacher detector (see Figure 2 (a)), are required for supervising the learning of the category-level discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' During the training procedure, the teacher de- tector is usually updated using exponential moving average (EMA) as follows, θη T = (1 − γ) · θη−1 T + γ · θη−1 S (14) where θη T represents the weights of the teacher detector at iteration η, θη−1 S denotes the weights of the student detector at iteration η − 1, and α is a constant coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' SUBMITTED TO IEEE JOURNAL 8 (a) EMA (b) AEMA (c) GT (d) Comparison of pseudo label quality Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Comparison of the pseudo label quality between EMA and AEMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Image (a) displays the pseudo label generated by EMA, image (b) the pseudo label by our AEMA, and image (c) the GT pseudo label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In image (d), we demonstrate the mAP scores of the generated pseudo labels of different strategies, and we can observe that AEMA produces better pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Despite simplicity, the EMA approach may lead to some low-quality pseudo labels (see Figure 5 (a)), because it does not consider the feedback from the two detectors, degrading the final detection performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' To address this problem for improving quality of pseudo labels, we propose an adaptive EMA (AEMA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Specifically, unlike EMA, AEMA considers the intermediate assessments of both teacher and student detectors during update by evaluating their performance on the source domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Using the assessments as a guidance, an update factor δ (as described later) is learned to adjust the coefficient α in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' More concretely, as in Figure 2, we maintain a memory bank, which is used for generating the assessments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' The memory is dynamically updated by storing the images and labels of source domain into it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In order to accurately assess the generalization ability of the detector, we introduce a domain shift simulation (DSS, see Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' 2 (a)) module, and apply it on the memory bank to generate discrepancy of data distribution on the source domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Specifically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' given the sampling data of images xm and labels ym of all categories from the memory bank,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' we randomly adjust the mean xu and variance σ2 of xm to generate the variant data distribution from the source domain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' as follows xm = �σ xm − xu √ σ2 + ϵ + �xu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' (15) Where the mean �xu and the standard deviation �σ for the new variant data distribution are obtained by using the uni- form distribution under xu and σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' as follows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' �xu = U(au,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' bu)xu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' (16) �σ = U(aσ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' bσ)σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' (17) Here,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' U(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' b) represents the uniform distribution between a and b and is predefined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Afterwards, the detection losses LT det and LS det, obtained by evaluating the teacher and stu- dent detectors on the above input date, are employed as the assessment results to derive the update factor δ, as follows, δ = � eτ1·(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='5−ρ), ρ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='5 eτ2·(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='5−ρ), ρ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='5 ρ = LS det LS det + LT det (18) where τ1 and τ2 are two constant values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Finally, the weights of the teacher detection model can be updated by our AEMA with δ as follows, θη T = (1 − γ · δ) · θη−1 T + γ · δ · θη−1 S (19) By using AEMA, we can take into account the assess- ments of two detectors to guide the update the teacher detection, resulting in better pseudo labels, as shown in Figure 5 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Furthermore, we show the statistic comparison of the pseudo labels obtained by teacher detector with EMA and our proposed AEMA in term of accuracy in Figure 5 (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' As demonstrated in Figure 5 (d), we can see that, the quality of the pseudo labels is clearly improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' We will further analyze the effectiveness of our AEMA in later experimental section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='5 Overall Loss Function and Optimization As discussed above, the omni-scale gated object detection network is supervised by Lgui and Ldet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Meanwhile, the multi-granularity discriminators are optimized in different granularities, including pixel-level Lpix, instance-level Lins and category-level Lcat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In summary, the overall loss func- tion is defined as L = (Lgui + Ldet � �� � object detection ) + α (Lpix + Lins + Lcat) � �� � multi-granularity discriminators (20) where α is the balancing factor between object detection and multi-granularity discriminators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' The training process of our proposed method is divided into two stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In stage 1 (S1), we train teacher detector by using SGD optimizer and random sampling with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' (6) on source domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Next, in stage 2 (S2), the student detector is optimized by using SDG optimizer and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' (20) on source with labels and target domains with pseudo labels, and the teacher detector is updated by adaptive exponential moving average (AEMA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' 4 EXPERIMENTS Extension of our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Our MGA framework is designed for general purpose and applicable to both one- and two-stage detection models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' To verify this, in addition to the representative one-stage FCOS [75], we further extend our MGA to the popular two-stage Faster-RCNN [7] that consists of Region Proposal Network (RPN) and RCNN with classification and regression branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' As shown in Figure 2 (b), we employ RPN as our coarse detection mod- ule, whose loss function is replaced by the original RPN loss, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=', Lgui = Lrpn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' and we use RCNN as object detection module with the loss defined as Ldet = Lcls+Lreg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' For omni- scale gate fusion, we first obtain the top K proposals by using RPN based on the backbone feature layer with stride 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Then, we further extract the pixel-level features with low-resolution and high-resolution streams, and generate instance features of 7 × 7 under the pixel-level feature map and original input feature maps by using the ROIAlign op- eration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Finally, the instance features are merged according to the RPN outputs and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' (5) after using a convolution of 3 × 3, as shown in Figure 3 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Quality of Pseudo Label 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='5 mAPAEMA mAPEMA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='3 mAP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='2 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='8 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='2 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='9 oracle FCOS VGG-16 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='7 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='9 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='5 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='1 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='8 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='0 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='8 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='0 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='7 oracle FCOS ResNet-101 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='3 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='0 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='8 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='8 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='3 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='2 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='4 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='9 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='6 Implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In this work, we implement our method based on different detectors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=', Faster-RCNN and FCOS) and backbones (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=', VGG-16 and ResNet-101) using PyTorch [87] to show generality of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Both VGG-16 and ResNet-101 are pre-trained on ImageNet [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' We utilize a unified optimization framework by using training process of two stages and warm-up followed the previous works [18] and [50] for different detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Similar to [23], we apply the Adam optimizer with an initial learning rate of 3e-4, a momentum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='9 and weight decay of 1e-4 in Faster-RCNN framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' For FCOS framework, we use SGD optimizer with an initial learning rate of 5e-3, a momentum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='9 and weight decay of 1e-4, being consistent with CFA [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' γ is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='1 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' The parameters au and bu are set respectively to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='4 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='5 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' (16), and the aδ and bδ respectively to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='8 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='9 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' The thresholds τprob and τnms for obtain- ing S are empirically set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='42 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' All our experiments are conducted on the machine with an Intel(R) Xeon(R) CPU and 4 Tesla V100 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Our code will be made publicly available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='com/tiankongzhang/MGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='1 Datasets To verify the proposed method, we conduct extensive exper- iments on different adaption settings, as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Weather adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' For weather adaptation, we explore generalization of the detector on Cityscapes [31] and Fog- gyCityscapes [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Cityscapes [31] is a popular street scene dataset with normal weather, which comprises 2,975 train- ing images and 500 validation images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' FoggyCityscapes [32] is synthesized on Cityscapes with different levels of fog (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='01 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='02).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' For fair comparison, we choose the level of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='02 for experiment as in other methods in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In weather adaptation, we use Cityscapes [31] as the source domain and FoggyCityscapes [32] as the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Cross-Camera adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In Cross-Camera adaptation, we evaluate our algorithm on KITTI [34] and Cityscapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' KITTI [34] is a popular traffic scene dataset containing 7,481 train- ing images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In this adaption experiment, KITTI is the source domain and Cityscapes is the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Following previous works [50], [81], we only report the results on the category of car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' SUBMITTED TO IEEE JOURNAL 10 TABLE 2 Results of our approach and comparison to state-of-the-arts on real-to-artistic adaptation from PASCAL VOC to Clipart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' The best two results are highlighted in red and blue fonts, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Note, there are no oracle results for Clipart because all images in Clipart are used for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Method Detector Backbone acro bicycle bird boat bottle bus car cat chair cow Baseline Faster-RCNN ResNet-101 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='6 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='5 24.' metadata={'source': 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Faster-RCNN ResNet-101 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='3 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='7 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='7 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='3 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='6 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='7 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='1 oracle Faster-RCNN ResNet-101 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='1 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='4 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='9 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='3 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='5 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='8 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='8 Synthetic-to-Real adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' For Synthetic-to-Real adapta- tion, we utilize SIM10k [33] and Cityscapes for experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' SIM10k [33] is a synthetic scene dataset from the game video Grand Theft Auto V (GTA5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' It contains 10k training images, and we conduct comparisons on the car class, similar to [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In this adaptation experiment, we utilize the SIM10k as the source domain and Cityscapes as the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Real-to-Artistic adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In Real-to-Artistic adaptation, we verify our method on PASCAL VOC [35], Clipart [36] and Watercolor [36] datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' PASCAL VOC [35] is a real- scene dataset including two sub-datasets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=', PASCAL VOC 2007 and PASCAL VOC 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' PASCAL VOC 2007 consists of 2,501 images for training and 2,510 images for validation, and PASCAL VOC 2012 contains 5,717 images for training and 5,823 mages for validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Clipart [36] is a carton dataset with 1k images and has the same categories as PASCAL VOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Watercolor [36] is a watercolor style dataset containing 1,000 training images and 1,000 testing images, and it shares 6 classes with PASCAL VOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In this setting, we use PASCAL VOC as the source domain and Clipart or Watercolor as the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='2 State-of-the-Art Comparison In this section, we demonstrate our results and comparison with state-of-the-art methods using different base detectors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=', Faster-RCNN [7] and FCOS [8]) and backbones (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=', VGG-16 [4] and ReseNet-101 [2]) on different adaptation scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In all comparison table, “Baseline (ours)” means that the baseline detector is equipped with our OSGF and trained using data augmentation as in our method but without adaption strategy, and “oracle” indicates that the baseline detector is trained and tested on the target domain without any adaptation strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Weather adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In Table 1, we report the results from Cityscapes to FoggyCityscapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' As displayed in Table 1, for Faster-RCNN detector, our method achieves the best mAP SUBMITTED TO IEEE JOURNAL 11 Baseline MGA-DA MGA GT Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Qualitative results and comparison (from left column to right column: weather adaptation from Cityscapes to FoggyCityscapes, real-to- artistic adaptation from PASCAL VOC to Clipart and Watercolor, cross-camera adaption from Kitti to Cityscapes, and synthetic-to-real adaption from SIM10k to Cityscapes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' We can see that MGA achieves superior results than MGA-DA in our conference version and the baseline method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' TABLE 4 Results of our approach and comparison to state-of-the-arts on cross-camera/synthetic-to-real adaptation detection results from Kitti/SIM10k to Cityscapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' The best two results are highlighted in red and blue fonts, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Method Detector Backbone APcar Baseline Faster-RCNN VGG-16 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='2/30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='1 DAF [22] Faster-RCNN VGG-16 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='5/39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='0 MAF [75] Faster-RCNN VGG-16 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='0/41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='1 ATF [70] Faster-RCNN VGG-16 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='1/42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='8 SC-DA [74] Faster-RCNN VGG-16 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='5/43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='0 UMT [84] Faster-RCNN VGG-16 /43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='1 SFOD-Mosaic [80] Faster-RCNN VGG-16 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='6/43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='1 CST [77] Faster-RCNN VGG-16 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='6/44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='5 MeGA-CDA [46] Faster-RCNN VGG-16 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='0/44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='8 SAPNet [23] Faster-RCNN VGG-16 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='4/44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='9 CDN [79] Faster-RCNN VGG-16 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='9/49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='3 TIA [47] Faster-RCNN VGG-16 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='0/ – DSS [83] Faster-RCNN ResNet-50 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='7/44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='5 SSD [88] Faster-RCNN ResNet-50 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='6/49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='3 SIGMA [50] Faster-RCNN VGG-16 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='8/53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='7 TDD [59] Faster-RCNN VGG-16 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='4/53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='4 MGA-DA [1] Faster-RCNN VGG-16 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='2/49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='8 Baseline (ours) Faster-RCNN VGG-16 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='7/44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='1 MGA (ours) Faster-RCNN VGG-16 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='3/55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='5 oracle Faster-RCNN VGG-16 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='1 SST-AL [69] FCOS 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='6/51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='8 CFA [86] FCOS VGG-16 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='2/49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='0 SCAN [86] FCOS VGG-16 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='8/52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='6 CFA [18] FCOS ResNet-101 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='0/51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='2 MGA-DA [1] FCOS VGG-16 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='5/54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='6 MGA-DA [1] FCOS ResNet-101 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='5/54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='1 Baseline (ours) FCOS VGG-16 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='1/43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='0 Baseline (ours) FCOS ResNet-101 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='3/43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='7 MGA (ours) FCOS VGG-16 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='9/55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='8 MGA (ours) FCOS ResNet-101 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='6/55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='4 oracle FCOS VGG-16 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='4 oracle FCOS ResNet-101 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='8 of 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='4% and outperforms the second best SDA [85] with 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='2% mAP by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Compared to the baseline (ours) of 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='1%, MGA obtains 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='3% performance gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' For FCOS detector, we also obtain the best mAP scores of 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='9% with VGG-16 and 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='9% with ResNet-101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Compared with the approaches of SCAN [86] with VGG-16 and CFA [18] with ResNet-101, our MGA respectively shows the 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='8% and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='7% gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In addition, our method observes obvious improvements over the baseline (ours) with 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='2% using VGG-16 and 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='7% gains using ResNet-101, which verifies the effectiveness of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Furthermore, compared with MGA-DA [1] with 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='3% mAP score for Fatser-RCNN and 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='6% and 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='8 % mAP scores for FCOS on VGG-16 and ResNet-101 backbone in our conference version, our MGA shows 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='1%, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='3% and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='1% gains, showing the effectiveness of our new contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Real-to-Artistic adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Table 2 and 3 show the results and comparison in real-to-artistic adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' As in Table 2, from PASCAL VOC to Clipart, our method achieves the second mAP score of 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='0%, and D-adapt [62] obtains the best performance of 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='0%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Compared to our baseline, we achieve 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='0% performance gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In Table 3, our MGA performs the best with 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='1% mAP score and surpasses the second best AT [60] with 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='9% by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Besides, our method outperforms the oracle, indicating that our MGA makes full use of the information between source and target domains for robust UDA detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Cross-Camera adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Table 4 displays the results and comparison from Kitti to Cityscapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' As shown in Table 4, on Faster-RCNN detector, our method shows the best result of 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='3% APcar with VGG-16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In contrast to the baseline (ours), we obtain 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='6% gain.' metadata={'source': 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car:81%car:100% ar:oo% car:100% 100 car:100%100% car:10 ar100%car:79% 8% car:47%SUBMITTED TO IEEE JOURNAL 12 gains with VGG-16 and ResNet-101, respectively, showing its advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Synthetic-to-Real adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Table 4 shows the result from Sim10k to Cityscapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' On Faster-RCNN detector, our MGA achieves the best result of 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='5% APcar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In contrast to the baseline (ours), it obtains a 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='4% gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' On FCOS detector, our method shows the best APcar of 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='8% with VGG-16 and 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='4% with ResNet-101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In comparison with SCAN [86] with VGG-16 and CFA [18] with ResNet-101, we ob- tains 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='2% and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='2% gains with VGG-16 and ResNet-101, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Compared to baselines (ours), it demonstrates gains of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='8% and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='7% with VGG-16 and ResNet-101, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Besides quantitative results, we demonstrate qualitative results our method and comparison to other approaches in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' From Figure 6, we can observe that MGA achieves superior detection results than MGA-DA in our conference version and the baseline method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='3 Ablation Study To further analyze our approach, we conduct ablation ex- periments on different components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' The results are reported under the weather adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Effectiveness of different components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In order to further validate the effectiveness of different components including omni-scale gated fusion (OSGF), multi-granularity discrim- inators (MGD) and adaptive exponential moving average (AEMA) in MGA, we demonstrate the results by gradually adding them to the baseline, which is FCOS with VGG16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Table 5 shows the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' As shown in Table 5, with our OSGF, the performance of the baseline is improved from 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='0% to 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='7% with a gain of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='7%, showing the effec- tiveness of OSGF in discriminative representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' When employing the proposed MGD for adaption, we achieve significant improvement by boosting the result from 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='7% to 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='3% with a 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='6%, which clearly evidences the effectiveness of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Further, when adopting AEMA for better pseudo labels, the final result is improved from 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='3% to 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='9%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Comparison of category-level discriminators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' As displayed in the Table 6, we compare our category-level discriminator Dcat and other related class-level discriminators, including Dcen [86], Dgrp [26] and Dcls [28] using FCOS with VGG16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' The “baseline” indicates that the detector is learned under all strategies but the class-level discriminator is removed from MGA module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Dcen focuses on reducing the differences on the center-aware distributions of source and target do- mains, which is composed of features of the central positions of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Dgrp aligns the feature distributions by building a sole domain discriminator for each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Dcls simultane- ously takes domain and class information and expands the binary domain labels by inserting the binary class labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' From Table 6, we obverse that the performance of de- tector is improved from 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='7% to 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='2%/46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='0%/45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='2% with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='5%/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='3%/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='5% gains by Dcen/Dgrp/Dcls over the base- line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In contrast, our Dcat improves the baseline detector performance from 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='7% to 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='9%, which shows 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='2% per- formance gains over the baseline and outperforms other three category-level discriminators by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='7%, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='9% and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='7%, respectively, evidencing the effectiveness of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' TABLE 5 Effectiveness of different components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' OSGF MGD AEMA mAP 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='0 ✓ 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='7 ✓ ✓ 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='3 ✓ ✓ ✓ 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='9 TABLE 6 Comparison between different category-level discriminators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' discriminator baseline Dcen Dgrp Dcls Dcat (ours) mAP 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='7 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='2 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='0 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='2 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='9 TABLE 7 Comparison between different fusion methods for OSGF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' S1 S2 w/o OSGF Conv fusion Average fusion Gated fusion (ours) ✓ 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='0 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='2 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='8 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='7 ✓ ✓ 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='6 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='2 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='1 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='9 TABLE 8 Comparison of different OSGF in this work (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=', OSGF (ours)) and conference version (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=', OSGF (conference)) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' The backbone of the detectors in this table is VGG16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' method detector # Params (M) S1 S1&S2 OSGF (conference) [1] FCOS 17 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='4 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='8 OSGF (ours) FCOS 14 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='7 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='9 OSGF (conference) [1] Faster-RCNN 55 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='8 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='1 OSGF (ours) Faster-RCNN 50 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='1 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='4 Analysis on gated fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In this paper, we introduce a novel gated fusion approach for improved discriminative feature representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In order to further analyze the gated fusion, we compare it with other commonly used fusion strategies including Conv fusion and Average fusion in Table 7 using FCOS with VGG16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In Table 7, “w/o OSGF” means that the OSGF module is removed from our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' “Conv fusion” indicates the fusion weights in OSGF are obtained by using 1 × 1 convolution, and “Average fu- sion” means averaging features of convolution kernels in OSGF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' From Table 7, we can observe that, in S1, all three fusion methods can improve the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In specific, the OSGF with our gated fusion strategy improves the mAP score from 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='0% to 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='7% with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='7% performance gains, outperforming those using Conv and Average fusion methods with 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='2% and 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='8% mAP scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Likewise, in S2, OSGF with our gated fusion achieves the best performance, evidencing the effectiveness of gated mechanism for repre- sentation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In addition, from Table 8, we can observe that the im- proved OSGF in this work contains less parameters and meanwhile achieves better performance compared to that in the preliminary conference version [1], which indicates the advantages of our improved OSGF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Comparison of EMA and AEMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' To show the effectiveness of our proposed AEMA, we compare it with EMA using FCOS with VGG16 as shown in Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' From Table 9, we obverse that EMA brings in a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='2% performance gain by improving the result from 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='3% to 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' However, when using the proposed AEMA, the performance is improved to SUBMITTED TO IEEE JOURNAL 13 TABLE 9 Comparison between EMA and AEMA for detection performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' EMA AEMA mAP 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='3 ✓ 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='5 ✓ 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='9 TABLE 10 Comparison of computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' The backbone of the detectors in this table is VGG16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' method detector # Total Params (M) FPS CFA [18] FCOS 177 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='5 MGA (ours) FCOS 429 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='1 SCL [81] Faster-RCNN 580 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='8 SAPNet [23] Faster-RCNN 556 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='2 MGA (ours) Faster-RCNN 535 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='0 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='9% with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content='6% gains, which demonstrates the superiority of our AEMA in learning better pseudo labels for detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In Table 10, we show the com- parison of computational complexity between our method and other state-of-the-art approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' 10, on FCOS, our MGA performs better with reasonable increased complexity over its primary contender CFA [18];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' while our method has less parameters on the top of Faster- RCNN than two recent methods SCL [81] and SAPNet [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' 5 CONCLUSION In this paper, we propose a novel unified multi-granularity alignment (MGA) framework, which encodes dependencies among different pixel-, instance-, and category-levels to achieve alignment of feature distributions for unsupervised domain adaptive detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Notably, we design the omni- scale gated fusion module with different scales and aspect rations to extract discriminative instance-level feature rep- resentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' In order to improve the quality of pseudo labels and mitigate the local misalignment problem in our MGA framework, we further propose a simple but effective dy- namic adaptive exponential moving average strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Exten- sive experiments evidence the effectiveness and superiority of our MGA on different detectors for unsupervised domain adaptive object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' REFERENCES [1] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Zhou, D.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} +page_content=' Schiele, “Seeking similarities over differences: Similarity-based domain alignment for adaptive object detection,” in ICCV, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfhPgA/content/2301.00371v1.pdf'} diff --git a/Q9E2T4oBgHgl3EQfWAeQ/content/tmp_files/2301.03829v1.pdf.txt b/Q9E2T4oBgHgl3EQfWAeQ/content/tmp_files/2301.03829v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..15db87f4fa57167f0936c3b839f95eb0b82183e4 --- /dev/null +++ b/Q9E2T4oBgHgl3EQfWAeQ/content/tmp_files/2301.03829v1.pdf.txt @@ -0,0 +1,1100 @@ +A Dietary Nutrition-aided Healthcare Platform via Effective +Food Recognition on a Localized Singaporean Food Dataset +Kaiping Zheng1, Thao Nguyen1, Jesslyn Hwei Sing Chong2, Charlene Enhui Goh3 +Melanie Herschel4, Hee Hoon Lee5, Beng Chin Ooi1, Wei Wang6, James Yip7 +1School of Computing, National University of Singapore +2Ng Teng Fong General Hospital, Dietetics and Nutrition Department, Singapore +3Faculty of Dentistry, National University of Singapore +4Universit¨at Stuttgart, Germany +5Ng Teng Fong General Hospital, Allied Health Division, Singapore +6TikTok Singapore +7National University Heart Centre, Singapore +Abstract +Localized food datasets have profound meaning in revealing a country’s special +cuisines to explore people’s dietary behaviors, which will shed light on their health +conditions and disease development. In this paper, revolving around the demand for +accurate food recognition in Singapore, we develop the FoodSG platform to incu- +bate diverse healthcare-oriented applications as a service in Singapore, taking into +account their shared requirements. We release a localized Singaporean food dataset +FoodSG-233 with a systematic cleaning and curation pipeline for promoting future +data management research in food computing. To overcome the hurdle in recognition +performance brought by Singaporean multifarious food dishes, we propose to inte- +grate supervised contrastive learning into our food recognition model FoodSG-SCL +for the intrinsic capability to mine hard positive/negative samples and therefore boost +the accuracy. Through a comprehensive evaluation, we share the insightful experi- +ence with practitioners in the data management community regarding food-related +data-intensive healthcare applications. +The FoodSG-233 dataset can be accessed via: https://foodlg.comp.nus.edu.sg/. +1 +Introduction +Healthcare analytics aims to conduct numerous analytic tasks with a broad range of health- +care data spanning cost and claim data, pharmaceutical data, research and development +data, clinical data, and patients’ behavior and sentiment data. Based upon these, health- +care analytics serves as a basis for different data-intensive healthcare applications, such as +chronic disease progression modeling [37, 45, 47, 44], and disease diagnosis [15, 26, 43, 42]. +Among the diverse data sources for healthcare analytics, large-scale food data plays a +significant role in revealing the dietary information and knowledge of patients, posing a +profound impact on humans’ health and well-being [30, 11]. For instance, long-term con- +sumption of unhealthy foods increases the risk of developing diseases such as diabetes, +hypertension, and hyperlipidemia. Therefore, there is an increasing demand for food data +for diet assessment and disease management. +Naturally, food computing emerges as a subfield of computing, being accorded for its +importance. It involves computational methods to analyze heterogeneous food data for +solving disparate food-related problems and thereby, contributes to diverse data-intensive +1 +arXiv:2301.03829v1 [cs.LG] 10 Jan 2023 + +salmon - grilled +yuseng (raw fish salad) +sushi +sushi +sushi +intra-class variation +inter-class resemblance +Figure 1: Two common issues in real-world food datasets. +healthcare applications [30]. Food recognition, which identifies food types from food im- +ages, is a cornerstone as it enables the effective assessment of people’s diets whereby it +provides much-needed intake information for disease prevention. Notably, combined with +nutrition information, many other food-related applications become possible, such as es- +timating the intake of calories and nutrients. +Prior studies either make use of hand-crafted features for food recognition [17, 39, 12] +or more recently turn to deep neural network models for boosted performance [22, 38], +benefiting from the remarkable advancement of deep learning. As a consequence, they +fuel the releasing of public food datasets [14, 16, 29, 23]. Although these datasets are +highly useful in numerous studies, they lay emphasis on general food types or different +geographical regions and hence, may not be applicable to more specific areas, in our case, +Singapore. We note that localized food datasets are indispensable, as they manage to take +into account each country’s particular cuisines in ingredients, cooking styles, and varieties. +Therefore, the localized food datasets could unveil valuable insights into the corresponding +local human groups in dietary behaviors, which could ultimately cast light on their disease +development. +Focusing on Singapore, we collect and curate a localized Singaporean food dataset in a +systematic fashion, and the derived FoodSG-233 dataset is of high quality in terms of vol- +ume and diversity compared with its counterparts. Built upon this FoodSG-233 dataset, +we collaborate closely with different healthcare sectors, clinicians, and dietitians on food +recognition-based applications, and identify their shared requirements. We subsequently +design and establish a full-fledged platform FoodSG driven by effective food recognition +and dietary nutrition analysis, for supporting diverse data-intensive healthcare applica- +tions in Singapore. With the FoodSG platform as a service, we manage to bring huge +benefits to healthcare management in Singapore from various aspects. +Probing into the FoodSG-233 dataset, we observe two critical issues in this food dataset +that are prevalent in real-world food datasets exemplified in Figure 1: (i) intra-class varia- +tion meaning that the food images in the same category may exhibit high dissimilarity in +appearance, which gives rise to hard positive samples, and (ii) inter-class resemblance indi- +cating that some dishes may look similar to each other despite being in different categories, +which results in hard negative samples. The existence of such hard positive/negative sam- +ples tends to impede the food recognition approaches to achieve satisfactory performance. +To address this issue, we introduce supervised contrastive learning (SCL) that encour- +ages hard positive/negative sample mining into the food recognition task and propose the +FoodSG-SCL model for recognizing food categories accurately. We summarize our main +contributions below. +2 + +• Stemming from the shared requirements of diverse dietary management programs, we +design and develop the FoodSG platform as a service for food-related analytics. +• FoodSG is currently serving a variety of healthcare-oriented applications and healthy +living programs in Singapore, bringing benefits to the country and her people in the +long run. +• We present real use cases to highlight the importance of localized datasets to the health- +care management of specific countries, and collect, clean, and release a localized Singa- +porean food dataset FoodSG-233 with a systematic curation pipeline in order to foster +data management research in food computing. +• We identify the particular challenges of food images, namely intra-class variation and +inter-class resemblance, which result in hard positive/negative samples hindering the +food recognition performance. To address these, we introduce SCL into the model design +and devise the FoodSG-SCL model to facilitate the mining of hard positive/negative +samples for boosted performance. +• We conduct an extensive experimental evaluation to validate the effectiveness of FoodSG-SCL, +which confirms the accuracy of FoodSG-SCL and simultaneously offers practitioners +fresh insights into data-intensive healthcare applications. +Despite our focus on Singaporean food, our design of the FoodSG platform for accommo- +dating different applications, and our systematic data preparation pipeline can be used +for other countries’ food computing and related research, exhibiting enormous potential +in promoting similar research directions and improving healthcare. +2 +FoodSG Overview +In this section, we present an overview of FoodSG. The architecture of FoodSG’s platform +as a service (PaaS) for supporting various applications is illustrated in Figure 2. FoodSG +is designed for the cloud to enable adoption by local healthcare providers. We develop +its backend with node.js, and adopt Redis for caching. We use PostgreSQL as FoodSG’s +backend database system for users’ diet and exercise information collection and storage, +etc. We also rely on FoodSG’s backend server to schedule FoodSG’s core component - food +recognition driven by our devised FoodSG-SCL model for analytics. Specifically, on our +curated and released FoodSG-233 dataset, we train FoodSG-SCL integrating the Data +Augmentation Module, the Encoder Module, the Projection Module, and the Prediction +Module in a two-stage manner to gain the contrastive power of discriminating food images +and thus, generating accurate predictions. We shall elaborate on the FoodSG-233 dataset +and the FoodSG-SCL model in Section 4 and Section 5, respectively. Further, the data +stream is managed by Apache Kafka, and we adopt ejabberd [7] to embed the instant +messaging service in FoodSG for facilitating communication between patients and their +clinicians or dietitians. +Upwards, FoodSG supports diverse applications through Angular [6], which provides ap- +pealing and informative user interfaces. +In order to support different users’ terminals +covering browsers, iOS, and Android, we further introduce Ionic [10] for a consistent and +enjoyable user experience. Such a design provides the functionalities shown in Figure 3, +spanning dietary intake recording, nutrition information retrieval, communication within +the community, etc. +3 + +Webapp +(Browser-based) +Mobile App +(iOS, Android) +Client +(Ionic, AngularJS) +HTTP Server +(Nginx) +Chat Server +(ejabberd) +pm2 +(node.js process manager) +Backend Server +(node.js/express) +Cache +(Redis) +FoodSG DB +(PostgreSQL) +Image +Storage +Chat DB +(PostgreSQL) +Supported Applications +FoodSG PaaS +Food Recognition +FoodSG-233 +Projection +Prediction +Encoder +Data Augmentation +FoodSG-SCL +Kafka +Figure 2: Overview of FoodSG’s architecture. +In summary, the FoodSG platform is equipped with the capability and flexibility of incu- +bating diverse, multifunction applications with varied focuses and concerns. +3 +Application Scenarios +In face of the growing prevalence of the diabetes epidemic and its resultant enormous +burden on patients and economics, the Singapore Ministry of Health launched a campaign +to combat diabetes in April 2016. The goal of this campaign is to facilitate a shared whole- +of-nation effort to reduce the diabetes burden in Singapore [13, 32]. Afterward, diabetes +as well as hypertension, and hyperlipidemia are all considered in urgent need of attention +in Singapore [3]. +Motivated by this, we explore and investigate food image analysis aided by dietary nutri- +tion, as an infrastructure to assist in stopping or slowing the progression and complication +development in these prevalent diseases. +Developed up till now, our food recognition- +based diet tracking and management platform FoodSG has supported a broad range of +healthcare-oriented application scenarios in Singapore as a service. In this section, we +present four representative ongoing scenarios, spanning three different levels from clinical +medicine to healthcare, and further to public use. +3H prevention. 3H refers to (i) hyperglycemia (diabetes), (ii) hypertension (high blood +pressure), and (iii) hyperlipidemia (high cholesterol) [3]. If a person develops these 3H +problems without proper management, his/her lifetime risk of heart disease, stroke, kidney +failure, etc could be increased greatly, leading to long-term medical attention and financial +burden, and diminished quality of life. A practical strategy for combating 3H problems +4 + +(a) Diary functionality. +(b) Community functionality. +Figure 3: Screenshots of Food(lg) powered by FoodSG. +lies in advocating healthier lifestyles, e.g., to exercise more, to eat healthier food, so that +we can manage or even prevent 3H. +Let’s take hyperglycemia, i.e., diabetes as an example, the tackling of which is a top +priority in Singapore [2]. Since we believe that prevention is better than cure, we aim +to incentivize people to adopt healthy lifestyles. In the Lifestyle Intervention (LIVEN) +program in the Ng Teng Fong General Hospital in Singapore, we develop the JurongHealth +Food Log (JHFoodlg) app powered by FoodSG to urge prediabetic patients to make +behavioral lifestyle changes that are sustainable in the long term. We enroll the patients +who are diagnosed with prediabetes, which is a pre-diagnosis of diabetes, indicating that +the patient’s blood sugar exceeds a normal range, but not reaching diabetic levels. 14.4% +of Singapore’s population is affected by prediabetes, among whom one-third will progress +to type 2 diabetes in eight years eventually [1]. Fortunately, prediabetes is reversible with +long-term lifestyle changes, such as disciplined diets and exercises. +In the program, we require the patients to use JHFoodlg in their everyday life to record +their diets and exercises. +JHFoodlg harnesses state-of-the-art food recognition mod- +els as the driving force, detects the food type and analyses the corresponding nutrients +based on patients’ uploaded photos, compiles their diets and exercise amount per day +into a diary so that the patients can review their diet summaries against their weight-loss +goals and exercise targets. JHFoodlg also embeds a health coaching component, which +connects patients with the hospital’s dietitians and physiotherapists to provide valuable +information, from education on nutrition and fitness, to personalized feedback, and further +5 + +4:39 +Diary +口 +十 +W39123 Sep -29 Sep 2019 +Lll +-0 +MON +TUE +WED +THU +FRI +SAT +SUN +23 +24 +25 +26 +27 +28 +29 +Kcal +Total Fats +SaturatedFat +Sodium +Sugar +Carbohydrate +MONDAY +23rd +SatFat +35% +600kcal +31% +Sodium +38% +Sugars +6% +Fat·Carbs +Protein +28% +52% +21 % +1318kcal remaining +Lunch +Rice,withstewed beef +12:51 PM +600kcal +Plate-23cm (392g) × 1 +Fat +SatFat +Sodium +Sugars +Carb +18.4 +7.4 +764.4 +2.4 +77.6 +①(1) +TUESDAY +24th +SatFat +45% +855kcal +Sodium +园 +5 +Home +Diary +Capture +Fit:ness4:40 +Community +Foodlg +21mago +Meesiam +Plate-23cm (577g) X 1 +20.ChimMohRoad,SingaporeSingapore270020 +- +Mee Siam with freshly squeezed lime for a refreshing lunch. +Nice for a light hazy day. +meesiambeehoonspicy +Berry likes this. +Writeacomment. +Meow +LastMondayat1:35PM +Home +Diary +Capture +Fitness +Communityencouragement and recommendations. In this way, they can monitor the patients’ progress +and communicate with patients seamlessly via JHFoodlg, during the journey to reverse +prediabetes. +We obtain promising results from a 6-month study in the LIVEN program. Almost all the +patients who participate in the program experience a weight loss of between 4% to 5% of +their initial body weight supported by JHFoodlg [1]. As an effective and essential app +for 3H prevention in clinical medicine, we plan to roll out JHFoodlg for a larger cohort +of patients in the future. +Dental care management. We develop a dental care management system eDental [46] +powered by FoodSG to record, detect and analyze users’ daily diets for potential risk +factors for dental decay, with dentists and oral surgeons from the Faculty of Dentistry +in National University of Singapore and National University Hospital. eDental’s key +features include: (i) assisting users in logging their diets for comprehensive analysis, (ii) +monitoring with attractive user interfaces and informative visual reports, (iii) triggering +diet risk alerts for dental decay when necessary, (iv) incentivizing users to continue using +the system through goal setting functions, and (v) providing essential educational support +to users. Thereby, with eDental, we manage to facilitate dentists and patients to co- +manage patients’ daily dietary risk factors by analyzing their diet diaries. +This scenario demonstrates the applicability of FoodSG’s service among clinical medicine, +healthcare, and public use in that the end users cover dentists, their patients, and the +general public who wishes to practice healthy diets for preventing dental problems. We +are currently working on a user study to assess the effectiveness of eDental in dental +practice. +Athletes’ diet planning. Our FoodSG platform also provides services for Singapore +Sport Institute (SSI) [4], under Sport Singapore [5] the core purpose of which is to trans- +form Singapore through sports and encourage individuals to live better through sports. In +SSI, we have a more specific goal, i.e., we endeavor to provide the best support to athletes +so that they can achieve their full potential and further fulfill their sporting aspirations. +To achieve this, we turn to FoodSG for its capability of planning the athletes’ diets, guar- +anteeing their food diversity to achieve nutrient balance, and facilitating their exercise +accordingly to maintain appropriate body weights, body mass index, etc. On the basis +of the athletes’ diets and exercise, we can also suggest nutrients for them to supplement. +Therefore, supported by FoodSG, we are able to unlock the potential in the athletes and +help them pursue high-performance sports. In this scenario, we target to deliver high- +quality healthcare to athletes via FoodSG. We are currently working on a 22-month user +study to evaluate FoodSG’s positive influence on athletes’ sports careers. +Public use. In order to serve public users, we release a public version of FoodSG as +the “Food(lg)” app in iOS, Android, and Web browsers, establishing a uniform and +consistent user experience. We detect the food types from users’ journaled entries in images +or text via state-of-the-art food recognition models and calculate users’ daily nutrient +estimates against the standard nutritional guidelines and food composition data from the +Singapore Health Promotion Board (HPB) [9]. We also encourage users to exercise more, +share food photos, and comment on them. This social feature lets the users motivate one +another for maintaining a healthy lifestyle. In this way, Food(lg) helps general users to +achieve a well-balanced diet, train their physical fitness, and hence, improve their health +conditions and their quality of life in the long run. +6 + +In a nutshell, the aforementioned deployed scenarios showcase that our developed FoodSG +platform manages to cater to diverse applications while relying on a robust data collection +and curation pipeline that will be described in the next section. +4 +Data Collection and Curation +Localized food datasets capture the unique characteristics of each country’s dish varieties, +cooking styles, and food ingredients, and are therefore highly indispensable and medically +meaningful in contributing to people’s health management in the specific country. +In +this section, we elaborate on the data collection and curation process of our released +localized Singaporean food dataset FoodSG-233, which is crucial for food recognition as +in Figure 2. Although our released dataset is specific to Singaporean food dishes, our +proposed pipeline for data preparation is general and applicable to other types of food +dishes, fostering similar research in other countries. +4.1 +Data Collection +We incorporate popular ready-to-eat Singaporean food dishes by referring to HPB [9] for +both food groups and dishes’ names. HPB is a government organization committed to +promoting healthy living in Singapore, which provides a large database of local Singa- +porean food and their corresponding nutrition facts [8]. The database is hierarchically +structured with different coarse-grained food groups, each of which contains fine-grained +food categories, e.g., the group “Sugars, sweets and confectionery” contains categories +such as “Parfait” and “Popcorn”. To build a comprehensive Singaporean food dataset, we +select 233 popular ready-to-eat Singaporean dishes from 13 main food groups and then +crawl candidate images from different search engines. Further, to expand the range of +images retrieved, we increase the number of crawled images using a set of synonyms for +each food category in our queries. Finally, we collect 226, 809 images for further curation. +4.2 +Data Curation Pipeline +In the data curation process, we follow three rules of thumb. +• Release as many images as possible so that the released dataset can support different +users’ practical needs. +• Add necessary preprocessing to curate and clean the dataset in order to guarantee +that after cleaning, (i) the images are of high quality, and (ii) the food recognition +performance is satisfactory. +• Reduce human participation as much as possible, i.e., since manual efforts are generally +expensive, involve human participation only when necessary. +Driven by these principles above, we design a pipeline shown in Figure 4 to curate our +collected data and derive the FoodSG-233 dataset. Next, we introduce each curation +step in detail. +7 + +Data +Collection +FoodSG-233 +Dataset +Consistent +Formatting +Deduplication +Foodness +Classification +Human +Calibration +226,809 +images +226,791 +images +212,765 +images +209,861 +images +211,536 +images +Figure 4: Data curation pipeline to derive the FoodSG-233 dataset. +4.2.1 +Consistent Formatting +Given the 226, 809 images collected, we first convert their formats to JPEG, and then +remove 10 truncated images in the process. Meanwhile, we constrain that each image has +a minimum size of 32 × 32 to guarantee the quality of retrieved images, and thus remove +8 images thereafter. After consistent formatting, 226, 791 images are fed to the next step. +4.2.2 +Deduplication +We then remove exact or near duplicate images within each food category. Specifically, we +calculate three different image hashes for each image, namely average hashing, perceptual +hashing [28], and difference hashing. Based on the concatenation of these three hashes, +we filter out similar images and keep a single image with the largest size when there are +exact or near duplicate ones. We have 212, 765 images after deduplication. +4.2.3 +Foodness Classification +Due to the existence of non-food images in the dataset, we further build a separate classifier +to differentiate food and non-food images for filtering out the non-food ones, i.e., a foodness +classification model. Specifically, we construct the training dataset via collecting 189, 705 +non-food images from the Imagenet dataset [19] as negative images, and 226, 037 food +images from prevailing food datasets including VIREO Food-172 [16], UEC-Food256 [23] +and Food101 [14] as positive images. We then train a DenseNet169 model [21] for foodness +classification and test the model on the current curated dataset. In the meantime, we +conduct a first pass of manual checking to determine if each image is food, and such +manual labels are then utilized as the ground truth to evaluate the foodness model’s +performance. Through this foodness classification step, 211, 536 food images are passed +to the next step of the curation pipeline, with the foodness model yielding an accuracy of +82.9%. +4.2.4 +Human Calibration +To guarantee the data quality, we conduct a second pass of manual checking as human +calibration to correct the images that are assigned to the wrong food categories. We end +up with 209, 861 images, which is the FoodSG-233 dataset. +4.2.5 +FoodSG-233 +After curation, FoodSG-233 has at least 400 food images per food category. The sorted +distribution of food image number per food category in FoodSG-233 is illustrated in +8 + +Food category +0 +1000 +2000 +3000 +4000 +5000 +6000 +Image number +Figure 5: +Sorted distribution of +food image number per food cate- +gory in FoodSG-233. +0.60 +0.61 +0.62 +0.63 +0.64 +Perceptual distance +Food101 +VIREO Food-172 +UEC-Food100 +UEC-Food256 +FoodSG-233 +Figure 6: Diversity comparison in +the perceptual distance (the larger, +the more diverse). +Table 1: FoodSG-233 vs. Existing Food Datasets. +Dataset +Category # Image # +Area +Image #/Category +Food101 [14] +101 +101,000 +Western +1000 +VIREO Food-172 [16] +172 +110,241 +Chinese +641 +UEC-Food100 [29] +100 +14,361 +Japanese +144 +UEC-Food256 [23] +256 +25,088 +Multiple +98 +FoodSG-233 +233 +209,861 +Singaporean +901 +Figure 5, which tends to follow a power-law distribution, exhibiting the popularity of +different food categories in the real world. +4.3 +Comparison with Existing Datasets +We conduct a comparison between FoodSG-233 and several widely adopted food datasets +in Table 1. FoodSG-233 is superior in terms of the total image number compared with +its counterparts. In addition, as a localized Singaporean food dataset, FoodSG-233 has +a competitively large category number and image number per category. +As illustrated in Figure 1, the same food category tends to exhibit varied appearances +in the real world. Hence, it is desired to release a representative dataset that covers the +variations per food category and thereby, assists in building the food recognition model +to tackle all sorts of user-uploaded food images effectively. With this goal, FoodSG-233 +is constructed to contain food images with a high intra-class variation in terms of color, +shape, and texture in the collection process. To validate the superiority of FoodSG-233 in +this aspect, we conduct a quantitative diversity comparison from two perspectives below. +To start with, we employ a novel diversity measure based on a perceptual distance cal- +culated via deep visual representations [41]. We use the open-source model to calculate +the pairwise distance within each food category, and then average different categories’ +distances as the diversity metric. This metric has a value range of [0, 1], and the larger +the value is, the more diverse the dataset is in appearance. The comparison is shown in +Figure 6, where FoodSG-233 exhibits the highest value among all the datasets in the +perceptual distance, confirming its diversity. +9 + +1650 +1675 +1700 +Bak kuh +teh +Waffle +Parfait +Tortilla +Tacos and +nachos +Lossless JPEG file size (bytes) +UEC-Food256 +FoodSG-233 +(a) Diversity comparison +(b) FoodSG-233 +(c) UEC-Food256 +Figure 7: FoodSG-233 contains diversified images. (a) Diversity comparison in the loss- +less JPEG file sizes of five food categories’ average images, i.e., the smaller the value, the +more diverse. (b) Each compared food category’s example images and the average image +in FoodSG-233. (c) Each compared food category’s example images and the average +image in UEC-Food256. +We take a step further to calculate the average image of each food category, upon which +we can compute the lossless JPEG file size to measure the dataset diversity from another +perspective. This metric reflects the information amount in an image. The underlying +rationale is that a food category with more diversified images corresponds to an average +image that is vaguer, whereas a category with less diversified images corresponds to a +clearer average image, e.g., with a more structured shape or a sharper appearance [19]. +The average images are 256 × 256 and compressed into the lossless JPEG file format. +Therefore, a more diversified food image set should exhibit a smaller size of the average +image’s lossless JPEG. +We compare FoodSG-233 with UEC-Food256 which has the most shared food categories. +The results are illustrated in Figure 7, including the comparison in the lossless JPEG file +sizes of the average images for five food categories, the corresponding example food images, +and the average images, respectively. As shown, FoodSG-233 provides more diversified +food images than UEC-Food256. +5 +SCL-based Food Recognition +In this section, we introduce our proposed FoodSG-SCL model, which is essential to +facilitate effective food recognition in FoodSG as discussed in Section 2. +5.1 +Model Overview +The overview of FoodSG-SCL based on SCL [24] for food recognition is shown in Figure 8. +FoodSG-SCL consists of four modules for data augmentation, encoding, projection, and +prediction, respectively. The design rationale is that we first augment each data sam- +ple in the Data Augmentation Module to generate its multiple views, which facilitates +FoodSG-SCL to learn generalizable and robust representations. The augmented samples +10 + +Rico's Roomuie-ca-Outitflic +KTco +POCCPei CaoenHEN +TABASOkeropok.comSt.Patty'sDay +XLNIW +PUDDING PARFAITS +pandousdzais.comAVOO +ume +atsSoftmax +Encoder +Data Augmentation +Projection +Prediction ++SCL Loss +Stage1 +128-D +233-D +“Bak Kut Teh” +Stage2 +2048-D +Finetuned on +FoodSG-233 +Pretrained on +ImageNet +Figure 8: FoodSG-SCL model for effective food recognition. +are then input to the Encoder Module to unveil the underlying information in the aug- +mented samples, using the state-of-the-art models pretrained on the ImageNet dataset, +and then finetuning them on our FoodSG-233 dataset. The Encoder Module generates +a compact representation that is fed to both the Projection Module for further abstrac- +tion and integration of the SCL loss and the Prediction Module for providing the final +prediction result. +5.2 +Model Architecture +Data Augmentation Module Aug(·). Given a batch of data as input, we first adopt +two advanced data augmentation mechanisms to generate the “multiviewed batch” [24] +composed of 2N samples, where N is the original batch size. We denote this multiviewed +batch as K = {1, . . . , 2N}. As discussed above, such augmented data will be fed to the +Encoder Module for further modeling. +In contrastive learning (CL) [18], a contrastive loss is proposed to maximize the agreement +between each sample’s different augmented view representations in the latent space. For +instance, given an example xi, after the two data augmentations Aug1(·) and Aug2(·), we +have ˜xi = Aug1(xi), and ˜xρ(i) = Aug2(xi), where i and ρ(i) denote the two augmented +samples from the same source sample. +Hence, the index i corresponds to the anchor +sample, the positive sample set P = {ρ(i)} contains only one sample, and the remaining +samples constitute the negative sample set. +SCL, short for supervised contrastive learning [24], extends the CL formulation to include +the samples falling into the same class (with the same y label) as i in its positive sample +set: +P = {p|p ∈ K ∧ yp = yi ∧ p ̸= i} +(1) +In comparison, SCL extends the self-supervised setting in CL to a fully-supervised setting, +which effectively leverages the label information and meanwhile, contributes to mining +hard positive/negative samples for boosted performance. +11 + +Encoder Module Enc(·). We then employ a state-of-the-art neural network model as the +Encoder Module to extract the compact representations from the augmented data samples. +For a higher capacity of this module, we use the model pretrained on ImageNet, and then +finetune it on FoodSG-233, grounded in the widely-acknowledged ability of convolutional +neural networks in learning transferable features generalizable to diverse computer vision +tasks [40]. After encoding, each sample ˜xi is converted to: +ei = Enc(˜xi) +(2) +where ei ∈ Rde and de is the dimension size of the generated representation. de is set +to 2048 in our evaluation. +We note that ei is further normalized to fall in the unit +hypersphere in Rde as suggested in [24], i.e., ∥ei∥ = 1. Such normalization is considered +to bring performance improvement to prediction tasks. +Projection Module Pro(·). Next, we project the derived ei into a more compact rep- +resentation in this Projection Module: +si = Pro(ei) = ϕMLP (ei), ϕMLP : Rde �−→ Rdp +(3) +where a multi-layer perceptron (MLP) model ϕMLP with one hidden layer is for projecting +and abstracting the representation from de-dimensional to dp-dimensional. We set dp to +128 in the following experiments. Similarly, si is normalized as ∥si∥ = 1. +The SCL loss is then applied to pull positive samples close to each other and push them +away from negative samples. As shown in Figure 8, the integration of SCL corresponds to +“Stage1” in FoodSG-SCL, which learns a strong backbone for FoodSG-SCL in order to +support the food recognition task. The learned si is discarded at the end of Stage1, while +the output of the Encoder Module ei will be used for prediction. +Prediction Module Pre(·). We input ei to a linear prediction model ψ for food recog- +nition: +qi = Pre(ei) = ψ(ei), ψ : Rde �−→ R|C| +(4) +where C denotes the set of food categories for prediction, |C| = 233 in the FoodSG-233 +dataset. We further parameterize ˆyi = σ(qi), where σ(·) is the softmax function. The +derived ˆyi is the predicted probability distribution over |C| classes and will be used for +training FoodSG-SCL in the optimization process. Besides, this Prediction Module corre- +sponds to “Stage 2” as in Figure 8, the focus of which lies in employing the well-trained +FoodSG-SCL model for food recognition in an effective manner. +5.3 +Optimization +Stage1. The SCL loss for each sample ˜xi is: +lscl +i += − 1 +|P| +� +p∈P +log +exp(sim(si, sp)/T) +� +k∈K\{i} exp(sim(si, sk)/T) +(5) +The contrastive prediction task is: given a sample ˜xi, identify the positive samples in P out +of all remaining samples K \ {i}. In Equation 5, sim(·, ·) is the cosine similarity function, +i.e., sim(si, sp) = s⊤ +i sp/(∥si∥∥sp∥). As both representations are normalized, sim(si, sp) is +equivalent to s⊤ +i sp. Moreover, T denotes the temperature parameter. The overall SCL loss +in Stage1 sums all samples’ losses: +Lscl = +� +i∈K +lscl +i +(6) +12 + +With this loss function in Stage1, FoodSG-SCL (except the Prediction Module) can be +effectively trained via gradient-based optimizers such as stochastic gradient descent (SGD). +Stage2. For the food recognition task as a multi-class classification problem, we adopt +the cross-entropy loss in Stage2. +Lce = − 1 +|K| +� +i∈K +� +c∈C +yc +i log(ˆyc +i ) +(7) +With the Encoder Module’s output ei and the Prediction Module, we can readily train +the Predictor Module by optimizing this cross-entropy loss function. +Finally, after both stages, FoodSG-SCL is equipped with the contrastive power, con- +tributing to the learning of hard positive/negative samples and further to the prediction +performance. +6 +Experimental Evaluation +In this section, we first introduce the experimental set-up and then describe the experi- +mental results. After that, we analyze the experimental results and discuss the derived +insights for practitioners on using the FoodSG platform for their specific data-intensive +applications, which is imperative to the data management research community in setting +its foot in food computing and starting to play an active role. +6.1 +Experimental Set-up +Evaluated Models. +We select five state-of-the-art image models for evaluation, and +compare their performance against integrating them into FoodSG-SCL as the Encoder +Module, in order to validate the efficacy of SCL in FoodSG-SCL. +• ResNet50 [20] presents a residual learning framework that is easier to train and opti- +mize, bringing considerably improved accuracy on numerous visual recognition tasks. +• DenseNet169 [21] establishes direct connections between each layer and every other +layer in a feed-forward manner, which enables the model to scale to more layers easily +without causing optimization difficulties. +• MobileNetV2 [33] improves the efficiency of mobile models via an inverted residual +structure and emphasizes the importance of linear bottlenecks to maintain the repre- +sentation power. +• EfficientNetV2 [35] employs training-aware neural architecture search, model scaling, +and progressive learning, for accuracy, training speed, and parameter efficiency. +• ConvNeXt [27] retains the advantages of standard ConvNets in simplicity and effi- +ciency, but redesigns the components in a ConvNet towards the modern Transformer- +based models. Therefore, ConvNeXt achieves superior performance matching up to such +Transformer-based models in accuracy and scalability. +Implementation Details. The food recognition on FoodSG-233 is formulated as a 233- +class classification. We randomly split the food images per category into 80%, 10%, and +10% for training, validation, and testing, respectively. We adopt both the top-1 and the +13 + +65 +70 +75 +80 +Percentage (%) +Top-1 Accuracy +80 +85 +90 +95 +Top-5 Accuracy +ResNet50 + (25.6M) +DenseNet169 + (14.1M) +MobileNetV2 + (3.5M) +EfficientNetV2 + (21.5M) +ConvNeXt + (88.6M) +w/o SCL +w/ SCL +Figure 9: Comparison results of evaluated models w/o SCL and their advanced variants +w/ SCL in FoodSG-SCL. Each model’s parameter number is shown in brackets. +top-5 accuracy as evaluation metrics, and an accurate recognition model should yield high +results in both metrics. We select the hyperparameters that achieve the best performance +on validation data and apply such settings to the testing data for reporting the average +experimental results of three independent runs. +For each of the aforementioned evaluated models, we adopt its model weights pretrained +on ImageNet and then finetune them on FoodSG-233 for evaluation. We train these +models via SGD [34] with a learning rate of 0.05 and a weight decay of 0.0001. We adopt +batch size differently across different models, as a larger model corresponds to a smaller +batch and we set the batch size as large as possible within the device memory capacity. +Specifically, the batch size is 512 in EfficientNetV2, 1024 in DenseNet169 and ConvNeXt, +2048 in ResNet50 and MobileNetV2, respectively. +For validating the effectiveness of FoodSG-SCL, we integrate the evaluated models as the +Encoder Module of FoodSG-SCL, start with the same model parameters pretrained on +ImageNet and then finetune on FoodSG-233 with the other essential modules. SGD is +used as the optimizer with the learning rate set to 0.1 and the epoch number set to 200. +As FoodSG-SCL utilizes two views for each sample, its batch size needs to be halved +accordingly. +Experimental Environment. We conduct the experiments in a server with Intel(R) +Xeon(R) Gold 6248R × 2, 3.0GHz, 24 cores per chip. The server has 768GB memory, and +8 NVIDIA V100 with 300GB/s NVLINK. We implement the models with PyTorch 1.12.1. +6.2 +Experimental Results +In Figure 9, we illustrate the experimental results of the evaluated image models without +SCL and their respective advanced variants with SCL as well as other modules in our +proposed FoodSG-SCL. +As shown, the advanced methods with SCL outperform the ones without SCL in top-1 +accuracy by a large margin consistently, which confirms the effectiveness of the SCL mech- +anism. However, the integration of SCL leads to a slight degradation in top-5 accuracy. +The underlying reasons for these phenomena will be explained in Section 6.3 in detail. +Among the five evaluated models, EfficientNetV2 is the most accurate in terms of both +top-1 and top-5 accuracy. In addition, despite the parameter number being one magni- +tude smaller than the others, MobileNetV2 still achieves moderate performance in both +evaluation metrics. +14 + +a +b +c +d +e +0 +9 +Probability (%) +laksa +w/o SCL +w/ SCL +a. tom yum noodle soup +laksa +b. mee siam +c. lontong with sayur lodeh +d. mee rebus +e. satay bee hoon +(a) Laksa +a +b +c +d +e +0 +2 +Probability (%) +sushi +w/o SCL +w/ SCL +a. salmon - grilled +sushi +b. yuseng +c. rice, korean bulgogi beef +d. pung kueh +e. dumpling +(b) Sushi +Figure 10: Representative food categories and their top 5 misclassified categories. +To further investigate the intrinsic property of SCL in FoodSG-SCL, we probe into the +cases where food images are misclassified. +We choose the following two ground truth +food categories to explore: (i) the representative local food category “laksa” which is a +spicy noodle dish popular in Singapore, and (ii) “sushi” which is relatively more widely +known. For each ground truth category, we select the best-performing checkpoint among +all evaluated models without SCL, i.e., EfficientNetV2 without SCL (according to Figure 9), +and evaluate it on all the testing samples to derive the predicted probability distribution +per testing sample. We then calculate the average of all testing samples’ results, based +on which we illustrate the top 5 misclassified categories and their respective probabilities +predicted. For comparison, we display the results of the same five food categories derived +by the most accurate model checkpoint of EfficientNetV2 with SCL as well. +The experimental results of both ground truth categories, including the predicted proba- +bilities of their top 5 misclassified categories and the corresponding example food images +are shown in Figure 10. For both laksa and sushi, the predicted top 5 misclassified cat- +egories generated by EfficientNetV2 without SCL are highly similar to the ground truth +category in terms of color, texture, and shape. On the contrary, EfficientNetV2 with SCL +effectively reduces the predicted probabilities of these five misclassified categories in both +cases, the reason for which will be explicated in Section 6.3. +6.3 +Discussions +Top-1 accuracy vs. +top-5 accuracy. +Generally, the top 1 predicted class is used +as the final result. +However, in practice, the top 1 prediction may not be completely +15 + +OTV SMITHOWHATTOCOOKTODAY.COMcorrect. To alleviate this issue, sometimes users are offered a set of five food category +candidates and allowed to select the correct one with “one more touch”. If the true food +category still cannot be located within the candidate set, then the user experience will be +negatively affected. In this sense, both top-1 accuracy and top-5 accuracy are crucial in +food recognition, and they depict how accurately we can recognize the food dishes on the +released FoodSG-233 dataset. +In Figure 9, the highest top-1 accuracy is 78.87% yielded by EfficientNetV2 with SCL. +Besides, EfficientNetV2 without SCL performs best in top-5 accuracy, i.e., 94.06%, whereas +EfficientNetV2 with SCL achieves 93.13% with a minor decrease. This means that in most +cases, the returned five candidate food categories successfully contain the ground truth, +and users could select the correct one with one more touch conveniently. Both findings +validate the effectiveness of FoodSG-SCL for food recognition. +SCL in food recognition. Food recognition is challenging in that certain food categories +highly resemble each other and it could be difficult to distinguish between them even for +humans. +Furthermore, such subtle differences in the appearance of the two categories +may correspond to a large difference in their constituent nutrients, which will have a +huge impact on diet assessment. This consequently bolsters the demand for accurate food +recognition when handling visually similar food images. +SCL is able to meet this demand. Specifically, SCL’s underlying principle lies in drawing +samples with the same label close while separating them away from samples with other +labels. This in nature improves top-1 accuracy, which is shown in Figure 9 where the +integration of SCL brings performance benefits consistently. +However, such benefits of SCL do not come for free. The models without SCL adopt the +standard cross-entropy loss and generate the top 5 categories that are the most similar to +the ground truth category in appearance shown in Figure 10, due to the representation +learning capability of convolutional neural networks. +Different from them, SCL-based +models tend to push the negative samples with different labels away from the positive +samples, in spite of their resemblance in appearance. Therefore, with SCL, the predicted +probabilities of the same five similar categories will be reduced to a large extent illustrated +in Figure 10, resulting in slightly compromised top-5 accuracy as observed in Figure 9. +According to our real-world practice, we firmly believe that top-1 accuracy should outweigh +top-5 accuracy in food recognition regardless of whether we allow users to further select +from a candidate category set. Hence, we introduce the SCL mechanism into FoodSG-SCL +to reduce the disturbing influence exerted by similar yet incorrect categories, providing +excellent gain in top-1 accuracy. +Edge Computing. In healthcare-oriented applications, there are certain scenarios where +users’ data cannot be processed in a centralized manner, e.g., users are not willing to +upload their food photos or share their nutrition intake information for privacy concerns. +This requires the computation involved in food recognition to be performed on edge de- +vices, such as mobile phones, which is thus affected by the computing capacity and space +limitation of the devices. Under such scenarios, more lightweight models are preferred as +the Encoder Module of FoodSG-SCL. For instance, MobileNetV2 exhibits a prominent +advantage of parameter number being one magnitude smaller than other models, at the +expense of accuracy decreasing from 78.87% (by EfficientNetV2 with SCL) to 70.55%. +16 + +7 +Related Work +As an interdisciplinary area, food computing targets to apply computational methods to +analyze heterogeneous sources of food data for supporting diverse food-related investi- +gations in healthcare, gastronomy, and agronomy [30]. Among the involved tasks, food +recognition is a cornerstone, which predicts the food items contained in food images. +Prior studies in the early stage exploit hand-crafted features for food recognition, such as +color histograms and SIFT features [17], statistics of pairwise local features [39], and bag of +features [12]. Recently, due to the record-breaking performance achieved by deep learning +models in numerous areas such as computer vision [25, 31, 36], various convolutional neural +network models [22, 38] are employed in food recognition for extracting the features in +food images and hence, delivering more satisfactory recognition performance than the +traditional approaches based on hand-crafted features. +In the meantime, a large number of food-related datasets are released to further promote +the development of food computing. For example, Food101 [14] is constructed to cover +101 food categories (mostly western), containing 101, 000 images in total. Different from +Food101, VIREO Food-172 [16] turns the focus to Chinese food and includes 172 food +categories, and 110, 241 images, respectively. Moreover, a Japanese food dataset UEC- +Food100 [29] is built to incorporate 100 popular categories. This UEC-Food100 dataset +is then expanded to UEC-Food256 [23] introducing more categories from other countries, +namely French, Italian, American, Chinese, Thai, Vietnamese, and Indonesian. +These +released multifarious food-related datasets present researchers with ample opportunity +for food computing and analysis. However, these datasets have their specific geographic +areas and therefore, only include particular types of cuisines. Different from existing work, +driven by FoodSG, we investigate Singaporean food dishes and release the localized dataset +FoodSG-233, which is of vital significance to FoodSG’s provided healthcare services for +Singaporeans. +8 +Conclusions +Focusing on the cuisines and healthcare-oriented applications in Singapore, we abstract the +shared requirements among diverse applications and develop the dietary nutrition-aided +platform FoodSG as a service to support all sorts of application scenarios in Singapore. +We collect, curate, and release a Singaporean food dataset FoodSG-233 systematically to +promote future research directions for the data management community in food computing. +Through delving into FoodSG-233 to analyze its issues of intra-class dissimilarity and +inter-class similarity, we propose to integrate SCL into food recognition and devise the +FoodSG-SCL model accordingly to facilitate the learning from hard positive/negative +samples for performance gains. 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In CIKM, pages 1579–1588. +ACM, 2017. +20 + diff --git a/Q9E2T4oBgHgl3EQfWAeQ/content/tmp_files/load_file.txt b/Q9E2T4oBgHgl3EQfWAeQ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d9f3f600b6a0ba06bc2ecf2545e4cfabd7b9a16d --- /dev/null +++ b/Q9E2T4oBgHgl3EQfWAeQ/content/tmp_files/load_file.txt @@ -0,0 +1,819 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf,len=818 +page_content='A Dietary Nutrition-aided Healthcare Platform via Effective Food Recognition on a Localized Singaporean Food Dataset Kaiping Zheng1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Thao Nguyen1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Jesslyn Hwei Sing Chong2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Charlene Enhui Goh3 Melanie Herschel4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Hee Hoon Lee5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Beng Chin Ooi1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Wei Wang6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' James Yip7 1School of Computing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' National University of Singapore 2Ng Teng Fong General Hospital,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Dietetics and Nutrition Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Singapore 3Faculty of Dentistry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' National University of Singapore 4Universit¨at Stuttgart,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Germany 5Ng Teng Fong General Hospital,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Allied Health Division,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Singapore 6TikTok Singapore 7National University Heart Centre,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Singapore Abstract Localized food datasets have profound meaning in revealing a country’s special cuisines to explore people’s dietary behaviors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' which will shed light on their health conditions and disease development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' In this paper, revolving around the demand for accurate food recognition in Singapore, we develop the FoodSG platform to incu- bate diverse healthcare-oriented applications as a service in Singapore, taking into account their shared requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We release a localized Singaporean food dataset FoodSG-233 with a systematic cleaning and curation pipeline for promoting future data management research in food computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' To overcome the hurdle in recognition performance brought by Singaporean multifarious food dishes, we propose to inte- grate supervised contrastive learning into our food recognition model FoodSG-SCL for the intrinsic capability to mine hard positive/negative samples and therefore boost the accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Through a comprehensive evaluation, we share the insightful experi- ence with practitioners in the data management community regarding food-related data-intensive healthcare applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' The FoodSG-233 dataset can be accessed via: https://foodlg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='nus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='sg/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' 1 Introduction Healthcare analytics aims to conduct numerous analytic tasks with a broad range of health- care data spanning cost and claim data, pharmaceutical data, research and development data, clinical data, and patients’ behavior and sentiment data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Based upon these, health- care analytics serves as a basis for different data-intensive healthcare applications, such as chronic disease progression modeling [37, 45, 47, 44], and disease diagnosis [15, 26, 43, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Among the diverse data sources for healthcare analytics, large-scale food data plays a significant role in revealing the dietary information and knowledge of patients, posing a profound impact on humans’ health and well-being [30, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' For instance, long-term con- sumption of unhealthy foods increases the risk of developing diseases such as diabetes, hypertension, and hyperlipidemia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Therefore, there is an increasing demand for food data for diet assessment and disease management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Naturally, food computing emerges as a subfield of computing, being accorded for its importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' It involves computational methods to analyze heterogeneous food data for solving disparate food-related problems and thereby, contributes to diverse data-intensive 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='03829v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='LG] 10 Jan 2023 salmon - grilled yuseng (raw fish salad) sushi sushi sushi intra-class variation inter-class resemblance Figure 1: Two common issues in real-world food datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' healthcare applications [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Food recognition, which identifies food types from food im- ages, is a cornerstone as it enables the effective assessment of people’s diets whereby it provides much-needed intake information for disease prevention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Notably, combined with nutrition information, many other food-related applications become possible, such as es- timating the intake of calories and nutrients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Prior studies either make use of hand-crafted features for food recognition [17, 39, 12] or more recently turn to deep neural network models for boosted performance [22, 38], benefiting from the remarkable advancement of deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' As a consequence, they fuel the releasing of public food datasets [14, 16, 29, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Although these datasets are highly useful in numerous studies, they lay emphasis on general food types or different geographical regions and hence, may not be applicable to more specific areas, in our case, Singapore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We note that localized food datasets are indispensable, as they manage to take into account each country’s particular cuisines in ingredients, cooking styles, and varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Therefore, the localized food datasets could unveil valuable insights into the corresponding local human groups in dietary behaviors, which could ultimately cast light on their disease development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Focusing on Singapore, we collect and curate a localized Singaporean food dataset in a systematic fashion, and the derived FoodSG-233 dataset is of high quality in terms of vol- ume and diversity compared with its counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Built upon this FoodSG-233 dataset, we collaborate closely with different healthcare sectors, clinicians, and dietitians on food recognition-based applications, and identify their shared requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We subsequently design and establish a full-fledged platform FoodSG driven by effective food recognition and dietary nutrition analysis, for supporting diverse data-intensive healthcare applica- tions in Singapore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' With the FoodSG platform as a service, we manage to bring huge benefits to healthcare management in Singapore from various aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Probing into the FoodSG-233 dataset,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' we observe two critical issues in this food dataset that are prevalent in real-world food datasets exemplified in Figure 1: (i) intra-class varia- tion meaning that the food images in the same category may exhibit high dissimilarity in appearance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' which gives rise to hard positive samples,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' and (ii) inter-class resemblance indi- cating that some dishes may look similar to each other despite being in different categories,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' which results in hard negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' The existence of such hard positive/negative sam- ples tends to impede the food recognition approaches to achieve satisfactory performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' To address this issue, we introduce supervised contrastive learning (SCL) that encour- ages hard positive/negative sample mining into the food recognition task and propose the FoodSG-SCL model for recognizing food categories accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We summarize our main contributions below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' 2 Stemming from the shared requirements of diverse dietary management programs, we design and develop the FoodSG platform as a service for food-related analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' FoodSG is currently serving a variety of healthcare-oriented applications and healthy living programs in Singapore, bringing benefits to the country and her people in the long run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We present real use cases to highlight the importance of localized datasets to the health- care management of specific countries, and collect, clean, and release a localized Singa- porean food dataset FoodSG-233 with a systematic curation pipeline in order to foster data management research in food computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We identify the particular challenges of food images, namely intra-class variation and inter-class resemblance, which result in hard positive/negative samples hindering the food recognition performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' To address these, we introduce SCL into the model design and devise the FoodSG-SCL model to facilitate the mining of hard positive/negative samples for boosted performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We conduct an extensive experimental evaluation to validate the effectiveness of FoodSG-SCL, which confirms the accuracy of FoodSG-SCL and simultaneously offers practitioners fresh insights into data-intensive healthcare applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Despite our focus on Singaporean food, our design of the FoodSG platform for accommo- dating different applications, and our systematic data preparation pipeline can be used for other countries’ food computing and related research, exhibiting enormous potential in promoting similar research directions and improving healthcare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' 2 FoodSG Overview In this section, we present an overview of FoodSG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' The architecture of FoodSG’s platform as a service (PaaS) for supporting various applications is illustrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' FoodSG is designed for the cloud to enable adoption by local healthcare providers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We develop its backend with node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='js, and adopt Redis for caching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We use PostgreSQL as FoodSG’s backend database system for users’ diet and exercise information collection and storage, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We also rely on FoodSG’s backend server to schedule FoodSG’s core component - food recognition driven by our devised FoodSG-SCL model for analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Specifically, on our curated and released FoodSG-233 dataset, we train FoodSG-SCL integrating the Data Augmentation Module, the Encoder Module, the Projection Module, and the Prediction Module in a two-stage manner to gain the contrastive power of discriminating food images and thus, generating accurate predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We shall elaborate on the FoodSG-233 dataset and the FoodSG-SCL model in Section 4 and Section 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Further, the data stream is managed by Apache Kafka, and we adopt ejabberd [7] to embed the instant messaging service in FoodSG for facilitating communication between patients and their clinicians or dietitians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Upwards, FoodSG supports diverse applications through Angular [6], which provides ap- pealing and informative user interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' In order to support different users’ terminals covering browsers, iOS, and Android, we further introduce Ionic [10] for a consistent and enjoyable user experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Such a design provides the functionalities shown in Figure 3, spanning dietary intake recording, nutrition information retrieval, communication within the community, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' 3 Webapp (Browser-based) Mobile App (iOS, Android) Client (Ionic, AngularJS) HTTP Server (Nginx) Chat Server (ejabberd) pm2 (node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='js process manager) Backend Server (node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='js/express) Cache (Redis) FoodSG DB (PostgreSQL) Image Storage Chat DB (PostgreSQL) Supported Applications FoodSG PaaS Food Recognition FoodSG-233 Projection Prediction Encoder Data Augmentation FoodSG-SCL Kafka Figure 2: Overview of FoodSG’s architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' In summary, the FoodSG platform is equipped with the capability and flexibility of incu- bating diverse, multifunction applications with varied focuses and concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' 3 Application Scenarios In face of the growing prevalence of the diabetes epidemic and its resultant enormous burden on patients and economics, the Singapore Ministry of Health launched a campaign to combat diabetes in April 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' The goal of this campaign is to facilitate a shared whole- of-nation effort to reduce the diabetes burden in Singapore [13, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Afterward, diabetes as well as hypertension, and hyperlipidemia are all considered in urgent need of attention in Singapore [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Motivated by this, we explore and investigate food image analysis aided by dietary nutri- tion, as an infrastructure to assist in stopping or slowing the progression and complication development in these prevalent diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Developed up till now, our food recognition- based diet tracking and management platform FoodSG has supported a broad range of healthcare-oriented application scenarios in Singapore as a service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' In this section, we present four representative ongoing scenarios, spanning three different levels from clinical medicine to healthcare, and further to public use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' 3H prevention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' 3H refers to (i) hyperglycemia (diabetes), (ii) hypertension (high blood pressure), and (iii) hyperlipidemia (high cholesterol) [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' If a person develops these 3H problems without proper management, his/her lifetime risk of heart disease, stroke, kidney failure, etc could be increased greatly, leading to long-term medical attention and financial burden, and diminished quality of life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' A practical strategy for combating 3H problems 4 (a) Diary functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' (b) Community functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Figure 3: Screenshots of Food(lg) powered by FoodSG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' lies in advocating healthier lifestyles, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=', to exercise more, to eat healthier food, so that we can manage or even prevent 3H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Let’s take hyperglycemia, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=', diabetes as an example, the tackling of which is a top priority in Singapore [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Since we believe that prevention is better than cure, we aim to incentivize people to adopt healthy lifestyles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' In the Lifestyle Intervention (LIVEN) program in the Ng Teng Fong General Hospital in Singapore, we develop the JurongHealth Food Log (JHFoodlg) app powered by FoodSG to urge prediabetic patients to make behavioral lifestyle changes that are sustainable in the long term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We enroll the patients who are diagnosed with prediabetes, which is a pre-diagnosis of diabetes, indicating that the patient’s blood sugar exceeds a normal range, but not reaching diabetic levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='4% of Singapore’s population is affected by prediabetes, among whom one-third will progress to type 2 diabetes in eight years eventually [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Fortunately, prediabetes is reversible with long-term lifestyle changes, such as disciplined diets and exercises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' In the program, we require the patients to use JHFoodlg in their everyday life to record their diets and exercises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' JHFoodlg harnesses state-of-the-art food recognition mod- els as the driving force, detects the food type and analyses the corresponding nutrients based on patients’ uploaded photos, compiles their diets and exercise amount per day into a diary so that the patients can review their diet summaries against their weight-loss goals and exercise targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' JHFoodlg also embeds a health coaching component,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' which connects patients with the hospital’s dietitians and physiotherapists to provide valuable information,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' from education on nutrition and fitness,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' to personalized feedback,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' and further 5 4:39 Diary 口 十 W39123 Sep -29 Sep 2019 Lll 0 MON TUE WED THU FRI SAT SUN 23 24 25 26 27 28 29 Kcal Total Fats SaturatedFat Sodium Sugar Carbohydrate MONDAY 23rd SatFat 35% 600kcal 31% Sodium 38% Sugars 6% Fat·Carbs Protein 28% 52% 21 % 1318kcal remaining Lunch Rice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='withstewed beef 12:51 PM 600kcal Plate-23cm (392g) × 1 Fat SatFat Sodium Sugars Carb 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='4 764.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='4 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='6 ①(1) TUESDAY 24th SatFat 45% 855kcal Sodium 园 5 Home Diary Capture Fit:ness4:40 Community Foodlg 21mago Meesiam Plate-23cm (577g) X 1 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='ChimMohRoad,SingaporeSingapore270020 Mee Siam with freshly squeezed lime for a refreshing lunch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Nice for a light hazy day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' meesiambeehoonspicy Berry likes this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Writeacomment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Meow LastMondayat1:35PM Home Diary Capture Fitness Communityencouragement and recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' In this way, they can monitor the patients’ progress and communicate with patients seamlessly via JHFoodlg, during the journey to reverse prediabetes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We obtain promising results from a 6-month study in the LIVEN program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Almost all the patients who participate in the program experience a weight loss of between 4% to 5% of their initial body weight supported by JHFoodlg [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' As an effective and essential app for 3H prevention in clinical medicine, we plan to roll out JHFoodlg for a larger cohort of patients in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Dental care management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We develop a dental care management system eDental [46] powered by FoodSG to record, detect and analyze users’ daily diets for potential risk factors for dental decay, with dentists and oral surgeons from the Faculty of Dentistry in National University of Singapore and National University Hospital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' eDental’s key features include: (i) assisting users in logging their diets for comprehensive analysis, (ii) monitoring with attractive user interfaces and informative visual reports, (iii) triggering diet risk alerts for dental decay when necessary, (iv) incentivizing users to continue using the system through goal setting functions, and (v) providing essential educational support to users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Thereby, with eDental, we manage to facilitate dentists and patients to co- manage patients’ daily dietary risk factors by analyzing their diet diaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' This scenario demonstrates the applicability of FoodSG’s service among clinical medicine, healthcare, and public use in that the end users cover dentists, their patients, and the general public who wishes to practice healthy diets for preventing dental problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We are currently working on a user study to assess the effectiveness of eDental in dental practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Athletes’ diet planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Our FoodSG platform also provides services for Singapore Sport Institute (SSI) [4], under Sport Singapore [5] the core purpose of which is to trans- form Singapore through sports and encourage individuals to live better through sports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' In SSI, we have a more specific goal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=', we endeavor to provide the best support to athletes so that they can achieve their full potential and further fulfill their sporting aspirations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' To achieve this, we turn to FoodSG for its capability of planning the athletes’ diets, guar- anteeing their food diversity to achieve nutrient balance, and facilitating their exercise accordingly to maintain appropriate body weights, body mass index, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' On the basis of the athletes’ diets and exercise, we can also suggest nutrients for them to supplement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Therefore, supported by FoodSG, we are able to unlock the potential in the athletes and help them pursue high-performance sports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' In this scenario, we target to deliver high- quality healthcare to athletes via FoodSG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We are currently working on a 22-month user study to evaluate FoodSG’s positive influence on athletes’ sports careers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Public use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' In order to serve public users, we release a public version of FoodSG as the “Food(lg)” app in iOS, Android, and Web browsers, establishing a uniform and consistent user experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We detect the food types from users’ journaled entries in images or text via state-of-the-art food recognition models and calculate users’ daily nutrient estimates against the standard nutritional guidelines and food composition data from the Singapore Health Promotion Board (HPB) [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We also encourage users to exercise more, share food photos, and comment on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' This social feature lets the users motivate one another for maintaining a healthy lifestyle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' In this way, Food(lg) helps general users to achieve a well-balanced diet, train their physical fitness, and hence, improve their health conditions and their quality of life in the long run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' 6 In a nutshell, the aforementioned deployed scenarios showcase that our developed FoodSG platform manages to cater to diverse applications while relying on a robust data collection and curation pipeline that will be described in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' 4 Data Collection and Curation Localized food datasets capture the unique characteristics of each country’s dish varieties, cooking styles, and food ingredients, and are therefore highly indispensable and medically meaningful in contributing to people’s health management in the specific country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' In this section, we elaborate on the data collection and curation process of our released localized Singaporean food dataset FoodSG-233, which is crucial for food recognition as in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Although our released dataset is specific to Singaporean food dishes, our proposed pipeline for data preparation is general and applicable to other types of food dishes, fostering similar research in other countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='1 Data Collection We incorporate popular ready-to-eat Singaporean food dishes by referring to HPB [9] for both food groups and dishes’ names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' HPB is a government organization committed to promoting healthy living in Singapore, which provides a large database of local Singa- porean food and their corresponding nutrition facts [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' The database is hierarchically structured with different coarse-grained food groups, each of which contains fine-grained food categories, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=', the group “Sugars, sweets and confectionery” contains categories such as “Parfait” and “Popcorn”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' To build a comprehensive Singaporean food dataset, we select 233 popular ready-to-eat Singaporean dishes from 13 main food groups and then crawl candidate images from different search engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Further, to expand the range of images retrieved, we increase the number of crawled images using a set of synonyms for each food category in our queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Finally, we collect 226, 809 images for further curation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='2 Data Curation Pipeline In the data curation process, we follow three rules of thumb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Release as many images as possible so that the released dataset can support different users’ practical needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Add necessary preprocessing to curate and clean the dataset in order to guarantee that after cleaning, (i) the images are of high quality, and (ii) the food recognition performance is satisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Reduce human participation as much as possible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=', since manual efforts are generally expensive, involve human participation only when necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Driven by these principles above, we design a pipeline shown in Figure 4 to curate our collected data and derive the FoodSG-233 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Next, we introduce each curation step in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' 7 Data Collection FoodSG-233 Dataset Consistent Formatting Deduplication Foodness Classification Human Calibration 226,809 images 226,791 images 212,765 images 209,861 images 211,536 images Figure 4: Data curation pipeline to derive the FoodSG-233 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='1 Consistent Formatting Given the 226, 809 images collected, we first convert their formats to JPEG, and then remove 10 truncated images in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Meanwhile, we constrain that each image has a minimum size of 32 × 32 to guarantee the quality of retrieved images, and thus remove 8 images thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' After consistent formatting, 226, 791 images are fed to the next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='2 Deduplication We then remove exact or near duplicate images within each food category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Specifically, we calculate three different image hashes for each image, namely average hashing, perceptual hashing [28], and difference hashing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Based on the concatenation of these three hashes, we filter out similar images and keep a single image with the largest size when there are exact or near duplicate ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We have 212, 765 images after deduplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='3 Foodness Classification Due to the existence of non-food images in the dataset, we further build a separate classifier to differentiate food and non-food images for filtering out the non-food ones, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=', a foodness classification model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Specifically, we construct the training dataset via collecting 189, 705 non-food images from the Imagenet dataset [19] as negative images, and 226, 037 food images from prevailing food datasets including VIREO Food-172 [16], UEC-Food256 [23] and Food101 [14] as positive images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We then train a DenseNet169 model [21] for foodness classification and test the model on the current curated dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' In the meantime, we conduct a first pass of manual checking to determine if each image is food, and such manual labels are then utilized as the ground truth to evaluate the foodness model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Through this foodness classification step, 211, 536 food images are passed to the next step of the curation pipeline, with the foodness model yielding an accuracy of 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='9%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='4 Human Calibration To guarantee the data quality, we conduct a second pass of manual checking as human calibration to correct the images that are assigned to the wrong food categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We end up with 209, 861 images, which is the FoodSG-233 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='5 FoodSG-233 After curation, FoodSG-233 has at least 400 food images per food category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' The sorted distribution of food image number per food category in FoodSG-233 is illustrated in 8 Food category 0 1000 2000 3000 4000 5000 6000 Image number Figure 5: Sorted distribution of food image number per food cate- gory in FoodSG-233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='64 Perceptual distance Food101 VIREO Food-172 UEC-Food100 UEC-Food256 FoodSG-233 Figure 6: Diversity comparison in the perceptual distance (the larger, the more diverse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Table 1: FoodSG-233 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Existing Food Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Dataset Category # Image # Area Image #/Category Food101 [14] 101 101,000 Western 1000 VIREO Food-172 [16] 172 110,241 Chinese 641 UEC-Food100 [29] 100 14,361 Japanese 144 UEC-Food256 [23] 256 25,088 Multiple 98 FoodSG-233 233 209,861 Singaporean 901 Figure 5, which tends to follow a power-law distribution, exhibiting the popularity of different food categories in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='3 Comparison with Existing Datasets We conduct a comparison between FoodSG-233 and several widely adopted food datasets in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' FoodSG-233 is superior in terms of the total image number compared with its counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' In addition, as a localized Singaporean food dataset, FoodSG-233 has a competitively large category number and image number per category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' As illustrated in Figure 1, the same food category tends to exhibit varied appearances in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Hence, it is desired to release a representative dataset that covers the variations per food category and thereby, assists in building the food recognition model to tackle all sorts of user-uploaded food images effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' With this goal, FoodSG-233 is constructed to contain food images with a high intra-class variation in terms of color, shape, and texture in the collection process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' To validate the superiority of FoodSG-233 in this aspect, we conduct a quantitative diversity comparison from two perspectives below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' To start with, we employ a novel diversity measure based on a perceptual distance cal- culated via deep visual representations [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We use the open-source model to calculate the pairwise distance within each food category, and then average different categories’ distances as the diversity metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' This metric has a value range of [0, 1], and the larger the value is, the more diverse the dataset is in appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' The comparison is shown in Figure 6, where FoodSG-233 exhibits the highest value among all the datasets in the perceptual distance, confirming its diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' 9 1650 1675 1700 Bak kuh teh Waffle Parfait Tortilla Tacos and nachos Lossless JPEG file size (bytes) UEC-Food256 FoodSG-233 (a) Diversity comparison (b) FoodSG-233 (c) UEC-Food256 Figure 7: FoodSG-233 contains diversified images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' (a) Diversity comparison in the loss- less JPEG file sizes of five food categories’ average images, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=', the smaller the value, the more diverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' (b) Each compared food category’s example images and the average image in FoodSG-233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' (c) Each compared food category’s example images and the average image in UEC-Food256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We take a step further to calculate the average image of each food category, upon which we can compute the lossless JPEG file size to measure the dataset diversity from another perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' This metric reflects the information amount in an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' The underlying rationale is that a food category with more diversified images corresponds to an average image that is vaguer, whereas a category with less diversified images corresponds to a clearer average image, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=', with a more structured shape or a sharper appearance [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' The average images are 256 × 256 and compressed into the lossless JPEG file format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Therefore, a more diversified food image set should exhibit a smaller size of the average image’s lossless JPEG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We compare FoodSG-233 with UEC-Food256 which has the most shared food categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' The results are illustrated in Figure 7, including the comparison in the lossless JPEG file sizes of the average images for five food categories, the corresponding example food images, and the average images, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' As shown, FoodSG-233 provides more diversified food images than UEC-Food256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' 5 SCL-based Food Recognition In this section, we introduce our proposed FoodSG-SCL model, which is essential to facilitate effective food recognition in FoodSG as discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='1 Model Overview The overview of FoodSG-SCL based on SCL [24] for food recognition is shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' FoodSG-SCL consists of four modules for data augmentation, encoding, projection, and prediction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' The design rationale is that we first augment each data sam- ple in the Data Augmentation Module to generate its multiple views, which facilitates FoodSG-SCL to learn generalizable and robust representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=" The augmented samples 10 Rico's Roomuie-ca-Outitflic KTco POCCPei CaoenHEN TABASOkeropok." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='comSt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content="Patty'sDay XLNIW PUDDING PARFAITS pandousdzais." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='comAVOO ume atsSoftmax Encoder Data Augmentation Projection Prediction +SCL Loss Stage1 128-D 233-D “Bak Kut Teh” Stage2 2048-D Finetuned on FoodSG-233 Pretrained on ImageNet Figure 8: FoodSG-SCL model for effective food recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' are then input to the Encoder Module to unveil the underlying information in the aug- mented samples, using the state-of-the-art models pretrained on the ImageNet dataset, and then finetuning them on our FoodSG-233 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' The Encoder Module generates a compact representation that is fed to both the Projection Module for further abstrac- tion and integration of the SCL loss and the Prediction Module for providing the final prediction result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='2 Model Architecture Data Augmentation Module Aug(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Given a batch of data as input, we first adopt two advanced data augmentation mechanisms to generate the “multiviewed batch” [24] composed of 2N samples, where N is the original batch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We denote this multiviewed batch as K = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' , 2N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' As discussed above, such augmented data will be fed to the Encoder Module for further modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' In contrastive learning (CL) [18], a contrastive loss is proposed to maximize the agreement between each sample’s different augmented view representations in the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' For instance, given an example xi, after the two data augmentations Aug1(·) and Aug2(·), we have ˜xi = Aug1(xi), and ˜xρ(i) = Aug2(xi), where i and ρ(i) denote the two augmented samples from the same source sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Hence, the index i corresponds to the anchor sample, the positive sample set P = {ρ(i)} contains only one sample, and the remaining samples constitute the negative sample set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' SCL, short for supervised contrastive learning [24], extends the CL formulation to include the samples falling into the same class (with the same y label) as i in its positive sample set: P = {p|p ∈ K ∧ yp = yi ∧ p ̸= i} (1) In comparison, SCL extends the self-supervised setting in CL to a fully-supervised setting, which effectively leverages the label information and meanwhile, contributes to mining hard positive/negative samples for boosted performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' 11 Encoder Module Enc(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We then employ a state-of-the-art neural network model as the Encoder Module to extract the compact representations from the augmented data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' For a higher capacity of this module, we use the model pretrained on ImageNet, and then finetune it on FoodSG-233, grounded in the widely-acknowledged ability of convolutional neural networks in learning transferable features generalizable to diverse computer vision tasks [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' After encoding, each sample ˜xi is converted to: ei = Enc(˜xi) (2) where ei ∈ Rde and de is the dimension size of the generated representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' de is set to 2048 in our evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We note that ei is further normalized to fall in the unit hypersphere in Rde as suggested in [24], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=', ∥ei∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Such normalization is considered to bring performance improvement to prediction tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Projection Module Pro(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Next, we project the derived ei into a more compact rep- resentation in this Projection Module: si = Pro(ei) = ϕMLP (ei), ϕMLP : Rde �−→ Rdp (3) where a multi-layer perceptron (MLP) model ϕMLP with one hidden layer is for projecting and abstracting the representation from de-dimensional to dp-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We set dp to 128 in the following experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Similarly, si is normalized as ∥si∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' The SCL loss is then applied to pull positive samples close to each other and push them away from negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' As shown in Figure 8, the integration of SCL corresponds to “Stage1” in FoodSG-SCL, which learns a strong backbone for FoodSG-SCL in order to support the food recognition task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' The learned si is discarded at the end of Stage1, while the output of the Encoder Module ei will be used for prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Prediction Module Pre(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We input ei to a linear prediction model ψ for food recog- nition: qi = Pre(ei) = ψ(ei), ψ : Rde �−→ R|C| (4) where C denotes the set of food categories for prediction, |C| = 233 in the FoodSG-233 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We further parameterize ˆyi = σ(qi), where σ(·) is the softmax function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' The derived ˆyi is the predicted probability distribution over |C| classes and will be used for training FoodSG-SCL in the optimization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Besides, this Prediction Module corre- sponds to “Stage 2” as in Figure 8, the focus of which lies in employing the well-trained FoodSG-SCL model for food recognition in an effective manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='3 Optimization Stage1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' The SCL loss for each sample ˜xi is: lscl i = − 1 |P| � p∈P log exp(sim(si, sp)/T) � k∈K\\{i} exp(sim(si, sk)/T) (5) The contrastive prediction task is: given a sample ˜xi, identify the positive samples in P out of all remaining samples K \\ {i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' In Equation 5, sim(·, ·) is the cosine similarity function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=', sim(si, sp) = s⊤ i sp/(∥si∥∥sp∥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' As both representations are normalized, sim(si, sp) is equivalent to s⊤ i sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Moreover, T denotes the temperature parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' The overall SCL loss in Stage1 sums all samples’ losses: Lscl = � i∈K lscl i (6) 12 With this loss function in Stage1, FoodSG-SCL (except the Prediction Module) can be effectively trained via gradient-based optimizers such as stochastic gradient descent (SGD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Stage2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' For the food recognition task as a multi-class classification problem, we adopt the cross-entropy loss in Stage2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Lce = − 1 |K| � i∈K � c∈C yc i log(ˆyc i ) (7) With the Encoder Module’s output ei and the Prediction Module, we can readily train the Predictor Module by optimizing this cross-entropy loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Finally, after both stages, FoodSG-SCL is equipped with the contrastive power, con- tributing to the learning of hard positive/negative samples and further to the prediction performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' 6 Experimental Evaluation In this section, we first introduce the experimental set-up and then describe the experi- mental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' After that, we analyze the experimental results and discuss the derived insights for practitioners on using the FoodSG platform for their specific data-intensive applications, which is imperative to the data management research community in setting its foot in food computing and starting to play an active role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='1 Experimental Set-up Evaluated Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We select five state-of-the-art image models for evaluation, and compare their performance against integrating them into FoodSG-SCL as the Encoder Module, in order to validate the efficacy of SCL in FoodSG-SCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' ResNet50 [20] presents a residual learning framework that is easier to train and opti- mize, bringing considerably improved accuracy on numerous visual recognition tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' DenseNet169 [21] establishes direct connections between each layer and every other layer in a feed-forward manner, which enables the model to scale to more layers easily without causing optimization difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' MobileNetV2 [33] improves the efficiency of mobile models via an inverted residual structure and emphasizes the importance of linear bottlenecks to maintain the repre- sentation power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' EfficientNetV2 [35] employs training-aware neural architecture search, model scaling, and progressive learning, for accuracy, training speed, and parameter efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' ConvNeXt [27] retains the advantages of standard ConvNets in simplicity and effi- ciency, but redesigns the components in a ConvNet towards the modern Transformer- based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Therefore, ConvNeXt achieves superior performance matching up to such Transformer-based models in accuracy and scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Implementation Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' The food recognition on FoodSG-233 is formulated as a 233- class classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We randomly split the food images per category into 80%, 10%, and 10% for training, validation, and testing, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We adopt both the top-1 and the 13 65 70 75 80 Percentage (%) Top-1 Accuracy 80 85 90 95 Top-5 Accuracy ResNet50 (25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='6M) DenseNet169 (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='1M) MobileNetV2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='5M) EfficientNetV2 (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='5M) ConvNeXt (88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='6M) w/o SCL w/ SCL Figure 9: Comparison results of evaluated models w/o SCL and their advanced variants w/ SCL in FoodSG-SCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Each model’s parameter number is shown in brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' top-5 accuracy as evaluation metrics, and an accurate recognition model should yield high results in both metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We select the hyperparameters that achieve the best performance on validation data and apply such settings to the testing data for reporting the average experimental results of three independent runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' For each of the aforementioned evaluated models, we adopt its model weights pretrained on ImageNet and then finetune them on FoodSG-233 for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We train these models via SGD [34] with a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='05 and a weight decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We adopt batch size differently across different models, as a larger model corresponds to a smaller batch and we set the batch size as large as possible within the device memory capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Specifically, the batch size is 512 in EfficientNetV2, 1024 in DenseNet169 and ConvNeXt, 2048 in ResNet50 and MobileNetV2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' For validating the effectiveness of FoodSG-SCL, we integrate the evaluated models as the Encoder Module of FoodSG-SCL, start with the same model parameters pretrained on ImageNet and then finetune on FoodSG-233 with the other essential modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' SGD is used as the optimizer with the learning rate set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='1 and the epoch number set to 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' As FoodSG-SCL utilizes two views for each sample, its batch size needs to be halved accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Experimental Environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We conduct the experiments in a server with Intel(R) Xeon(R) Gold 6248R × 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='0GHz, 24 cores per chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' The server has 768GB memory, and 8 NVIDIA V100 with 300GB/s NVLINK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We implement the models with PyTorch 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='2 Experimental Results In Figure 9, we illustrate the experimental results of the evaluated image models without SCL and their respective advanced variants with SCL as well as other modules in our proposed FoodSG-SCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' As shown, the advanced methods with SCL outperform the ones without SCL in top-1 accuracy by a large margin consistently, which confirms the effectiveness of the SCL mech- anism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' However, the integration of SCL leads to a slight degradation in top-5 accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' The underlying reasons for these phenomena will be explained in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='3 in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Among the five evaluated models, EfficientNetV2 is the most accurate in terms of both top-1 and top-5 accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' In addition, despite the parameter number being one magni- tude smaller than the others, MobileNetV2 still achieves moderate performance in both evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' 14 a b c d e 0 9 Probability (%) laksa w/o SCL w/ SCL a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' tom yum noodle soup laksa b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' mee siam c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' lontong with sayur lodeh d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' mee rebus e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' satay bee hoon (a) Laksa a b c d e 0 2 Probability (%) sushi w/o SCL w/ SCL a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' salmon - grilled sushi b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' yuseng c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' rice, korean bulgogi beef d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' pung kueh e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' dumpling (b) Sushi Figure 10: Representative food categories and their top 5 misclassified categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' To further investigate the intrinsic property of SCL in FoodSG-SCL, we probe into the cases where food images are misclassified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We choose the following two ground truth food categories to explore: (i) the representative local food category “laksa” which is a spicy noodle dish popular in Singapore, and (ii) “sushi” which is relatively more widely known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' For each ground truth category, we select the best-performing checkpoint among all evaluated models without SCL, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=', EfficientNetV2 without SCL (according to Figure 9), and evaluate it on all the testing samples to derive the predicted probability distribution per testing sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We then calculate the average of all testing samples’ results, based on which we illustrate the top 5 misclassified categories and their respective probabilities predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' For comparison, we display the results of the same five food categories derived by the most accurate model checkpoint of EfficientNetV2 with SCL as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' The experimental results of both ground truth categories, including the predicted proba- bilities of their top 5 misclassified categories and the corresponding example food images are shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' For both laksa and sushi, the predicted top 5 misclassified cat- egories generated by EfficientNetV2 without SCL are highly similar to the ground truth category in terms of color, texture, and shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' On the contrary, EfficientNetV2 with SCL effectively reduces the predicted probabilities of these five misclassified categories in both cases, the reason for which will be explicated in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='3 Discussions Top-1 accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' top-5 accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Generally, the top 1 predicted class is used as the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' However, in practice, the top 1 prediction may not be completely 15 OTV SMITHOWHATTOCOOKTODAY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='COMcorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' To alleviate this issue, sometimes users are offered a set of five food category candidates and allowed to select the correct one with “one more touch”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' If the true food category still cannot be located within the candidate set, then the user experience will be negatively affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' In this sense, both top-1 accuracy and top-5 accuracy are crucial in food recognition, and they depict how accurately we can recognize the food dishes on the released FoodSG-233 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' In Figure 9, the highest top-1 accuracy is 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='87% yielded by EfficientNetV2 with SCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Besides, EfficientNetV2 without SCL performs best in top-5 accuracy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=', 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='06%, whereas EfficientNetV2 with SCL achieves 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='13% with a minor decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' This means that in most cases, the returned five candidate food categories successfully contain the ground truth, and users could select the correct one with one more touch conveniently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Both findings validate the effectiveness of FoodSG-SCL for food recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' SCL in food recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Food recognition is challenging in that certain food categories highly resemble each other and it could be difficult to distinguish between them even for humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Furthermore, such subtle differences in the appearance of the two categories may correspond to a large difference in their constituent nutrients, which will have a huge impact on diet assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' This consequently bolsters the demand for accurate food recognition when handling visually similar food images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' SCL is able to meet this demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Specifically, SCL’s underlying principle lies in drawing samples with the same label close while separating them away from samples with other labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' This in nature improves top-1 accuracy, which is shown in Figure 9 where the integration of SCL brings performance benefits consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' However, such benefits of SCL do not come for free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' The models without SCL adopt the standard cross-entropy loss and generate the top 5 categories that are the most similar to the ground truth category in appearance shown in Figure 10, due to the representation learning capability of convolutional neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Different from them, SCL-based models tend to push the negative samples with different labels away from the positive samples, in spite of their resemblance in appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Therefore, with SCL, the predicted probabilities of the same five similar categories will be reduced to a large extent illustrated in Figure 10, resulting in slightly compromised top-5 accuracy as observed in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' According to our real-world practice, we firmly believe that top-1 accuracy should outweigh top-5 accuracy in food recognition regardless of whether we allow users to further select from a candidate category set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Hence, we introduce the SCL mechanism into FoodSG-SCL to reduce the disturbing influence exerted by similar yet incorrect categories, providing excellent gain in top-1 accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Edge Computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' In healthcare-oriented applications, there are certain scenarios where users’ data cannot be processed in a centralized manner, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=', users are not willing to upload their food photos or share their nutrition intake information for privacy concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' This requires the computation involved in food recognition to be performed on edge de- vices, such as mobile phones, which is thus affected by the computing capacity and space limitation of the devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Under such scenarios, more lightweight models are preferred as the Encoder Module of FoodSG-SCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' For instance, MobileNetV2 exhibits a prominent advantage of parameter number being one magnitude smaller than other models, at the expense of accuracy decreasing from 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='87% (by EfficientNetV2 with SCL) to 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='55%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' 16 7 Related Work As an interdisciplinary area, food computing targets to apply computational methods to analyze heterogeneous sources of food data for supporting diverse food-related investi- gations in healthcare, gastronomy, and agronomy [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Among the involved tasks, food recognition is a cornerstone, which predicts the food items contained in food images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Prior studies in the early stage exploit hand-crafted features for food recognition, such as color histograms and SIFT features [17], statistics of pairwise local features [39], and bag of features [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Recently, due to the record-breaking performance achieved by deep learning models in numerous areas such as computer vision [25, 31, 36], various convolutional neural network models [22, 38] are employed in food recognition for extracting the features in food images and hence, delivering more satisfactory recognition performance than the traditional approaches based on hand-crafted features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' In the meantime, a large number of food-related datasets are released to further promote the development of food computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' For example, Food101 [14] is constructed to cover 101 food categories (mostly western), containing 101, 000 images in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Different from Food101, VIREO Food-172 [16] turns the focus to Chinese food and includes 172 food categories, and 110, 241 images, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Moreover, a Japanese food dataset UEC- Food100 [29] is built to incorporate 100 popular categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' This UEC-Food100 dataset is then expanded to UEC-Food256 [23] introducing more categories from other countries, namely French, Italian, American, Chinese, Thai, Vietnamese, and Indonesian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' These released multifarious food-related datasets present researchers with ample opportunity for food computing and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' However, these datasets have their specific geographic areas and therefore, only include particular types of cuisines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Different from existing work, driven by FoodSG, we investigate Singaporean food dishes and release the localized dataset FoodSG-233, which is of vital significance to FoodSG’s provided healthcare services for Singaporeans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' 8 Conclusions Focusing on the cuisines and healthcare-oriented applications in Singapore, we abstract the shared requirements among diverse applications and develop the dietary nutrition-aided platform FoodSG as a service to support all sorts of application scenarios in Singapore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' We collect, curate, and release a Singaporean food dataset FoodSG-233 systematically to promote future research directions for the data management community in food computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Through delving into FoodSG-233 to analyze its issues of intra-class dissimilarity and inter-class similarity, we propose to integrate SCL into food recognition and devise the FoodSG-SCL model accordingly to facilitate the learning from hard positive/negative samples for performance gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' By evaluating FoodSG-SCL from multiple perspectives, we deliver fresh insights and valuable experience in data-intensive healthcare applications to practitioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' 17 References [1] https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='todayonline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content='com/singapore/get-snap-happy-new-app- pre-diabetics-help-manage-health-condition, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' [2] https://www.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' Capturing feature-level irregularity in disease progression modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' In CIKM, pages 1579–1588.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' ACM, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} +page_content=' 20' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E2T4oBgHgl3EQfWAeQ/content/2301.03829v1.pdf'} diff --git a/QtE4T4oBgHgl3EQflA0t/content/tmp_files/2301.05155v1.pdf.txt b/QtE4T4oBgHgl3EQflA0t/content/tmp_files/2301.05155v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3c1d36137b0b8400cb72bbe3b50643b0db764a0e --- /dev/null +++ b/QtE4T4oBgHgl3EQflA0t/content/tmp_files/2301.05155v1.pdf.txt @@ -0,0 +1,2613 @@ +Computing m-Eternal Domination Number of +Cactus Graphs in Linear Time +Václav Blažej +Faculty of Information Technology, Czech Technical University in Prague, Prague, Czech Republic +Jan Matyáš Křišťan +Faculty of Information Technology, Czech Technical University in Prague, Prague, Czech Republic +Tomáš Valla +Faculty of Information Technology, Czech Technical University in Prague, Prague, Czech Republic +Abstract +In m-eternal domination attacker and defender play on a graph. Initially, the defender places +guards on vertices. In each round, the attacker chooses a vertex to attack. Then, the defender +can move each guard to a neighboring vertex and must move a guard to the attacked vertex. The +m-eternal domination number is the minimum number of guards such that the graph can be defended +indefinitely. +In this paper, we study the m-eternal domination number of cactus graphs. We consider two +variants of the m-eternal domination number: one allows multiple guards to occupy a single vertex, +the second variant requires the guards to occupy distinct vertices. We develop several tools for +obtaining lower and upper bounds on these problems and we use them to obtain an algorithm which +computes the minimum number of required guards of cactus graphs for both variants of the problem. +2012 ACM Subject Classification Graph Theory +Keywords and phrases Graphs, Algorithms, Eternal domination +1 +Introduction +Consider the following game, played by an attacker and a defender on graph G. The defender +controls a set of guards, which he initially places on the vertices of G. Each vertex can be +occupied by at most one guard. +In each round, the attacker first chooses one vertex, which he attacks. The defender then +must defend against the attack by moving some or all of his guards along their adjacent +edges, so that one of the guards moves to the attacked vertex. +If the attacked vertex is not occupied by a guard after the attack, the attacker wins. The +defender wins if he can defend indefinitely. +Defending a graph from attacks using guards for an infinite number of steps was introduced +by Burger et al. [3]. In this paper, we study the concept of m-eternal domination, which was +introduced by Goddard et al. [6] (eternal domination was originally called eternal security). +Here, the notion of the letter “m” emphasizes that multiple guards may move during each +round. There is also a variant of the problem studied by Goddard et al. [6] where only one +guard may move during each round, which is not considered in this paper. +The m-eternal domination number γ∞ +m (G) is the minimum number of guards which defend +against all attacks indefinitely. Goddard et al. [6] established γ∞ +m exactly for paths, cycles, +complete graphs and complete bipartite graphs. Since then, several results have focused on +The +authors +acknowledge +the +support +of +the +OP +VVV +MEYS +funded +project +CZ.02.1.01/0.0/0.0/16_019/0000765 “Research Center for Informatics”. This work was supported by +the Grant Agency of the Czech Technical University in Prague, grant No. SGS20/208/OHK3/3T/18. +Supported by the grant 22-19557S of the Czech Science Foundation. +arXiv:2301.05155v1 [cs.DS] 12 Jan 2023 + +2 +Computing m-Eternal Domination Number of Cactus Graphs in Linear Time +finding bounds on γ∞ +m under different conditions or graph classes. Among the studied graph +classes are trees [12, 8, 14], grids [4, 18, 16, 10, 5, 9, 15], and interval graphs [2, 17]. For a +good survey of other related results and topics, see Klostermeyer and Mynhardt [13]. +Very little is known regarding the algorithmic aspects of m-eternal domination. The +decision problem (asking if γ∞ +m (G) ≤ k) is NP-hard and belongs to EXPTIME, however, it is +not known whether it lies in the class PSPACE [13]. +1.1 +Original Results +In this paper, we focus on the class of cactus graphs (connected graphs where each edge +lies in at most one cycle) and provide an algorithm for computing γ∞ +m in cactus graphs. In +Section 4, we provide a set of tools with more general applications to proving upper and lower +bounds of γ∞ +m . Those tools are then used in Section 5 to describe a set of reductions, which +allow us to compute γ∞ +m of cactus graphs. This is a significant expansion of basic principles +which were introduced by Klostermeyer and MacGillivray [11], in which they provide an +algorithm for computing γ∞ +m of trees. +Our main result is summarized in the following theorem. +▶ Theorem 1.1. Let G be a cactus graph on n vertices. Then there exists a polynomial +algorithm which computes γ∞ +m (G). +1.2 +Preliminaries +Let us now review all the standard concepts formally. A graph is a cactus if its every edge +lies on at most one cycle. For an undirected graph G let a configuration be a multiset of +its vertices C = {c1, . . . , cn | ci ∈ V (G)}. We will refer to the elements of configurations as +guards. If a vertex is an element of a configuration, then it is occupied (by a guard). Two +configurations C1 and C2 of G are mutually traversable if there is some set of pairs T (C1, C2) = +{(v1, u1), (v2, u2), . . . , (vn, un)} such that C1 = {v1, . . . , vn} and C2 = {u1, . . . , un} and +{vi, ui} ∈ E(G) for all i from 1 to n. We perceive the guards as tokens which move through the +graph. The elements of T (C1, C2) are called movements and a single ordered pair among them +is a move of a guard. A guard that moves in T (C1, C2) to the same vertex where he started +is called stationary. A strategy in G is a graph SG = (C, F) where C is a set of configurations +over V (G) such that all of the configurations have the same size and F ⊆ C2 describe possible +transitions between the configurations. The order of a strategy is the number of guards in +each of its configurations. In papers on this topic it is often assumed that the strategy edges +are given implicitly as F = +� +{C1, C2} ∈ C2 | C1 and C2 are mutually traversable in G +� +. For +our purposes, we want to prescribe the strategy explicitly. We introduce the notions for +exact strategy prescription in Section 4.2. +We call the strategy SG to be defending against vertex attacks if for any C ∈ C the +configuration C and its neighbors in SG cover all vertices of G, i.e., when a vertex v ∈ V (G) +is “attacked” one can always respond by changing to a configuration which has a guard at +the vertex v. Formally, SG = (C, F) is defending if +(∀C ∈ C) (∀v ∈ V (G)) +� +v ∈ C ∨ (∃C′ ∈ C)({C, C′} ∈ F ∧ v ∈ C′) +� +. +Note that every configuration in a strategy which defends against vertex attacks induces +a dominating set in G as otherwise, the attacker would win in the next round. +We investigate two variants of the game. The variants differ in whether they allow +multiple guards to occupy the same vertex. Let an m-Eternal Guard Strategy in G be a +strategy defending against vertex attacks in G. + +V. Blažej, J. M. Křišťan, and T. Valla +3 +Input: +An undirected graph G = (V, E). +Question: What is the minimum number of guards γ∞ +m such that there exists an m-Eternal +Guard Strategy SG where each vertex is occupied by at most one guard that defends +against vertex attacks in G? +m-Eternal Domination +Input: +An undirected graph G = (V, E). +Question: What is the minimum number of guards Γ∞ +m such that there exists an m-Eternal +Guard Strategy SG that defends against vertex attacks in G? +m-Eternal Guard Configuration +The open neighborhood of u in G will be denoted as NG(u). By Pn we denote a path +with n edges and n + 1 vertices. By G[U] we denote the subgraph of G induced by the set of +vertices U ⊆ V (G). +2 +High-level Overview of the Proof +In order to solve the m-Eternal Domination and the m-Eternal Guard Configuration +on cactus graphs, we use induction on the number of vertices. Base cases will be presented +in Definition 5.6. In the induction step, we show how to reduce cactus graph G to a smaller +cactus graph G′ while showing lower bound and upper bound in the following ways. Reduction +from G to G′ is done using Observations 4.3 and 4.8–4.10. These directly show a lower bound +Γ∞ +m (G) ≥ Γ∞ +m (G′) + K for some constant K. Then, we show an expansion from G′ to G. We +assume that G′ has an optimal defending strategy that holds several nice properties from the +induction. We show that a part of the graph G′ along with its strategy can be exchanged for +a different one by showing that Definition 4.25 holds for them. Such parts are then exchanged +using Definition 4.27 which expands G′ into G while showing that an upper bound devised +by Observation 4.29 applies. This gets us an upper bound γ∞ +m (G) ≤ γ∞ +m (G′) + K (the same +K as in the lower bound). Combining the lower and upper bound using Lemma 4.2 gets us +the optimal number of guards for G. +The used reduction depends on a leaf component that the cactus graph contains by +Observation 5.3. One case is that the subgraph is a tree and the second case is that there +is a leaf cycle – a cycle with leaves which is connected to the rest of the graph via a single +articulation. We split the reductions into three groups. +The first group called leaf reductions shown in Section 3 has a few simple reductions of +leaves which are not incident to a leaf cycle. These were shown to be sufficient to determine +the γ∞ +m for any tree by Goddard, Hedetniemi, and Hedetniemi [6]. We reintroduce these +reductions in our framework and show more general results so that the reductions can be used +over tree subgraphs of non-tree graph classes. They also serve as an introductory example of +how to use the tools from Section 4. +Further reductions are more involved and require non-trivial manipulation with strategies. +It is beneficial to establish strategies with nice properties in the induction to allow a stronger +induction step. In Section 5.2, we show the properties which are used in the two other groups +of reductions. +The last two groups called cycle reductions and constant component reductions are shown +in Sections 5.3 and 5.4. Cycle reductions concern substructures that appear on leaf cycles. +We fix a leaf cycle and use these reductions repeatedly on it. Each reduction shortens the leaf +cycle. Eventually, the cycle is very short and is reduced by constant component reductions. + +4 +Computing m-Eternal Domination Number of Cactus Graphs in Linear Time +After these reductions, the leaf cycle is removed entirely and only zero, one, or two leaves are +left in its place. Such leaves are then processed either as tree leaves or leaf vertices adjacent +to another leaf cycle. +3 +Reducing Trees +In this section, we intuitively present tools to achieve lower and upper bounds and which will +be formally introduced in Section 4. We focus on tree reductions, which were first described +by Goddard, Hedetniemi, and Hedetniemi [6] as a part of the linear algorithm for computing +the m-eternal domination number γ∞ +m on trees. We now show this set of reductions along +with the proofs of their correctness in Lemmas 3.1–3.3. +For graph G, let us have a vertex u ∈ V (G) which is adjacent to ℓ ≥ 1 leaves and has +degree d. Let v be one of the leaves adjacent to u. We define three leaf reductions of G to +G′ as follows. See Table 1 for an illustration of respective bound proofs. +▶ Reduction 1. t1 If ℓ = 1 and d ≤ 2, let G′ = G \ {u, v}. +▶ Reduction 2. t2 If ℓ > 2, let G′ = G \ {v}. +▶ Reduction 3. t3 If ℓ = 2 and d = 3, let G′ = G \ {all leaves adjacent to u}. +Reductions t2 and t3 can be joined to a single reduction which removes all leaves of a +vertex with ℓ ≥ 2 and d = ℓ + 1 (used in [6]). However, Reduction t2 may be used in a wider +range of scenarios as it does not require a specific value of d. +Assume now, that we know the optimum number of guards for G′ (for both Γ∞ +m and γ∞ +m ). +Our goal is to show two things. By showing that G always uses at least K more guards than +G′ we get a lower bound on the number of guards necessary for G. By showing that there is +a strategy for G which uses at most K more guards than an optimum strategy on G′ we get +an upper bound on the number of guards on G. Together, these bounds give us an optimum +number of guards for G. This concept is formally introduced in Lemma 4.2. +Table 1 Leaf reductions; Lower bound side depicts clique reductions (removal of marked vertices +and joining its neighborhood with a clique); Upper bound side labels vertices with Greek letters +of states where they belong, and arrows show how one state transitions to another. The marked +groups of vertices are created with Definition 4.30. +Reduction +Lower bound +Upper bound +t1 +−1 +v +u +a +a +v +α +β +u ++1 +a +a +t2 +−0 +u +u +v ++0 +u Ω′ +α′ +β′ +w +u Ω +α +β +w +vγ +t3 +u +u +−1 +v +a +a +u +β +u ++1 +a +a α′ +β′ +α +Ω +v +w + +V. Blažej, J. M. Křišťan, and T. Valla +5 +−2 +−2 +−1 ++1 ++2 ++2 ++1 +Ω +Ω +α β +α +α +β1 +γ +δ1 +γ +δ +Ω +Ω +Ω +Ω +µ +ν +δ2 +Ω +β2 +β1β2β3 +β4 +t2 +t2 +t1 +t3 +t3 +t1 +t3 +Figure 1 An example application of leaf reductions on a tree graph. Dotted lines signify that the +strategy does not use that edge, and the strategies on subtrees are independent, which is caused by +Reduction t1. Note how Reduction t3 can be used even when there is no vertex a. +Having the reductions in hand see Figure 1 for an example of how the reductions are +used to construct a strategy for a tree. +We now proceed to show the bounds obtained from these reductions. Generally, the proofs +contain lower bound and upper bound portions, see Table 1 for accompanying illustrations. +Lower bounds can be shown quite easily – delimit a connected part of a graph which is +guaranteed to contain K guards, remove it, and join its neighborhood with a clique. Upper +bounds are more tricky – we assume some optimal strategy on G′, which has nice properties, +and then we expand it to G while preserving the properties. The notation used in the +following proofs is defined in Section 4. +We say that graph G is defended with k guards if k = γ∞ +m (G) = Γ∞ +m (G) and the strategy +using k guards is proper in the sense of Property 3, which is defined in Section 5. This allows +us the state the lemmas concisely. Let us now see the proofs for the three leaf reductions. +▶ Lemma 3.1. Let G′ be G after application of Reduction t1. +G is defended with 1 more +guard than G′. +Proof. As {u, v} is a leaf and its neighbor, there is always at least one guard so we may apply +Observation 4.8 on {u, v} to get a lower bound of Γ∞ +m (G) ≥ Γ∞ +m (G′) + 1. To get an upper +bound γ∞ +m (G) ≤ γ∞ +m (G′) + 1, we dedicate one new guard to defend {u, v} independently on +the rest of the strategy. Putting the lower bound and upper bound together using Lemma 4.2 +we get that G is defended with 1 more guard than G′. +◀ +Note, that the final strategy graph after Reduction t1 is a Cartesian product of the +strategy graph on G′ and a graph with a single edge. Cartesian product is a basis for +Definition 4.30 where we introduce an operation which joins strategies even if the strategies +are not entirely independent. We shall use this operation along with a property shown in +Lemma 4.37 – that a strategy can be altered so that a vertex adjacent to multiple leaves is +always occupied. +To ease notation, we shall reserve the prime symbol (′) to denote structures of the reduced +instance such as the graph G′, defending strategy B′, strategy graph S′ +G′, its states (vertices) +Ω′ and transitions (edges) F′, etc. +▶ Lemma 3.2. Let G′ be G after application of Reduction t2. +G is defended with the same +number of guards as G′. +Proof. Lower bound of 0 is obtained by using Observation 4.3 to identify v with u so +Γ∞ +m (G) ≥ Γ∞ +m (G′). + +6 +Computing m-Eternal Domination Number of Cactus Graphs in Linear Time +For upper bound, from induction we have a defending labelled strategy B′ of G′. We +wish to alter it so it defends vertex v as well. We apply Lemma 4.37 to alter B′ so that u is +occupied in each state of Ω′. Let w be a leaf adjacent to u distinct from v. We partition all +states (vertices) of the strategy S′ +G′ as follows. A state α′ belongs to S′(w) if in α′ vertex w +is occupied. +Now we perform graph Cartesian product of S′ +G′ with a single edge {α, β} over subset +S′(w) (Definition 4.30). Written in short as SG′ = S′ +G′ □S′(w) {α, β}. This splits all vertices +of the strategy where w is occupied into two. We denote the new sets as α and β. This +got us a new strategy graph SG′ over the reduced graph G′. Now we expand from G′ to +G while altering the strategy slightly. In β we substitute the guard on w with a guard on +v. The guards shall transition between states of α and β as T (α, β) = {(w, u), (u, v)} while +the rest of them shall not move. As w and v are siblings it follows that we can transition +from any γ ∈ Ω to w the same way as to v if they were swapped. This remains defending by +Lemma 4.32. Hence, γ∞ +m (G) ≤ γ∞ +m (G′) and by Lemma 4.2 we get that G is defended with +the same number of guards as G′. +◀ +The shown strategy basically defends v in the “same way” it defends w. We can do this +when one can transition from one to the other in a single step while the remaining guards +remain stationary. For a detailed explanation see Definition 4.30 and its lemmas that show +its properties. +The previous reduction bounds were proven with an extensive explanation. +In the +following proofs, we just use the tools to arrive at the result directly. Note that a very similar +argument could be used to obtain an arbitrary number of leaves. +▶ Lemma 3.3. Let G′ be G after application of Reduction t3. +G is defended with 1 more +guard than G′. +Proof. Lower bound of 1 is obtained by using Observation 4.8 on vertices {u, v}, which +results in a graph isomorphic to one that is created by removing all leaves adjacent to u +which gets us Γ∞ +m (G) ≥ Γ∞ +m (G′) + 1. For upper bound, we apply Lemma 4.33 which adds the +two leaves to u using one extra guard which directly results in γ∞ +m (G) ≤ γ∞ +m (G′) + 1. Using +Lemma 4.2 we get that G is defended with 1 more guard than G′. +◀ +Note that the bounds devised for Reductions t1, t2, and t3 do not require the graph to +be a tree. We may use these reductions in any graph class. Hence, we may reduce any leaves +in subtrees which appear as parts of other graphs. In particular, Reduction t2 may be also +used to reduce the number of leaves adjacent to any vertex to 2 because connections of u to +other vertices do not interfere with the reduction. Note that in that case, we obtain lower +bounds for the m-Eternal Guard Configuration and upper bounds for the m-Eternal +Domination. +It was previously shown by Goddard, Hedetniemi, and Hedetniemi [6] that these reductions +(originally given in a slightly different form) are sufficient to solve any tree graph. Note that +this can be shown by rooting the tree and repeatedly applying Reductions t1, t2, and t3 on +the parent of the deepest leaf. +Also note, that the reductions t1 and t3 do not require the rest of the graph (signified by +vertex a) to be there at all, hence, these solve base cases where only a single edge or a star +remain. +When the reductions are used on a tree we get a partitioning of vertices into subtrees +which are defended independently. These components constitute a neo-colonization, a notion +introduced by Goddard et al. [6] and often used in contemporary papers. + +V. Blažej, J. M. Křišťan, and T. Valla +7 +4 +The m-eternal domination Toolbox +This section gives tools to show lower and upper bounds for the m-eternal domination +problem. Before we present the approach in detail we show several key ideas and a detailed +structure for the rest of this section. Throughout this paper, we reserve prime (e.g. G′ and +α′) to denote structures of the reduced instance. +▶ Observation 4.1. γ(G) ≤ Γ∞ +m (G) ≤ γ∞ +m (G) ≤ 2 · γ(G) for any graph G. +Proof. Every m-Eternal Domination strategy can be applied as an m-Eternal Guard Config- +uration strategy so γ∞ +m ≥ Γ∞ +m . Every configuration in each of these strategies must induce a +dominating set. Therefore, they are all lower bound by the domination number γ. +It is also known that an m-Eternal Domination strategy can be constructed by defending +neighborhood of each vertex in the dominating set independently of each other (with a +simple strategy for stars) that uses at most 2 · γ(G) guards as shown by Klostermeyer and +Mynhardt. [13]. +◀ +We now show the lemma which sums up how the bounds of the optimal strategies are +obtained. +▶ Lemma 4.2. Let us assume that for graphs G, G′, and an integer constant k +γ∞ +m (G) ≤ γ∞ +m (G′) + k, +(1) +Γ∞ +m (G) ≥ Γ∞ +m (G′) + k, +(2) +γ∞ +m (G′) = Γ∞ +m (G′). +(3) +Then γ∞ +m (G) = Γ∞ +m (G) = γ∞ +m (G′) + k = Γ∞ +m (G′) + k. +Proof. Given the assumptions, we have +γ∞ +m (G) +(1) +≤ γ∞ +m (G′) + k +(3)= Γ∞ +m (G′) + k +(2) +≤ Γ∞ +m (G) +Obs. 4.1 +≤ γ∞ +m (G). +As the first and the last term is identical all these values are equal. +◀ +Hence, it suffices to prove that for G and its reduction G′ we have γ∞ +m (G) ≤ γ∞ +m (G′)+k and +Γ∞ +m (G′) ≤ Γ∞ +m (G) − k. If we already have the optimal strategy for G′, then our constructive +upper bounds together with Lemma 4.2 give us an optimal strategy for G. +We present the tools for obtaining lower bounds in Section 4.1, terminology and new +concepts for upper bounds in Section 4.2, and tools which use the new concepts to obtain +upper bounds in Section 4.3. See Figure 2 for a detailed section overview. +4.1 +Lower Bounds +We start this section with a few elementary observations about strategies. Then, we show +a pair of lemmas which are the main tools in obtaining lower bounds. Last, using these +lemmas, we obtain three lower bound observations which we use frequently in Section 5. +We say that G′ is a result of identifying u with v in G if it is a result of removing u while +adding the edges so that NG′(v) = NG(u) ∪ NG(v). +▶ Observation 4.3 (Vertex identification). Let G be a graph and u and v be its two distinct +vertices. Then for a graph G′, which is a result of identifying u with v in G, Γ∞ +m (G′) ≤ Γ∞ +m (G). + +8 +Computing m-Eternal Domination Number of Cactus Graphs in Linear Time +Section 4.1 Lower bounds +Observation 4.3 Vertex identification +Observation 4.4 Substitute guard +Definition 4.5 Clique reduction +Lemma 4.6 Lower bound lemma +Lemma 4.7 Ink lemma +Observation 4.8 Leaf lower bound +Observation 4.9 Star lower bound +Observation 4.10 Path lower bound +Section 4.2 Upper bounds +Definitions 4.11–4.14, 4.17, and 4.19: +State, Movement, Interface, Transition, +(Partial) Defended graph and subgraph +Property 1 Symmetry +Definition 4.21 Compatible +Definition 4.22 Cutting +Definition 4.23 Composing +Lemma 4.24 Composing compatible +Definition 4.25 Interface equivalent +Lemma 4.26 Transferred compatibility +Definition 4.27 Expansion +Lemma 4.28 Equivalency constant +Observation 4.29 Upper bound +Section 4.3 Tools for altering strategies +Definition 4.30 Cartesian product over subset +Lemma 4.33 Leaves addition +Definition 4.35 Group state +Figure 2 Overview of Section 4. Boxes represent respective subsections; Arrows on the left side +show which notions are used to prove other notions; Right arrows show which notions are frequently +used in Section 5 to obtain results for cactus graphs. +Proof. Let v′ ∈ V (G′) be the vertex created by identifying u with v in G. Let SG be an +optimal strategy of G. Let SG′ be a strategy on G′ which is the same as SG except that in +every configuration each u and v is substituted by v′. +Any pair of traversable configurations in SG is still traversable in SG′ as in every movement +u and v can be replaced by v′. Any attack on V (G′) \ {v′} is defended by a configuration +in SG′ which was created from a respective configuration of SG, and v′ is defended by a +configuration which defended u in G. +◀ +▶ Observation 4.4. If a graph has a clique on distinct vertices u, v, w, then guard movements +(u, v) and (v, w) can be substituted with movement (u, w) and a stationary guard on v. +Along with vertex identification, the following reduction is the main tool for obtaining +lower bounds. +▶ Definition 4.5 (Clique reduction). Let G be a graph and H be its non-empty induced +connected subgraph. By clique reduction of H in G we mean the creation of a new graph +G′ that is the result of removing H from G and mutually connecting all neighbors of H in +G \ H by an edge. +See Figure 3 for an illustration of a clique reduction. Using Observations 4.3 and 4.4 we +now show that the clique reduction implies a lower bound on G which can be later used to +show tight strategy lower bounds. +▶ Lemma 4.6 (Lower bound lemma). Let G be a graph and H be its non-empty induced +connected subgraph such that in at least one optimal m-eternal guard strategy, there are always +at least k guards present on H. Let G′ be the result of a clique reduction of H in G. Then +Γ∞ +m (G′) ≤ Γ∞ +m (G) − k. +Proof. If there is no neighbor of H in G, then clique reduction removes H and adds no +edges. We can remove all the guards which were standing on H so G′ is clearly defended by +Γ∞ +m (G) − k guards. +Otherwise, there is a neighbor of H in G \ H, say v. Let SG = (C, F) be an optimal +strategy on G. We use Observation 4.3 to identify all vertices V (H) with v in G to obtain + +V. Blažej, J. M. Křišťan, and T. Valla +9 +a subgraph of G′ along with a strategy SG′. Note that in G′ each configuration of SG′ has +at least k guards on v because before identification H always contained k guards. Also, +configurations which defend v in SG′ have at least k + 1 guards on v by the same argument. +Let S− +G′ be a strategy which is the same as SG′, except it has k less guard on v in each +configuration. We see that each configuration which defended v in SG had at least k + 1 +guards on v so it defends v in S− +G′. Guards which defended V (G) \ (V (H) ∪ {v}) remain +unchanged. It remains to check whether configurations which were traversable in SG remain +traversable in S− +G′. +Our goal is to show that there exists a set of movements between each pair of configurations +of SG′ which have k stationary guards on v which are not needed for defending G′. Such +guards then may be removed to obtain S− +G′ and the remaining movements show that the +respective configurations are traversable. +Each movement (u, h) and (h, w) such that h ∈ V (H) in SG has its respective pair of +movements (u, v), (v, w) in SG′. As u and w are neighbors of V (H) then there is an edge +{u, w} ∈ E(G′) added by the construction of G′. By Observation 4.4 we may substitute +movements (u, v), (v, w) with (u, w) and a stationary guard on v. +Assume there are less than k stationary guards on v in SG′ after applying the substitution +exhaustively. Then there must be at most k − 1 stationary guards and at least one guard +which leaves v or at least one guard which arrives to v, but there may not be both (one +leaving and one arriving) as they would form (u, v), (v, w) pair and the substitution could +be applied. When the guard is leaving or arriving there are at most k − 1 guards in the final +or starting configuration, respectively, which is a contradiction because there are at least k +guards on v. +Removing k guards from v in SG′ yields S− +G′ where each configuration pair remains +traversable, which concludes the proof. +◀ +To use Lemma 4.6 we need to show that an induced subgraph H of G is always occupied +by at least k guards. To do that we have the following lemma. +▶ Lemma 4.7 (Ink lemma). Let H be an induced subgraph of G. Let (v1, v2, . . . , vk) be +a sequence of vertices in H such that it holds d(u, vi) > i for every i and for every u ∈ +V (G) \ V (H), and also for every j < i it holds that d(vj, vi) > i − j. Then there are at least +k guards on H in every defending m-eternal guard configuration. +Proof. Let C be any fixed configuration of a defending strategy. We show that C contains +at least k guards on H. +Assume that the attacker performed a sequence of attacks (v1, v2, . . . , vk) one by one. At +the i-th step of the attack sequence, the following is true. Guards who were standing on +V (G) \ V (H) at the beginning of the attack sequence are more than i edges far from vi so +they cannot reach vi in time to defend it. Similarly, any guard that defended vj with j < i +can not defend the attack on vi as their distance from vi is more than i − j at the time they +defended vj. Therefore, none of the guards can reach vi in time and we need an additional +guard placed on H. +In total, we need k guards on H in a configuration to be able to defend the attack +sequence. This is true for any configuration so every defending strategy must have k guards +on H in every configuration. +◀ +The operation in Lemma 4.6 together with the lower bound obtained from Lemma 4.7 +allows us to make a graph smaller while showing that the removed part required some +minimum number of guards. See an example usage of Lemmas 4.6 and 4.7 in Figure 3. + +10 +Computing m-Eternal Domination Number of Cactus Graphs in Linear Time +G +H +G′ +Figure 3 A graph G with an induced subgraph H. Graph G′ is obtained by a clique reduction +from Definition 4.5. By Lemma 4.7 we have that every configuration contains at least 1 guard on H. +Hence, by Lemma 4.6 we have Γ∞ +m (G) ≥ Γ∞ +m (G′) + 1. +▶ Observation 4.8. In graph G, let v be a leaf vertex and let u be its neighbor, and let +H = G[{u, v}]. By Lemma 4.7 with a sequence (v) we obtain that 1 guard is on H. In other +words, the closed neighborhood of every leaf must contain at least one guard otherwise an +attack on the leaf could not be defended. Let G′ be a graph obtained by using the operation of +Lemma 4.6 on H, this gives us Γ∞ +m (G) ≥ Γ∞ +m (G′) + 1. +▶ Observation 4.9. In graph G, let u be a vertex which is adjacent to at least two leaves +{v1, v2, . . . }, and let H = N[u] denote the closed neighborhood of u. By Lemma 4.7 with a +sequence (v1, v2) we obtain that 2 guards are on H. In other words, the closed neighborhood +of u must contain at least 2 guards otherwise two consecutive attacks on different leaves +adjacent to u could not be defended. Let G′ be a graph obtained by using the operation of +Lemma 4.6 on H, this gives us Γ∞ +m (G) ≥ Γ∞ +m (G′) + 2. +▶ Observation 4.10. Let us have graph G and its induced subgraph H that is isomorphic +to a path on three vertices. We label these three vertices of H as u1, u2, u3 (in order). By +Lemma 4.7 with a sequence (u2) we have the lower bound of 1 on the number of guards on +H. Let G′ be a graph obtained by using the operation of Lemma 4.6 on {u1, u2, u3}. This +gives us Γ∞ +m (G) ≥ Γ∞ +m (G′) + 1. +4.2 +Upper Bounds +This section introduces notation to describe strategies which are used to achieve upper +bounds for γ∞ +m . We assume that we have a graph G and its reduced copy G′. The main idea +is that a strategy for G′ can be locally changed to obtain a strategy for G. To accommodate +this local change, we show how to cut and compose parts of the graph while preserving its +strategy. At the end of this section, we present a set of sufficient rules that allow such a local +change. Then, in Section 4.3, we present tools which we use to obtain upper bounds. +In our constructions, we need to have control over the movements of the guards. We also +need a way to represent only part of the strategy over an induced subgraph of G. To do so, +we introduce states (labelled configurations) and labelled strategy that prescribes the guard +movements on state transition. +▶ Definition 4.11 (States). Let states be a set of labels Ω and let state vertex mapping P of +Ω to V (G) be P : Ω → 2V (G), i.e., a state α ∈ Ω represents a subset of vertices P(α) ⊆ V (G) +(also called guards) of a graph G. Let S(v) = {β | β ∈ Ω, v ∈ P(β)} (states that contain v) +for every v ∈ V (G). +We will use Greek letters such as α, β, γ, δ, φ to signify states or sets of states. Move of a +guard is still an ordered pair of vertices (u, v) such that {u, v} ∈ E(G) or u = v (stationary +guard). + +V. Blažej, J. M. Křišťan, and T. Valla +11 +Building towards a comprehensive definition of a labelled strategy, we first build a more +general concept – partial labelled strategy. This will allow us to do cutting and composing +with a well-defined strategy over a subgraph. +▶ Definition 4.12 (Interface). Let an interface R of a graph G with respect to its supergraph +H be a subset of vertices such that +R = +� +u | (∃v) u, v ∈ V (H), {u, v} ∈ E(H), u ∈ V (G), v ̸∈ V (G) +� +, +i.e., those vertices of G which have a neighbor in V (H) \ V (G) in H . +H +G +R +G +R +α +α +α +β +β +β +β +α +αβ +α +α +β +u +v +w +Figure 4 Left: An interface R of G with respect to H. Bold edges signify the cut between V (G) +and V (H) \ V (G) that is responsible for the vertices in R; Right: Transition T (α, β) from α to β. +Arrows signify movements of the transition. All vertices in states α and β must be paired up with a +movement if they are not in the interface R. In the interface, a guard moves from u to the rest of +the graph outside of G so a movement is missing for u. Note the difference between v and w: in w +the same guard stays on the vertex, in v (as there is no (v, v) movement) the guard on v moves out +and a different one moves to v. +See Figure 4 for an example of an interface. The interface marks the vertices where the +strategy may be incomplete. The transitions between states incorporate the interface by +allowing the moves to be incomplete in the following way. +▶ Definition 4.13 (Transition). For states α and β and a graph G with an interface R, let a +transition (from α to β) denoted by T (α, β) be a set of moves such that +T (α, β) ⊆ +� +(u, v) | u ∈ P(α), v ∈ P(β), {u, v} ∈ E(G) ∨ u = v +� +, +for each u ∈ P(α) \ R there exists exactly one (u, v) ∈ T (α, β), +for each v ∈ P(β) \ R there exists exactly one (u, v) ∈ T (α, β), +for each u ∈ P(α) ∩ R there exists at most one (u, v) ∈ T (α, β), and +for each v ∈ P(β) ∩ R there exists at most one (u, v) ∈ T (α, β). +See Figure 4 for an example of a transition and how it interacts with an interface. Note +that if the interface R is empty, then the transition yields a bijection between guards of the +states, which gives an exact prescription on how they move between the two states. +Transition gives us that each guard can be in relation with at most one other guard. In +our case, there is at most one guard on each vertex. Hence, we may use the standard relation +terminology for the set of pairs defined by a transition. +▶ Definition 4.14 (Partial labelled strategy). A partial labelled strategy is (G, SG, P, T , R) +where G is a graph, SG = (Ω, F) is a strategy graph such that Ω is a set of vertices (states) +and F is a set of edges, P is a state vertex mapping of Ω to V (G), R ⊆ V (G) is an interface +of G, and T maps orientations (α, β) and (β, α) of every edge {α, β} ∈ F to transitions +T (α, β) and T (β, α), respectively. + +12 +Computing m-Eternal Domination Number of Cactus Graphs in Linear Time +For various purposes, it may be beneficial to think of the strategy graph SG as an oriented +graph (allowing non-symmetric transitions, see Property 1), or even multigraph (allowing +multiple different transitions between the same set of states). +The labelled strategy may defend against attacks indefinitely if it is in accordance with +the following definition. +▶ Definition 4.15 (Defending). A partial labelled strategy (G, (Ω, F), P, T , R) is defending G +if for every state α ∈ Ω and each vertex v ∈ V (G) there is β ∈ Ω such that {α, β} ∈ F and +v ∈ P(β). +In other words, Definition 4.15 says that for each vertex of the graph every state is either +occupying it or a state which occupies it is reachable with only one transition. This directly +leads to the following observation. Recall that S(u) denotes a set of states that contain u. +▶ Observation 4.16. A strategy is defending a graph if for every u ∈ V (G) set S(u) is a +dominating set of the strategy graph SG. +▶ Definition 4.17 (Labelled strategy). A labelled strategy is D = (G, SG, P, T ) such that +(G, SG, P, T , ∅) is a partial labelled strategy. +Note, that all the states in the labelled strategy must contain the same number of guards +because the transitions are bijections. When the strategy is optimal the number of guards +corresponds to γ∞ +m . +Partial labelled strategies can have several nice properties, which we present now. When +the strategy graph is unoriented it is natural to require symmetry of transitions. +▶ Property 1 (Symmetry). A partial labelled strategy B = (G, (Ω, F), P, T , R) is symmetrical +if and only if T (α, β) is a converse relation to T (β, α) (i.e., T (α, β) = {(a, b) | (b, a) ∈ +T (β, α)}) for every {α, β} ∈ F. +▶ Lemma 4.18. For each partial labelled strategy B = (G, (Ω, F), P, T , R) there exists a +symmetrical partial labelled strategy B′ = (G, (Ω, F), P, T ′, R). +Proof. For each pair of states {α, β} ∈ F fix an arbitrary orientation (α, β) and take the +T (α, β) with an interface R which gives us T (α, β). Note that by swapping u with v and +α with β in Definition 4.13 we obtain the same definition but for T (β, α). We substitute +the transition T (β, α) for this newly found transition. Performing this substitution for every +pair of states in F gives us the desired T ′. +◀ +By Lemma 4.18, we will always assume that the partial labelled strategy is symmetrical. +We also use this property to infer transitions. We show only one direction of the transition +mapping and let the other direction be the converse transition given by symmetry. +It is not easy to grasp the labelled strategy description only from the formal notation +so we shall draw many auxiliary pictures. Vertices and edges shall be depicted by small +circles (or squares) and line segments, respectively; vertices may be labelled by their letter +name; by Greek letters we signify the states which contain respective vertices; guard moves +in transitions are depicted by differently styled arrows on edges which point between the +state labels (Note that the arrows are always shown only in one direction because we assume +Property 1.); the interface vertices are marked by gray-filled areas; see Figure 5 for an +example of a labelled strategy. +We will need to cut part of the labelled strategy and put something slightly different in +its place. To tackle that we put forward the following notions. + +V. Blažej, J. M. Křišťan, and T. Valla +13 +α +β +γ +a +b +c +u +v +a +b +c +u +v ++ +u +B +A +C +cutting +composing +γ +β +α +α +β +γ +γ +β +γ +β +α +Figure 5 Example of a labelled strategy B and two partial labelled strategies A and C; the full +formal description of the labelled strategy B = (G, (Ω, F), P, T ) is G = (V, E), V = {a, b, c, u, v}, +E = � +{a, b}, {b, u}, {u, v}, {a, c}, {c, u}� +, Ω = {α, β, γ}, F = � +{α, β}, {α, γ}, {β, γ}� +, P(α) = {v, a}, +P(β) = {u, b}, P(γ) = {u, c}, (R = ∅), T (α, β) = {(a, b), (v, u)}, T (α, γ) = {(a, c), (v, u)}, T (β, γ) = +{(b, u), (u, c)}; the formal descriptions of partial labelled strategies A and C are similar while restricted +to their subgraph and they contain an interface R = {u}. +▶ Definition 4.19 (Partial labelled substrategy). The partial labelled substrategy B′ of a +labelled strategy B = (G, (Ω, F), P, T ) for some induced subgraph G′ of G is a partial labelled +strategy B′ = (G′, (Ω, F), P ′, T ′, R) where R is an interface of G′ with respect to G, for all +α ∈ Ω it holds P ′(α) = P(α) ∩ V (G′), and for all β, γ ∈ Ω it holds T ′(β, γ) = {(a, b) | +(a, b) ∈ T (β, γ) ∧ a, b ∈ V (G′)}. +It is not immediately obvious that the Definition 4.19 made a partial labelled substrategy +in a way that it constitutes a partial labelled strategy; so we show that next. +▶ Observation 4.20. A partial labelled substrategy B′ = (G′, (Ω, F), P ′, T ′, R) of a labelled +strategy B = (G, (Ω, F), P, T ) is a partial labelled strategy. +Proof. G′ is a graph, Ω is a set of labels, and R is a subset of V (G′) which is in accordance +to Definition 4.14. P ′ was created by restricting P to the vertices of V (G′). We only removed +some guards from the mapping so this is okay by Definition 4.14. Last, the only guards +which are not included in T ′ are those whose moves in T went outside of G′. Assume such +guard on vertex u with a move (u, v) where v ̸∈ G′, hence, u ∈ R by Definition 4.12. As +stated in Definition 4.13 any guard in R does not have to be included in a move so any +partial labelled substrategy is a partial labelled strategy. +◀ +Now we present conditions which are necessary to be able to combine two partial labelled +strategies into one labelled strategy. Based on that, we show how to split a labelled strategy +into two partial labelled strategies. Figure 5 shows an example of the following operations. +▶ Definition 4.21 (Compatible). Two partial labelled strategies B1 and B2 (denoted as +Bi = (Gi, (Ωi, Fi), Pi, Mi, Ri)) are called compatible if the following conditions hold true. +R1 = R2 = V (G1) ∩ V (G2), i.e., their graphs overlap exactly in the interface, +(Ω1, F1) = (Ω2, F2), i.e., the strategy graphs are the same, +M1(α, β) ∪ M2(α, β) is a bijection between P1(α) ∪ P2(α) and P1(β) ∪ P2(β) for every +α, β ∈ Ω1. +The conditions for compatible partial labelled strategies ensure that the interfaces overlap +in a way that a composed function will be a bijection which allows us to cut and compose +them in the following way. +▶ Definition 4.22 (Cut). Let us have a labelled strategy B and a vertex cut R which partitions +the vertices of G(B) into R, A and C in such a way that there are no edges between A and +C. We say B is cut along R into two partial labelled substrategies A and C where A is a + +14 +Computing m-Eternal Domination Number of Cactus Graphs in Linear Time +partial labelled substrategy induced by V (G(A)) = R∪A and C is a partial labelled substrategy +induced by V (G(C)) = R ∪ C such that A and C are compatible. +▶ Definition 4.23 (Composing). By composing two partial labelled strategies B1 and B2 +(Bi = (Gi, (Ωi, Fi), Pi, Mi, Ri)) we mean getting (G∗, (Ω, F), P ∗, T ∗) where G∗ = +� +V (G1) ∪ +V (G2), E(G1) ∪ E(G2) +� +, Ω = Ω1 ∪ Ω2, F = F1 ∪ F2, ∀γ ∈ Ω we have P ∗(γ) = P1(γ) ∪ P2(γ), +and T ∗(α, β) = M1(α, β) ∪ M2(α, β) for every α, β ∈ Ω. +▶ Lemma 4.24. Composing two compatible partial labelled strategies yields a labelled strategy. +Proof. Let us use the notation of Definitions 4.21 and 4.23. We need to check whether +(G∗, (Ω, F), P ∗, T ∗, ∅) is a partial labelled strategy. First, G∗ is a graph where we unite +vertices and edges, while only the interface vertices are overlapping; this constitutes a well- +defined graph without multiedges and loops. The states Ω1 are the same for the compatible +B1 and B2. Next, mapping of the states to vertices is done by uniting the individual sets +P1(γ) ∪ P2(γ) for each γ ∈ Ω1. Each vertex is now guarded in the union of states it was +guarded before. Last, we check whether the union of M1 and M2 always maps to a well- +defined transition, however, this is ensured by compatibility conditions over Pi and Mi in +Definition 4.21. +◀ +We define an equivalency relation (reflexive, symmetric, and transitive) with respect to +the interfaces as follows. +▶ Definition 4.25 (Interface equivalent). Two partial labelled strategies B1 and B2 (Bi = +(Gi, (Ωi, Fi), Pi, Mi, Ri)) are interface equivalent if G[R1] = G[R2], Ω1 = Ω2, F1 = F2, for all +α ∈ Ω1 we have P1(α) ∩ R1 = P2(α) ∩ R2, and we have (a, b) ∈ M1(β, γ) ⇔ (a, b) ∈ M2(β, γ) +for all u such that a = u ∨ b = u for all β, γ ∈ Ω1. +Interface equivalent partial labelled strategies have the same states with respect to the +interface. This allows us to infer compatibility as stated in the following lemma. +▶ Lemma 4.26. For three partial labelled strategies B1, B2, and B3 if B1 is compatible +with B2, B2 is interface equivalent with B3, and V (G(B1)) ∩ V (G(B3)) = R(B3), then B1 is +compatible with B3. +Proof. Let Bi = (Gi, (Ωi, Fi), Pi, Mi, Ri). We will check the conditions stated in Defini- +tion 4.21. As V (G1) ∩ V (G3) = R3 and R3 = R2 by interface equivalency, and R2 = R1 by +compatibility, the first condition holds. As G3[R3] = G2[R2] and V (G1) ∩ V (G3) = R3 there +are no possible edges which would be shared by G1 and G3 outside of R3. Ω1 = Ω2 by their +compatibility, Ω2 = Ω3 by interface equivalency, so Ω1 = Ω3. The B3 is a partial labelled +strategy so each guard on a vertex in V (G3) \ R3 is covered by M3 exactly once. The guards +on R3 are covered exactly when they were covered on R2. As B1 and B2 are compatible the +guards on R2 were covered by M1 exactly when they were not covered by M2 and vice-versa. +Hence, this property still holds for B1 and B3. +◀ +The culmination of the previous notions and lemmas is the following procedure which we use +as one major part for proving upper bounds. +▶ Definition 4.27 (Expansion). Let us have a labelled strategy B with a partial labelled +substrategy C. Let us also have a partial labelled strategy C′ which is interface equivalent with +C. An expansion of B from C to C′ is the following sequence of operations. +Cutting B along R(C) into C and D (see Definition 4.22), +composing D with C′ into a labelled strategy R (see Definition 4.23). + +V. Blažej, J. M. Křišťan, and T. Valla +15 +Partial labelled strategies D and C′ are compatible due to Lemma 4.26. The result R is a +labelled strategy due to Lemma 4.24. +To establish the difference in the number of guards used to defend B and R we have the +following lemma. +▶ Lemma 4.28. For two interface equivalent partial labelled strategy B1 and B2 (as in +Definition 4.25) there is some constant K(P1, P2) ∈ Z such that for all α ∈ Ω1 we have +K(P1, P2) = |P2(α)| − |P1(α)|. +Proof. Suppose we have arbitrary states α, β ∈ Ω1 and let Kα = |P2(α)| − |P1(α)| and +Kβ = |P2(β)| − |P1(β)|. Let M1(α, β) be part of B1 and M2(α, β) part of B2. Each defines a +pairing of guards in respective states. However, the guards on the interface are not required +to participate in the pairing. So we have |P1(α)| + g1(α, β) = |P1(β)| + g1(β, α) where gi is +the number of guards that do not participate in the pairing of respective Mi (we assume +symmetric moves). Similarly for B2 we have |P2(α)| + g2(α, β) = |P2(β)| + g2(β, α). +As the partial labelled strategies are interface equivalent, the sets of guards which do not +participate in the pairings is the same, so g1(γ, δ) = g2(γ, δ) for all γ, δ ∈ Ω1. We get +Kα = |P2(α)| − |P1(α)| += |P2(β)| + g1(β, α) − g1(α, β) − (|P1(β)| + g2(β, α) − g2(α, β)) += |P2(β)| − |P1(β)| + (g1(β, α) − g2(β, α)) + (g2(α, β) − g1(α, β)) += |P2(β)| − |P1(β)| = Kβ. +We set K(P1, P2) = Kα as we showed that this value is the same irrespective of the chosen +α. +◀ +To be able to use an expansion we need to select a partial labelled substrategy C of B +and then show that C is interface equivalent with C′. The expansion then proceeds as in +Definition 4.27 and an upper bound is obtained from the following observation. +▶ Observation 4.29. Let us have an expansion of B from C to C′ (Definition 4.27) which +results in a labelled strategy R. The expansion increases the number of used guards by +K(P(C), P(C′)) due to Lemma 4.28. Assuming that B is an optimal strategy we obtain +γ∞ +m (G(R)) ≤ γ∞ +m (G(B)) + K(P(C), P(C′)). +We showed a way to describe a labelled strategy and how we can exchange the underlying +defended graph. However, to be able to do this we need the strategy to be the same for +the original and expanded graph. So before we start expansion we alter the strategy on the +original graph. This is discussed in the following section. +4.3 +Tools for Altering Strategies +In this section, we introduce further notions useful for working with strategies when building +upper bound constructions. The typical upper bound proof uses tools introduced in this +section to alter the strategy and then applies expansion (Definition 4.27) which gives the +upper bound by Observation 4.29. +First, let us note that all the notions can be thought of as “up to isomorphism” because +we can relabel graph or strategy vertices and relabeling does not fundamentally change +them. We skipped this in definitions for the sake of readability. Let us also set from now on + +16 +Computing m-Eternal Domination Number of Cactus Graphs in Linear Time +B = (G, SG, P, T , R) and SG = (Ω, F), and similarly for B′ and B∗ have respective graphs G′ +and G∗, strategies, mappings, etc. +Now we present the main operation for altering strategy graphs. +▶ Definition 4.30 (Graph Cartesian product over subset). Let us have graphs G1 and G2 while +A ⊆ V (G1). The graph Cartesian product over subset A denoted as G1 □A G2 is a graph H +such that +V (H) = {(u, ∅) | u ∈ V (G1) \ A} ∪ {(u, v) | u ∈ A, v ∈ V (G2)}, +{(a, b), (c, d)} ∈ E(H) ⇔ +� +(a = c) ∧ (a ∈ A) ∧ ({b, d} ∈ E(G2)) +� +∨ +∨ +� +{a, c} ∈ E(G1) ∧ ((b = d) ∨ (b = ∅) ∨ (d = ∅)) +� +. +The operation in Definition 4.30 can be thought of as a Cartesian product where the +sets of vertices created from G1 which are not present in A are identified to a single vertex. +Equivalently, H can be constructed by taking the graph Cartesian product of G[A] and +H, adding G[V (G) \ A], relabeling each new vertex u as (u, ∅) and connecting each such +(u, ∅) ∈ V (G) \ A to all (v, x) ∈ A × V (H) such that v ∈ NG1(u). This operations will prove +very useful when altering strategy graphs – we will see it used soon in Lemma 4.37 and +many times in Section 5. The aim of this operation is to defend parts of the graph almost +independently. The edges created from G1 represent changes of guard positions within one +part of the graph and edges from G2 represent changes in another part. While guards move +within one part of the graph then the guards in the other part will remain stationary. The +necessity of the set A comes from the fact that the strategy in one part assumes that a guard +occupies vertex (e.g. u) so then the altered part is restricted to vertices where the guard is +present on the vertex (A = S(u)). See Figure 6 for an example application of the Cartesian +product over subset. +A += +A +G1 +G2 +H +Figure 6 Example of a graph Cartesian product of G1 and G2 over subset A. +We shall use the Cartesian product of G′ and complete graph over subset very often so +we will use short notation that allows us to focus on what happens in the created strategy. +▶ Definition 4.31 (Short notation). Let G □A {α1, α2, . . . , αn} = G □A Kn where V (Kn) = +{β1, . . . , βn} and αi denotes sets of states created from βi, i.e., αi = {(a, βi) | a ∈ A}. +The Cartesian product over subset will be used to first alter the strategy graph. The +multiplied states shall defend the same set of vertices as before. Then, during expansion, the +guards shall be moved in order to defend new parts of the graph. There, we need to ensure +that the strategy remains defending. For this, we have the following lemma that tackles the +unchanged and changed states in separate cases. +▶ Lemma 4.32. Let us have H = G1 □A G2 with vertices labelled as in Definition 4.30. + +V. Blažej, J. M. Křišťan, and T. Valla +17 +1. If C is a dominating set of G1, then {(c, b) | (c, b) ∈ V (H), c ∈ C, b = ∅ ∨ b ∈ A} is a +dominating set of H. +2. If A is a dominating set of G1 and B is a dominating set of G2, then A × B (i.e., +{(a, b) | a ∈ A, b ∈ B}) is a dominating set of H. +Proof. Let us have a vertex (x, y) in V (H). +In Case 1, there is a c ∈ C that dominates x in G1. Therefore, (c, y′) with y′ = ∅ if c ̸∈ A +or with y′ = y if c ∈ A dominates (x, y). +In Case 2, if x ̸∈ A (so y = ∅), then there is some a ∈ A that dominates x in G1. As +|B| ≥ 1 there is (a, b) ∈ A×B that dominates (x, y) in H. Otherwise x ∈ A so there is b ∈ B +that dominates y in G2. Hence, (x, y) is dominated by (x, b) ∈ A × B in H. +◀ +When changing the strategy, we want to keep the properties of Lemma 4.32 to ensure +that labelled strategy is defending. +The following lemma shows the second major operation for changing strategies. It allows +us to add leaves to arbitrary vertex and defend the new graph with one more guard. +▶ Lemma 4.33 (Leaves addition). Let us have a graph G and let u ∈ V (G) such that it has +ℓ ≥ 1 adjacent leaf vertices v1, . . . , vℓ. Let G′ be a graph G with vertices v1, . . . , vℓ removed. +For any defending labelled strategy B′ there is a defending labelled strategy B with strategy +graph SG = S′ +G′ □S(u) Kℓ that uses one more guard than B′. +Proof. First, let SG′ = S′ +G′ □S(u) {α1, . . . , αℓ} (see short notation Definition 4.31). As u +must be defended S(u) ̸= ∅. By its construction, all guards of strategy SG′ are stationary on +T (αi, αj). Let δ′ = Ω′ \ {α1, . . . , αℓ}. We expand the strategy over SG′ to G by adding u to +δ (i.e., P(δ) = P ′(δ′) ∪ {u}) and adding vi to αi. We set T (αi, αj) = {(vi, u), (u, vj)} and +we extend T (δ, αi) with (u, vi). +As αi dominates the clique and S′(u) dominates S′ +G′ we have by Lemma 4.32 that B is a +defending labelled strategy for G. +◀ +▶ Observation 4.34. In Lemma 4.33 the construction works the same for any number of +leaves, hence, we may add additional leaves after its use retroactively. +We shall build strategies where vast majority of leaves are defended with Lemma 4.33. +This gives a merit to treat all such states in the same way as their transitions with respect +to the rest of the graph are isomorphic. To do this we put forward the following notion. +▶ Definition 4.35 (Group state). Let a group defense be a set of states which were created +by Cartesian product of G and a clique Kn over a subset. +We shall use group defense only to describe groups of leaves. By Observation 4.34 we +will be able to add new leaves to such group at any point of the construction. +In group states, but also in general strategies we investigated, it seems that vertices which +are adjacent to multiple leaves are often permanently occupied. To get a concrete result +from this observation let us show how to alter an m-Eternal Domination strategy such that +such vertices are permanently occupied. +▶ Definition 4.36 (Permanently defended). A vertex u is permanently defended (permanently +occupied) in B if S(u) = Ω, i.e., u ∈ P(α) for every α ∈ Ω. +▶ Lemma 4.37. For a graph G and its arbitrary defending labelled strategy B′ we may create +a defending labelled strategy B which uses the same number of guards and where each vertex +adjacent to at least 2 leaves is permanently defended. + +18 +Computing m-Eternal Domination Number of Cactus Graphs in Linear Time +Proof. Let vertex u be a vertex with at least 2 adjacent leaves such that u is not permanently +occupied in in strategy B′. Let α′ be a state where no guard occupies u. In P ′(α′), there +must be a guard on each leaf adjacent to u. Let v and w be two of the leaves adjacent +to u in G′. Let SG′ = S′ +G′ □Ω′\S′(u) {αv, αw} (see short notation Definition 4.31). Let +P ′(αv) = P(α′) ∪ {u} \ {w} and P ′(αw) = P(α′) ∪ {u} \ {v}, so P(αv) occupies v and P(αw) +occupies w. +To create transitions T , we keep all transitions between states within S′(u) the same. +For all β ∈ Ω \ S′(u), we set T (αv, β) as T (α′, β) with w in each movement substituted by +u. Such substitution still constitutes movements as N[w] ⊆ N[u]. Similarly, we create the +transitions for T (αw, β). Furthermore, we set T (αw, αv) = {(v, u), (u, w)}. Note, that the +transitions in reverse direction are derived from symmetry. This shows that there are valid +transitions for all edges of the created strategy graph SG. +In the obtained strategy B vertex u is permanently defended. As we use Cartesian product +with a complete graph over a dominating subset it follows from Lemma 4.32 that the strategy +is still defending. +We repeat the above procedure for each vertex adjacent to at least 2 leaves until all such +vertices are permanently defended. +◀ +5 +Reducing Cactus Graphs +In this section, we prove that Γ∞ +m (G) = γ∞ +m (G) for cactus graphs by showing optimal +strategies and unconditional lower bounds. The main idea is to repeatedly use reductions on +the cactus graph G to produce smaller cactus graph G′. Then we prove that a strategy for G +uses a fixed number of guards more than an optimal strategy for G′. Respective lower bound +then shows that the strategy for G is indeed also optimal. We will describe precise way we +get such results in Section 5.1 but before that, we show the overall structure of the proof. +The proof uses an induction on the number of vertices. The base case is a small graph (1 +or 2 vertices) where the optimal strategy is elementary (see Definition 5.6). The induction +step is described in detail later. Now we show several structural properties of cactus graphs +which allow us to do the induction. +▶ Definition 5.1 (Leaf cycle). Leaf cycle is a cycle which has at most one vertex (called +connecting vertex) which has neighbor such that it is not a vertex of the cycle nor a leaf. +See a leaf cycle on Figure 8. +▶ Definition 5.2 (Leaf component). By a leaf component we mean either a leaf cycle or a +leaf vertex which is not adjacent to a leaf cycle. +▶ Observation 5.3. Every cactus graph with at least 3 vertices contains a leaf component. +Proof. Let us obtain the block-cut tree T representation of the cactus graph and root it in +an arbitrary block node (see [7, block-cutpoint trees, page 36]). Each block node of T either +represents a single edge or a cycle. We observe the deepest nodes of T to get the following +three cases, see Figure 7. +A +There is a deepest node which represents a cycle. +B +A deepest node’s grandparent block is a single edge block. +C +A deepest node’s grandparent block is a cycle block. +D +No deepest node has a grandparent. + +V. Blažej, J. M. Křišťan, and T. Valla +19 +A +B +C +D +or +Figure 7 Example subtrees for structures which always appear in the block-cut tree of a cactus +graph. Big squares represent cycle nodes, small full circles represent articulations, and small empty +circles represent single edge nodes. +Note that in cases A and C the graph contains a leaf cycle, in case B it contains a leaf vertex +which is not adjacent to a leaf cycle. The case D is trivial and the graph is either a cycle or +a single edge. Hence, a cactus graph always contains a leaf component. +◀ +▶ Definition 5.4 (Vertex colors). A non-connecting vertex v of a leaf cycle C is labeled with +a color col(v) which depends on the number of adjacent leaves in the following way. +col(v) is + + + + + + + +�0 +if v is adjacent to 0 leaves +�1 +if v is adjacent to 1 leaf +�2 +if v is adjacent to at least 2 leaves +We shall label v as col(v) = �X if v is the connecting vertex of C. When a vertex can have +different colors (to cover several cases at once) we list them by set of colors. For that purpose, +we may write �0 instead of {�0}, and similarly for �1, �2, and �X. Also, we use a shortcut to +denote all colors �⋆ = {�0, �1, �2, �X}. +See Figure 8 for an example of �0, �1, and �2 vertices. +a +b +c +d +G +e +�0 +�1 +�2 +�X +�⋆ +Figure 8 A leaf cycle of a graph G with a partial +labelled strategy B containing vertices a, b, c, d, and +e with colors �0, �1, �2, �X, and �⋆, respectively. The +original graph was bigger and its rest was connected +to d. To look at the leaf cycle in isolation, the graph +was cut in vertex d that now constitutes interface of +B. This cycle would be denoted by a leaf sequence +(�X,�⋆,�0,�1,�2,�X) (or reversed). +To describe reductions over leaf components we will use a concise notation for the leaf +cycles which just lists the colors of consecutive vertices of the cycle as follows. See Figure 8 +for an example. +▶ Definition 5.5 (Leaf sequence). Let (v1, . . . , vn) be n consecutive vertices of a leaf cycle. +The leaf sequence of vertices (v1, . . . , vn) is (col(v1), . . . , col(vn)) where col(vi) ⊆ {�0, �1, �2, �X}. +Moreover, given two leaf sequences A and B and a graph G which contains a leaf cycle with +a leaf sequence A, let A → B denote a reduction of subgraph with leaf sequence A to one with +leaf sequence B in G to obtain G′. +Note that if the leaf sequence starts and ends with a connecting vertex and contains no +�⋆, then it describes the whole cycle because colors correspond to the number of leaves and +there is only one connecting vertex in a leaf cycle. +Now, we show the base case and the overview of the induction step. + +20 +Computing m-Eternal Domination Number of Cactus Graphs in Linear Time +▶ Definition 5.6 (Base cases). Let the base cases be the following graphs along with their +optimal defending labelled strategies. +A single isolated vertex with no edges defended by labelled strategy +� +({u}, ∅), ({α}, ∅), {α → {u}}, ∅ +� +. +A single isolated edge is defended by labelled strategy +�� +{u, v}, {{u, v}} +� +, +� +{α, β}, {{α, β}} +� +, +� +α → {u}, β → {v} +� +, +� +(α, β) → {(u, v)} +�� +. +5.1 +Technique and Overview +For the induction step, every reduction takes the cactus graph G and changes it to G′ which +has smaller number of vertices. Reductions will be performed on a leaf component which +by Observation 5.3 is always present in a cactus graph on at least three vertices. The two +cactus graphs which have at most two vertices are covered by base cases from Definition 5.6. +More precisely, every reduction shows lower bound and upper bound. Lower bound is +shown for the m-Eternal Guard Configuration and involves using Observations 4.3 +and 4.8–4.10. These tools make the graph smaller and show that in any defending strategy +the removed parts required some minimum number of guards; they give us lower bound +Γ∞ +m (G) ≥ Γ∞ +m (G′) + K for some constant K. +Upper bounds are shown for the m-Eternal Domination and usually involve two +separate steps. First step takes the reduced graph G′ and its optimal strategy S′ +G′ and +shows how to alter the strategy by tools shown in Section 4.3. This does not change the +number of guards, but only structure of the defense. Second step uses the framework shown +in Section 4.2. It takes part of the graph we intend to expand (Definition 4.27), cuts it, and +replaces with an interface equivalent (Definition 4.25) partial labelled strategy, as described +in Observation 4.29. During the expansion, the strategy graph SG does not change (so +SG = SG′), however the graph and mapping does change. The labelled strategy now maps +strategy graph states so that there are new guards and some states have guards moved to +other vertices of the graph. We also show how the transitions change between states that +were altered. When the defense of the graph is managed with K additional guards, this gives +us an upper bound γ∞ +m (G) ≤ γ∞ +m (G′) + K (the same K as in the lower bound). +Combining the lower and upper bound using Lemma 4.2 results in an optimal number of +guards for G for m-Eternal Domination and m-Eternal Guard Configuration. +The used reduction depends on a leaf component that the cactus graph contains by +Observation 5.3. +If the deepest node is not adjacent to a leaf cycle, then we use leaf +reductions shown in Section 3. Using these reduction exhaustively results in having a leaf +cycle (or a base case) – we will show this soon in Lemma 5.7. +To reduce leaf cycles we will need additional properties on edges of the leaf cycle. This +involves being able to forbid movement along an edge, and forcing move along an edge. We +achieve this by partitioning all states of the strategy graph into tree groups which ensure +these properties. The properties are established in Section 5.2. +Having the properties we take the leaf cycle and look at its vertex colors. If a color +pattern is listed among reductions then we have a way to remove it. The reductions are +split into two groups. First recognizes just a small part of the cycle, making it shorter – +these are called cycle reductions Section 5.3. The second recognizes the whole cycle and +removes it entirely, leaving just a few leaves in its place – these are constant component +reductions shown in Section 5.4. We show that one of these reductions may always be used + +V. Blažej, J. M. Křišťan, and T. Valla +21 +by exhaustive search of all possibilities in depicted in Figure 14; and doubling this function, +we show a slightly different proof in Lemma 5.17. +We end this section with the aforementioned proof of the cactus graph structure after +application of leaf reductions in Lemma 5.7 and a diagram overview of the remaining sections +in Figure 9. +▶ Lemma 5.7. Exhaustive application of Reductions t1, t2, and t3 on a cactus graph G +results in reaching the base case or it results in a cactus graph with a leaf cycle. +Proof. We saw in Observation 5.3 that in every cactus there either is a leaf cycle or there is +a set of ℓ ≥ 1 leaves with a common parent which is connected to the rest of the graph with +a single edge. The number ℓ directly implies which tree reduction may be applied. If ℓ = 1, +then we may use Reduction t1; if ℓ = 2, then we use Reduction t3; and last, if ℓ > 2, then we +use Reduction t2. After exhaustive application we either reach the base case or the other +case applies – we have a leaf cycle. +◀ +Section 4.1 Lower bounds +Observation 4.3 Vertex identification +Observation 4.8 Leaf lower bound +Observation 4.9 Star lower bound +Observation 4.10 Path lower bound +Section 5.2 Properties of cycle edges +Property 3 Proper labelled strategy +Section 4.3 Tools for altering strategies +Definition 4.30 Cartesian product over subset +Lemma 4.33 Leaves addition +Definition 4.35 Group state +Section 3 Leaf reductions +Lemma 3.1 Reduction t1 +Lemma 3.2 Reduction t2 +Lemma 3.3 Reduction t3 +Section 5.3 Cycle reductions +Lemma 5.10 Reduction c1 +Lemma 5.11 Reductions c2 and c3 +Lemma 5.12 Reduction c4 +Lemma 5.13 Reduction c5 +Lemma 5.16 Reduction c6 +Section 5.4 Constant component reductions +Definition 5.19 Cactus multigraph +Lemma 5.20 Reduction m1 +Lemma 5.21 Reduction m2 +Lemma 5.22 Reduction r1 +Lemma 5.23 Reduction r2 +Lemma 5.24 Reduction r3 +Lemma 5.25 Reduction r4 +Lemma 5.26 Reduction r5 +Figure 9 Overview of Section 5. Left boxes represent tools obtained in Section 4 (see Figure 2) +and properties we introduce in Section 5.2; Right box shows structure of Section 5; Left-to-right +arrows show which tools are used for which results. Right-to-right (green) arrows show that the +reduction is partially based on or uses another reduction. +5.2 +Properties of Cycle Edges +We shall assume that the built strategy over the graph holds some properties which allow us +to make stronger induction step. More precisely, these properties shall be necessary to show +Reductions c1, c4, and c5. +▶ Definition 5.8 (Edge states). By edge states of (u, v) in Ω (where {u, v} ∈ E(G)) we mean + +22 +Computing m-Eternal Domination Number of Cactus Graphs in Linear Time +creating sets Lu,v, Ru,v, and Nu,v such that +α ∈ Lu,v if ∃β ∈ Ω, (u, v) ∈ T (α, β), +α ∈ Ru,v if ∃β ∈ Ω, (v, u) ∈ T (α, β), +Nu,v = Ω \ (Lu,v ∪ Ru,v) +Note that because the orientation of the edge plays a role in these definitions, we have +La,b = Rb,a, Ra,b = Lb,a, and Na,b = Nb,a. The names of the sets reflect from which side a +guard can traverse the edge. Also note, that if we assume symmetry then when moving to +and from Nu,v the edge {u, v} cannot be traversed. We propose the following edge property +which is somewhat similar to Observation 4.16. +▶ Property 2 (Proper edge states). For a strategy B over graph G an edge {u, v} holds +Property 2 if and only if its edge states Lu,v, Ru,v, and Nu,v are all non-empty, Lu,v∩Ru,v = ∅, +and each of them is a dominating set over SG. +There are several ramifications of an edge having Property 2. Because of Lu,v ∩ Ru,v = ∅ +there is no state where we may choose to move over (u, v) or (v, u), i.e., at most one of +these movements is available. At the same time, as each of these sets is dominating SG, +it follows that we may get into any of these sets in one transition. Last, as each set is +non-empty we may force the strategy to forbid to move over {u, v} in the current and one +future transition by moving to Nu,v at any point. Additionally, we may force a movement +over (u, v) by moving first to some α ∈ Lu,v and then to β ∈ Ru,v such that (u, v) ∈ T (α, β) +as per Definition 5.8. +All the properties that proper edge states additionally have compared to non-proper edge +states are true irrespective of permutations of Lu,v, Ru,v, and Nu,v. Hence, we may use the +same sets on different edges by permuting them and checking that they constitute edge states +of the new edge. +▶ Observation 5.9. For edges {u, v} and {a, b} if we map proper edge states Lu,v, Ru,v, and +Nu,v to new sets La,b, Ra,b, and Na,b (with possibly permuting them) then these constitute +proper edge states of {a, b} if and only if they constitute edge states of {a, b}. +Proof. If the new edge states La,b, Ra,b, and Na,b do not constitute edge states then they +trivially cannot be proper edge states. When Lu,v, Ru,v, and Nu,v are proper edge states +then they are disjoint and nonempty. These properties do not depend on their order so as +long as the new states are edge states they will be proper. +◀ +Our goal will be to have Property 2 on all edges that lie on a leaf cycle that are incident +to at least one �0 or �X vertex. We shall also show that it holds in some special cases to make +several reductions easier. +In reductions, we will check that an edge has Property 2, however, the intuition about +it is as follows. We need to check whether each cycle edge is traversed at least once and +whether it is not traversed at all by at least one state. Also, it is usually trivial, but we +should check that the edge cannot be traversed in both directions from some state. +▶ Property 3 (Proper labelled strategy). A partial labelled strategy B over a cactus graph G +has Property 3 if and only if Property 2 holds for each edge that lie on a cycle and +is incident to a �0 or a �X vertex, +or is on a leaf cycle (�X, �2, �2, �X), +or is incident to a �X vertex while not being a edge which lies between �X and a �2 vertex on +leaf cycle (�X, �2, �0, �0, �2, �X) or (�X, �0, �0, �2, �X). + +V. Blažej, J. M. Křišťan, and T. Valla +23 +Our goal is to keep our cactus graph proper (as per Property 3) in all steps of reducing. +For simplicity, we shall work with reductions as if all edges on cycles which are incident to +�0 or �X vertex have Property 2 and we shall tackle the exceptions to this rule separately in +Observation 5.18 and Lemma 5.27. +5.3 +Cycle Reductions +Due to Lemma 5.7 we know that applying tree reductions may result in either solving the +instance entirely or we obtain a leaf cycle. In this section, we will tackle leaf cycles with +cycle reductions which results in a leaf cycle of constant size. Constant-sized leaf cycles are +then resolved in Section 5.4. +Let C denote a leaf cycle where vertices are labeled with colors according to Definition 5.4. +Cycle reductions consist of the following reductions (see notation in Definition 5.5). E.g., +Reduction c1 describes that a graph G with a leaf cycle that contains consecutive vertices +U with colors (�⋆, �1,�⋆) may be changed to G′ by substituting U with a vertices of colors +(�⋆,�⋆) (so just �1 was removed). At the same time, it claims that γ∞ +m (G) ≤ γ∞ +m (G′) + 1 and +Γ∞ +m (G) ≥ Γ∞ +m (G′) + 1. All of this is concisely written as (�⋆, �1,�⋆) → (�⋆,�⋆) + 1. +▶ Reduction 4. c1 (�⋆, �1,�⋆) → (�⋆,�⋆) + 1 where (�⋆,�⋆) has Property 2. +▶ Reduction 5. c2 (�2, �1,�⋆) → (�2,�⋆) + 1 +▶ Reduction 6. c3 (�2, �2,�⋆) → (�2,�⋆) + 1 +▶ Reduction 7. c4 (�⋆, �0, �0, �0,�⋆) → (�⋆,�⋆) + 1 where (�⋆,�⋆) has Property 2. +▶ Reduction 8. c5 (�⋆, �0, �2, �0,�⋆) → (�⋆,�⋆) + 2 where (�⋆,�⋆) has Property 2. +▶ Reduction 9. c6 (�X, �2, [�0, �2]2k, �X) → (�1) + 3k + 1 and (�X, �2, [�0, �2]2k+1, �X) → (�2) + 3k + 2 +Let a and b be the first and the last vertex of the leaf cycle in G′, respectively, that +are described by the reduction. It is clear that these reductions may be used in cases +where a and b are non-connected disjoint vertices. We note that the reductions will be used +when {a, b} ∈ E(G′) though the result contains a pair of multiedges between a and b in G. +Moreover, these reductions may be used even in case where a = b. Applying the reduction in +such a case results in a loop in a within G′. Though loops and multiedges may be created by +the process they will be immediately removed. These cases will be addressed in Section 5.4.1. +Reductions c1, c4, and c5 require the edge that is being expended (edge {a, b} in G′) +holds Property 2. We shall ensure this by keeping Property 3 for G′ while ensuring that +during every expansion this property is preserved. +▶ Lemma 5.10. Let G′ be G after application of Reduction c1. +G is defended with 1 more +guard than G′. +Proof. Let us label the vertices of colors (�⋆, �1,�⋆) by a, u, b, respectively. Let v be the leaf +adjacent to u. By using Observation 4.8 on vertices {u, v} we get lower bound Γ∞ +m (G) ≥ +Γ∞ +m (G′) + 1. +For upper bound, let La,b, Ra,b, and Na,b be edge states of the edge a, b in the strategy +of G′ obtained as stated in Definition 5.8. We extend all states of La,b and Ra,b by adding +u to them, and we add v to Na,b. We substitute movements (a, b) with {(a, u), (u, b)} in +T (La,b, Ra,b) and we add (u, v) to T (La,b ∪ Ra,b, Na,b). The new vertices are defended as +La,b and Na,b are dominating the strategy graph because Property 2 holds for {a, b} in G′. + +24 +Computing m-Eternal Domination Number of Cactus Graphs in Linear Time +Table 2 List of the cycle reductions; notation is the same as in Table 1 +Reduction +Lower bound +Upper bound +c1 +a +b +−1 +u +v +a +b +v +u +a +b +α +β +γ +γ +a +b +α′ +β′ ++1 +c2 +−1 +u +a +b +a +b +a +u +a +b +γ′ +γ ++1 +β′ +β +Ω +Ω +b +c3 +−1 +u +a +b +a +b +a +u +a +b +γ′ +γ ++1 +β′ +β +Ω +Ω +b +c4 +a +b +−1 +u +a +b +a +b +α +β +α +β +u +γ ++1 +c +d +a +b +α′ +β′ +c5 +a +b +−2 +u +b +a +a +b +α +β +α +β +u +Ω ++2 +c +d +γ +a +b +α′ +β′ +c6 +See Figures 10 and 11 +The edge states for the new edges {a, u} and {u, b} in G remain the same as for {a, b} in G′. +Therefore, these edges now hold Property 2 in G. By extending all states with one guard we +got a defending labelled strategy, so γ∞ +m (G) ≤ γ∞ +m (G′) + 1. By Lemma 4.2 we get that G is +defended with one more guard than G′. +◀ +Reductions c2 and c3 merge a group of consecutive red and pink vertices and defend +leaves adjacent to them by a group state (Definition 4.35). +▶ Lemma 5.11. Let G′ be G after application of Reduction c2 or c3. G is defended with 1 +more guard than G′. +Proof. The reductions are separate for the sake of future argument but they are proven in +the same way. Let us label the vertices of colors (�2, {�1, �2},�⋆) by a, u, b, respectively. Let R1 +denote all leaves adjacent to u. By applying Observation 4.8 on R1 ∪ {u} we get lower bound +Γ∞ +m (G) ≥ Γ∞ +m (G′) + 1. +For upper bound, let γ′ be the group state for leaves adjacent to a. We add to leaves +R1 two new vertices which are defended by γ′ as pointed out in Lemma 4.33. Now we split +u from a, taking its leaves with it that we now label by R2. Transitions between leaves is +extended to T (R1, R2) = {(R1, a), (a, u), (u, R2)} and similarly, we extend all transitions +which used a. The transitions that interacted with a and b are preserved, so the reduction +expands interface equivalent partial labelled strategies. Though we did not need Property 2 +the graph still has Property 3 because the new edge {u, b} takes on exact transitions that +{a, b} had. So if {a, b} held the property in G′, then {b, u} holds it in G. +We added one guard so γ∞ +m (G) ≤ γ∞ +m (G′)+1 and by Lemma 4.2 we get that G is defended +with one more guard than G′. +◀ +▶ Lemma 5.12. Let G′ be G after application of Reduction c4. +G is defended with 1 more +guard than G′. + +V. Blažej, J. M. Křišťan, and T. Valla +25 +Proof. Let us label the vertices of colors (�⋆, �0, �0, �0,�⋆) by a, c, u, d, b, respectively. Using +Observation 4.10 on {c, u, d} we get lower bound Γ∞ +m (G) ≥ Γ∞ +m (G′) + 1. +For upper bound, let L′ +a,b, R′ +a,b, and N ′ +a,b be edge states of the edge a, b in the strategy +of G′ obtained from Definition 5.8. These are proper edge states as {a, b} holds Property 2 +in G′. We extend the states by adding d to all states of L′ +a,b, c to R′ +a,b, and u to N ′ +a,b; this +creates sets La,b, Ra,b, and Na,b. We substitute movements along (a, b) with {(a, c), (d, b)} +in T (La,b, Ra,b), hence, the exchanged parts of the graph are interface equivalent. We add +(c, u) to T (Ra,b, Na,b) and (d, u) to T (La,b, Na,b). The new {c, u, d} vertices are defended by +the nonempty sets Ra,b, Na,b, and La,b, respectively. The edge states for the new edges are +as follows. +La,b = La,c = Ld,b = Nc,u = Ld,u +Ra,b = Ra,c = Rd,b = Lc,u = Nd,u +(4) +Na,b = Na,c = Nd,b = Rc,u = Rd,u +As these are only permutations of the edge sets by Observation 5.9 they hold Property 2. We +get γ∞ +m (G) ≤ γ∞ +m (G′) + 1. By Lemma 4.2 we get that G is defended with one more guard +than G′. +◀ +▶ Lemma 5.13. Let G′ be G after application of Reduction c5. G is defended with 2 more +guards than G′. +Proof. The proof goes very similarly as the proof of Lemma 5.12, but all states Na,b shall +group defend leaves adjacent to u while u will be permanently occupied. +We label the vertices of colors (�⋆, �0, �2, �0,�⋆) by a, c, u, d, b, respectively. Let R be the +leaves neighboring u. Using Observation 4.9 on u and its neighborhood we get lower bound +Γ∞ +m (G) ≥ Γ∞ +m (G′) + 2. +Repeat the same sequence of steps as in the proof of Lemma 5.12 which uses one guard +and then add leaves adjacent to u by Lemma 4.33 using one extra guard. This does not +change transitions over the edges which are not incident to the leaves so by Observation 5.9 +they still hold Property 2. We get γ∞ +m (G) ≤ γ∞ +m (G′) + 2. By Lemma 4.2 we get that G is +defended with two more guards than G′. +◀ +We remark that using reductions t1, t2, t3, c1, c4, and a small set of constant component +reductions is sufficient to solve so-called Christmas cactus graphs (graphs where each edge is +in at most one cycle and each vertex is in at most two 2-connected components) for which +the optimal strategy we presented in [1]. The remaining reductions tackle vertices of color �2, +which are not present in the class of Christmas cactus graphs. +The last cycle reduction is a curious special case, let us recall it first. +▶ Reduction 9. c6 (�X, �2, [�0, �2]2k, �X) → (�1) + 3k + 1 and (�X, �2, [�0, �2]2k+1, �X) → (�2) + 3k + 2 +We shall use all the other cycle reductions first and if none of them can be used, then we +use Reduction c6. This allows us to assume a particular structure which we define and prove +now. The reason behind this structure may also be well understood from decision diagram +of reduction application in Figure 14. +▶ Definition 5.14 (RW-cycle). A leaf cycle C is a RW-cycle if it consists of vertices with +alternating �2 and �0 colors such that the first and last is �2, i.e., C = (�X, �2, �0, �2, �0, . . . ,�2, �0, �2, �X). +▶ Lemma 5.15. Assume a leaf cycle C where Reductions c1, c2, c3, c4, and c5 cannot be +applied anywhere. Then, |C| ≤ 6 or C is a RW-cycle. + +26 +Computing m-Eternal Domination Number of Cactus Graphs in Linear Time +Proof. First, note that if the leaf cycle contains �1 vertices, then there always is a �1 vertex +which neighbors �0 or �X, in that case we use Reduction c1, or it neighbors �2, we use Reduction c2. +Hence, all �1 vertices are removed if Reductions c1 and c2 were exhaustively used. +Next, as only �2, �0, and �X vertices remain, exhaustively using Reduction c3 ensures that +there are no two adjacent �2 vertices. (Note that if only red vertices remained, than we end +up with (�X, �2, �X) which removes the multiedge by Reduction m2 and then uses Reduction t3.) +Last, if there is a (�2, �0, �0) part of a leaf cycle then the �2 vertex is either also adjacent to +�X or Reduction c5 can be used as would (�⋆, �0, �2, �0, �0) necessarily occur. Exhaustively using +Reduction c4 ensures that such cases do not occur. So whenever there are two �0 adjacent +vertices the �X vertex is at distance at most 2 from them. This means that either the cycle has +at most 5 vertices or the vertices of colors �2 and �0 alternate and constitute a RW-cycle. +◀ +The lower bound for an RW-cycle will not be much harder than for other reductions, +however, for the upper bound we will need a strategy made just right for such a cycle. +▶ Lemma 5.16. Reduction c6 is correct. +Proof. Let us label the vertices along the cycle as u1, u2, . . . , un with u1 being the connecting +vertex (so that �2 vertices are even). Let un+1 = u1. Let R2, R4, etc. be the leaves adjacent +to the red vertices u2, u4, and so on. +First, we use Observation 4.8 on {u6} ∪ R6; second, we use Observation 4.9 on N[u4]. +We shortened the cycle by 4 and got lower bound of 3. By repeating the argument k times +we end up with a cycle G′ of constant size 2 or 4. The cycle of size 2 gets reduced by m2 and +then t3 which results in a lower bound of 1. The RW-cycle of size 4 has form (�X, �2, �0, �2, �X) +and its lower bound is shown in Reduction r4 to be 2. Putting the 3k for every 4 vertices +together with 1 and 2 lower bound for respective sizes of the cycle, we get the desired lower +bounds. See Figure 10 for an illustration of shortening the cycle by 4. +−3 +u2 +u4 +R2 +u8 +R6 +R8 +u7 +u6 +u1 +u2 +R1 +u8 +R8 +u7 +u1 +R4 +u3 +u5 +Figure 10 Part of the lower bound proof for Reduction c6 +Strategy for the upper bound is quite tricky to describe so let us define a few new notions +just for its description. Let red-parity of an even number i be the parity of i/2. This divides +red vertices of the RW-cycle into red-odd and red-even, based on the red-parity. Let reverse +labeling be the labeling of the RW-cycle in opposite ordering, i.e., if u′ +1 = un+1, u′ +2 = un, +u′ +3 = un−1, . . . , u′ +n = u2, and u′ +n+1 = u1, then u′ +1, . . . , u′ +n is the reverse labeling with respect +to labeling u1, . . . , un. +For the upper bound, we distinguish two cases depending on the size of the RW-cycle. +First case is that the size is n = 4k + 2, the second has size n = 4k. +We now focus on the first case, where the RW-cycle has size n = 4k + 2. Note that in +RW-cycle of this size reverse labeling does not change the red-parity of red vertices. We alter +the strategy by gradually expanding the states as follows. +S∗ +G′ = S′ +G′ □S′(u1) {α1, α2, β4, β8, . . . , β4k} and SG′ = S∗ +G′ □Ω\S′(u1) {β2, β6, . . . , β4k+2} + +V. Blažej, J. M. Křišťan, and T. Valla +27 +Now we perform expansion from G′ to G and set the states Ω of SG as follows. +P(α1) = +k� +x=0 +{u4x+3} +P(α2) = +k� +x=0 +{u4x+1} +P(β4x) = +� +(P(α2) ∩ {uj}ji) +� +∪ {R4x} +(5) +P(β4x+2) = +� +(P(α1) ∩ {uj}ji) +� +∪ {R4x+2} \ {u1} +Notice that u1 ∈ P(α2) as u4x+1 = u1 for x = 0 and u1 ∈ P(α1) as u4x+3 = u4k+3 = un+1 = +u1 for x = k. For P(β4x+2) the intersections imply that u1 is not contained, but we mention +it explicitly for clarity (as we do not consider un+1 to be uj for j < i even though it equals +u1). The state β2i group defends leaves adjacent to u2i. Note that they behave differently +based on their their red-parity. +Now for the transitions. To make the notation concise let us shorten consecutive move- +ments through red vertices as Fi = {(ui, ui+1), (ui+1, ui+2)} and Bi = {(ui, ui−1), (ui−1, ui−2)} +(as forward and backward). Note that we use Fi and Bi only for odd values of i. +Let us have integers x and y and assume, without loss of generality, that x ≤ y. We set +the movements of transitions as follows. If 2x and 2y have the same red-parity, then +T (β2x, β2y) = {(R2x, u2x), (u2x, u2x+1)} ∪ +y−4 +� +i=x +F2i+3 ∪ {(u2y−1, u2y), (u2y, R2y)}. +(6) +Otherwise, 2x and 2y have different red-parity. If x is odd (so y is even), then their transition +is defined as follows. +T (β2x, β2y) = {(R2x, u2x), (u2x, u2x−1)}∪ +x� +i=2 +B2i−3 ∪ +2k−1 +� +i=y+1 +B2i+3 ∪{(u2y+1, u2y), (u2y, R2y)} +If the red-parity is different and x is even, then in reverse labeling and swapping x with y we +end up in the case where the red-parity is still different, but x is odd. This case was already +solved. The other direction of these transitions is filled in by symmetry (Property 1). +It remains to describe transitions with α1 and α2. +T (α1, α2) = {(u1, u1)} ∪ +k−1 +� +i=0 +F4i+3 +T (β4x, α1) = {(R4x, u4x), (u4x, u4x−1)} ∪ +x−1 +� +i=1 +B4i+1 +(7) +T (β4x+2, α2) = {(R4x+2, u4x+2), (u4x+2, u4x+1)} ∪ +x� +i=1 +B4i−1 +(8) +Note that α1 is α2 in reverse labeling. Hence, the case T (β4x, α2) is equivalent to T (β4x, α1) +in reverse labeling. Similarly, the case T (β4x+2, α1) is equivalent to T (β4x+2, α2) in reverse +labeling. See a part of this strategy on Figure 11. +For the interface equivalency, note that in α1, α2, and β4k occupy u1 and states β4k+2 do +not occupy u1. We showed how to transition between every pair of states, so the strategies +are interface equivalent with a single pink vertex with expanded states as in SG′. +Now for the second case, where the RW-cycle has size n = 4k. Note that in RW-cycle of +this size reverse labeling changes the red-parity of red vertices, which was not true in the +first case. The difference in the construction of the strategy is that now we expand from a + +28 +Computing m-Eternal Domination Number of Cactus Graphs in Linear Time +u1 u2 u3 u4 u5 +u6 u7 u8 u9 +u10u11u12u13 +u14u15 +u16u17 +u18 u19 = u1 +β2 +β4 +β8 +β10 +β12 +β14 +β16 +β18 +β6 +α1 +α1 +α1 +α1 +α1,2, β4k +α2 +α2 +α2 +α2 +β6 +β12 +α1 +α2 +Figure 11 Part of a strategy on a RW-cycle of red-odd size 18 with guards (shown purple) placed +on P(β6). A few selected transitions are shown as an example. +red vertex. Let u′ +1 and u′ +2 be the two leaves of the red vertex. Let δ′ = Ω′ \ (S(u′ +1) ∪ S(u′ +2)). +We gradually alter the strategy in the following way. +S1 +G′ = S′ +G′ □S(u′ +1) {γ, β4, β8, β12, . . . , βn} +S2 +G′ = S1 +G′ □S(u′ +2) {α1, β2, β6, . . . , βn−2} +SG′ = S2 +G′ □δ′ {α2} +We perform the expansion to get SG such that all the states have exactly the same definitions +as in the first case, see Equation (5). We note a major difference: in the second case, u1 is +not an element of P(α1). We added one extra state γ which has P(γ) = P(α1). There will +be a major significance for this state when proving edge properties. +Now we describe the transitions for the the strategy on G. For 2x and 2y of the same +red-parity, the transition Equation (6) still holds. In case 2x and 2y (with x < y) have +different red-parity, then we consider two separate cases based on red-parity of 2x. +T (β4x, β4y+2) = {(R4x, u4x), (u4x, u4x−1)} ∪ +x−1 +� +i=1 +B4i+1 ∪ +k� +i=y+2 +B4i−1 ∪ +∪ {(u4y+1, y4y+2, (u4y+2, R4y+2)} +T (β4x+2, β4y) = {(R4x+2, u4x+2), (u4x+2, u4x+1)} ∪ +x−1 +� +i=0 +B4i+3 ∪ +k+1 +� +i=y+2 +B4i−3 ∪ +∪ {(u4y−1, y4y, (u4y, R4y)} +Notice the difference in u1 – transition T (β4x, β4y+2) does not move through u1 so there +u1 is stationary during it; in T (β4x+2, β4y) movements {(u2, u1), (u1, un)} happen. To fill +all possibilities of mutual transitions among β2x we add transitions obtained by reversed +labeling and symmetry. +Now we show the transitions with α1 and α2. Note that reversed labeling does not change +these two states. For β4x+2 we can apply Equation (8) to get T (β4x+2, α2), and by reversing +the labeling this gives us also T (β4x, α2). Note that after this transition there is one less +guard on G as it leaves through the interface {u1}. In particular, T (β2x, α2) moves to u1 via +(u2, u1) if 2x is red-odd, and via (un, un+1) if 2x is red-even. +Similarly, for β4x we can apply Equation (7) to get T (β4x, α1), and by reversing the +labeling we get T (β4x+2, α1). This transition did not interact with the interface. The +transition among the two states is as follows. +T (α1, α2) = +k−1 +� +i=0 +F4i+3 + +V. Blažej, J. M. Křišťan, and T. Valla +29 +Note that this again results in a move (un, un+1). +Last, we introduce the new state γ which has the same guard configuration as α1, but +differs in one transition. +So T (γ, β2x) = T (α1, β2x), and T (γ, α2) = ∅ (all guards are +stationary), but T (γ, α2) shall be T (α1, α2) in reverse labeling. More precisely, +T (γ, α2) = +k−1 +� +i=0 +B4i+3. +This contains a move (u2, u1). See how movements interact with the interface in Figure 12. +u1 +u1 +red-odd +red-even +Figure 12 Movements through the �X vertex in red-odd and red-even RW-cycle. +We discussed the interface impact of all transitions and note that they are equivalent to +those in SG′, hence, the exchanged strategy is interface equivalent. +It remains to show that SG is a proper strategy in both cases. The strategy started S′ +G′ +was a clique and by Cartesian product over single vertices it remained a clique. Thus, it +suffices to say that there is at least one state in Lui,ui+1, Rui,ui+1, and Nui,ui+1, and that +Lui,ui+1 ∩ Rui,ui+1 = ∅ for every i ∈ {1, . . . , n}, as any non-empty subset of vertices of the +clique is dominating. +Now we show the partitioning of the states into Lux,ux+1, Rux,ux+1, and Nux,ux+1 for each +x ∈ {1, . . . , n}, see Figure 13 for an illustration. First, observe that all closed neighborhoods +of u2x for 4 ≤ 2x ≤ n − 2 contain exactly 2 guards in all the states we defined for this +strategy. Let x be an even integer such that 4 ≤ x ≤ n − 2. Let e = {ux+1, ux+2}. +N +R +L +R +L +R +L +R +L +R +L +e1 +e2 +Figure 13 The states of β2x that belong to Le, Re, Ne for e equal to the edges e1 and e2. +We show that βx ∈ Ne by a contradiction. Assume that in some transition from βx a +guard moved through e. As in βx vertex ux+1 is not occupied the guard must have moved +from ux+1. However, then N[ux] would have 3 guards after the transition which cannot +happen as we observed; a contradiction. +For edges that could not be addressed in the argument because they are too close to the �X +vertex – e1 = {u1, u2} and e2 = {u3, u4}. We observe that for red-even RW-cycles α1 ∈ Ne1 +and γ ∈ Ne2. For red-odd RW-cycles β2 ∈ Ne2 and βn ∈ Ne1. +Now we claim that for any even x such that 2 ≤ x ≤ n, e = {ux+1, ux+2}, the states βy +where y ̸= x are in Le if and only if ux+1 ∈ P(βy), and they are in Re otherwise. We shall +prove this more intuitively, as otherwise the claim can be proved by exhaustively listing all +edges in all the transitions. First notice, that all movements from βy which are not incident +to leaves are performed over a continuous part of the cycle which starts in uy, and that they +move “away” from uy towards the other end of the part. The movements always move an + +30 +Computing m-Eternal Domination Number of Cactus Graphs in Linear Time +occupied �0 to �2 and if the part continues then moves the �2 to the adjacent �0 (this is true +even when moving through the �X vertex). Hence, if e (that is not incident to �X) is included +in the part of the movement, then we move (ux+1, ux+2) if and only if xx+1 is occupied, and +we move (ux+2, ux+1) if and only if ux+1 was unoccupied. This proves the claim. +Remainder of the edges which start at even positions and their N, L, and R sets can be +obtained by the same argument on reversed labelling. +◀ +5.4 +Constant Component Reductions +The following lemma shows that considering the constant component cases completes the list +of all necessary reductions. +▶ Lemma 5.17. Let us have a cactus graph G. After an exhaustive application of leaf and +cycle reductions the reduced cactus graph G′ is either a base case or it contains a leaf cycle +of constant size. +Proof. First, by Observation 5.3 the cactus always contains a leaf component. +If we +exhaustively apply tree reductions, then by Lemma 5.7 we are either done or there is a leaf +cycle C. In 5.15 we saw that an exhaustive application of the cycle rules results either in a +base case or a cycle with alternating �2 and �0 vertices, which gets tackled by Reduction c6. +The cases that remain are cycles of constant sizes where none of the reductions may be +applied. +◀ +We obtain the list of constant leaf cycles by the following procedure. First, we apply +Reductions c1 and c2 exhaustively. This removes all pink vertices from the leaf cycle. Now, +let us scan over the vertices of the leaf cycle in a linear order of vertices along the cycle, +starting from the connecting vertex. On the one hand, whenever there is a cycle reduction +applicable on the vertices which were scanned so far, then we can apply it. Hence, such a +leaf cycle does not belong to constant leaf cycle cases. On the other hand, when the cycle +returns back to the connecting vertex and still no cycle reduction may be used, then this +cycle constitutes a constant leaf cycle. We present a full search diagram in Figure 14. +Again, we shall denote the reductions concisely as defined by Definition 5.5. However, +in constant component reductions the leaf sequence describes the whole leaf cycle and the +connecting vertex is listed as the first and the last vertex. The vertices of the leaf cycle +will be denoted by u, u1, u2, . . . , un−1, u where u is the connecting vertex. Let R1, . . . , Rn−1 +denote sets of all leaves adjacent to vertices u1, . . . , un−1, respectively. Note that the size +0 ≤ |Ri| ≤ 2 and directly coincides with color of respective vertex ui. See Figure 15 for an +example of a leaf sequence of constant leaf cycle and notation of its vertices. +Recall that the cycle reductions may be used even when the result does not create a +simple graph, which is resolved in Section 5.4.1. +▶ Observation 5.18. A strategy for a leaf cycle (u, u1, u2, u) with colors (�X, �2, �2, �X) is built +in such a way that the edge (u1, u2) holds Property 2 (even though it is not incident to a �0 +vertex) which makes an expansion of Reductions c4 or c5 over this edge possible. +Proof. This leaf cycle gets reduced by Reduction c3, then m2, and last with tree reduction +t3. We show that in the strategy resulting for expansions holds Property 2 on edges {u, u1} +and {u, u2}. See Figure 15 for an illustration. Checking the exact movements of this strategy, +we have that +(u1, u2) ̸∈ T (Lu,u1, Ru,u1), (u1, u2) ̸∈ T (Lu,u1, Nu,u1), (u1, u2) ∈ T (Ru,u1, Nu,u1). + +V. Blažej, J. M. Křišťan, and T. Valla +31 +m1 +m2 +r1 +c4 +r3 +c5 +c3 +r2 +c5 +c3 +r2 +r3 +c4 +c∗ +4 +c5 +c3 +r5 +c5 +c3 +r4 +c5 +c∗ +5 +c5 +c3 +c6 +c3 +c3 +c3 +�X +�X +�X +�X +�0 +�X +�0 +�2 +�2 +�0 +�X +�0 +�2 +�2 +�0 +�X +�X +�X�0 +�X +�0 +�2 +�2 +�0 +�X +�0 +�2 +�2 +�0 +�X +�X�0 +�X +�X�0 +�2 +�0 +�2 +�2 +�0 +�2 +�2 +�0 +�2 +�2 +�0 +�X +�2 +Figure 14 Case analysis of applied reductions on a leaf cycle. Vertices �1 were removed first by +exhaustively applying their reductions. Scanning over vertices of a leaf cycle in order from the +connecting vertex we identify these cases. The leaves show which reduction should be used for the +scanned leaf cycle. Labels c1 up to c5 (yellow leaves) signify cycle reductions; labels mi and ri (red +leaves) signify constant component reductions; nodes with a star ∗ require Observation 5.18. We +can check that all the cases are covered by seeing that all inner (empty) nodes have outgoing edges +labelled �0, �2, and �X. +In particular, we may set Lu,u1 = Nu1,u2, Ru,u1 = Lu1,u2, and Nu,u1 = Ru1,u2. As the edge +move sets for {u, u1} holds the properties which require all these sets to be non-empty, we +have that they hold for {u1, u2} as well. +◀ +β +γ +Ω +α +Ω +α +β +γ +u1 +u0 =un +u2 +γ +α +β +Ω +α +β +Ω +α +β +t3 +m2 +c3 +R1 +R2 +Figure 15 Left: Building the strategy for a (�X,�2,�2,�X) leaf cycle. The states α, β, and γ are +representants of sets Lu,a, Ru,a, and Nu,a, respectively. Right: Example guard configurations for +states α, β, and γ. +From Observation 5.18 we know that the cases (�X, �2, �0, �0, �0, �2, �X) and (�X, �2, �0, �2, �0, �2, �X) can +be reduced by Reductions c4 and c5, respectively. See these cases in Figure 14. +5.4.1 +Loops and Multiedges +Similarly to Observation 5.18, for the constant cases where we need to show that the properties +hold. By allowing cycle reductions to apply in cases where the vertices a and b are adjacent, +or even identical, we allowed the result of the reduction to contain multiedges or loops. This +intermediate form of the graph can be thought of as a generalized cactus graph. +▶ Definition 5.19 (Cactus multigraph). Let the cactus multigraph be a multigraph (possibly +with loops) that is connected and its every edge lies on at most one cycle. +A cactus multigraph differs from a cactus graph by allowing loops on arbitrary vertices +(cycles of size 1) and allowing 2 multiedges between some vertices (cycles of size 2). The + +32 +Computing m-Eternal Domination Number of Cactus Graphs in Linear Time +cactus multigraph may be changed to a cactus graph by removing multiedges and loops. The +following two reductions take care of that. +▶ Reduction 10. m1 Let G′ be G with one loop removed. +▶ Reduction 11. m2 Let G′ be G with a multiedge {u, v} (2 edges) where v has degree 2 (1 +neighbor) changed to a single edge. +u +u +u +v +u +v +Figure 16 Left: loop reduction m1; Right: multiedge reduction m2 +Observe these reductions on Figure 16. We now prove that they do not need any additional +guards. +▶ Lemma 5.20. Let G′ be G after application of Reduction m1. +G is defended with the +same number of guards as G′. +Proof. The strategy on G can be easily adapted to G′ by replacing any guard movement +along the loop of u by not moving the guard on u, thus Γ∞ +m (G′) ≤ Γ∞ +m (G). At the same +time, any strategy on G′ is applicable on G, so γ∞ +m (G) ≤ γ∞ +m (G′). The equality follows from +Lemma 4.2. However, we would like the loop in u to have Property 2. +Intuitively, to keep the properties, we could say that at any configuration where u is +occupied the guard can be moved along the loop in any direction or to be forbidden from +moving along it while the configuration stays the same. Formally, we can achieve the same by +setting SG = S′ +G′ □S′u {α, β, γ}. We now set that T (α, β) = {(u, u)}. This creates Lu,u = α, +Ru,u = β, and Qu,u = Ω \ {α, β}. This altered strategy holds Property 2 for the loop of u as +the sets Lu,u, Ru,u, and Nu,u are non-empty and dominating SG because S′(u) dominates +S′ +G′. +◀ +In our case, Reduction m1 gets used after Reduction c4 is used on (�X, �0, �0, �0, �X) or after +Reduction c5 is used on (�X, �0, �2, �0, �X). It could also be used on (�X, �1, �X) after Reduction c1; +but in that case we can remove the multiedge first. +▶ Lemma 5.21. Let G′ be G after application of Reduction m2. +G is defended with the +same number of guards as G′. +Proof. Let e1, e2 be the two different edges {u, v} oriented as (u, v) in G. We assume that +G′ is G with e2 removed. Lower and upper bound are clear as every move along e2 can be +changed to a move along e1 and the strategy on G′ is applicable to G without change. The +challenge is, again, to show that Property 2 holds for e1 and e2 in G. +Let β′ = S′(v) and α′ = Ω′ \ β. To prove the property on e1 and e2, we will modify the +strategy on G′ in the following way. If β′ ̸= Ω′, then there is a move along e1 in G′. In that +case, we set SG′ = S′ +G′ □β′ {β, γ} while we alter the movements T (α, γ) to move along e2 +instead of e1. The edge states have α ∈ Le1, β ∈ Re1, and γ ∈ Ne1, and similarly for e2 +(with swapped β and γ). +Second case is that β′ = Ω′ while α′ ̸= Ω′. Here, we alter the strategy such that for all +states where u is not occupied, we move the guard from v to u. This makes it so that v is +occupied in states α which we now split into α1 and α2 in the same way as in the previous +case. + +V. Blažej, J. M. Křišťan, and T. Valla +33 +The last case is β′ = Ω′ while α′ ̸= Ω′. Here we set SG′ = S′ +G′ □Ω′ {α1, α2, α3} and +setting T ({α1, α2}) = {(u, v), (v, u)}, i.e., transitioning along e1 and e2 in opposite directions. +Also T (α1, α3) and T (α2, α3) have all guards stationary. This makes edge states as α1 ∈ Le1, +α2 ∈ Re1, and α3 ∈ Ne1 while the exact same edge states work for e2. +In all the cases the edge states are non-empty, hence, Property 2 holds for e1 and e2 after +Reduction m2. +◀ +α +α′ +β′ +m2 β +γ +α1 +Ω +Ω +Ω +Ω +α2 +Ω +Ω +e1 +e1 +e2 +α′ +u +v +Figure 17 Cases of Reduction m2. Left: There is a movement along the edge. Middle: Leaf is +permanently occupied. Right: Leaf and its neighbor are permanently occupied. +We note that in our strategy the case where v is permanently defended shall not occur. +If we did not use Reduction m2 the number of constant size leaf cycle reductions would be +significantly bigger. It gets used after reduction of (�X, �0, �1, �X) or (�X, �1, �1, �X) by c1, (�X, �2, �1, �X) by +c2, (�X, �2, �2, �X) by c3, (�X, �0, �0, �0, �0, �X) or (�X, �0, �0, �0, �2, �X) by c4, (�X, �0, �0, �2, �0, �X) or (�X, �0, �2, �0, �2, �X) +by c5. Without Reduction m2 each of these cases would have to be analyzed separately. +5.4.2 +Constant Size Leaf Cycle Reductions +By Lemma 5.17 the last cases that have to be resolved are covered by the following reductions. +See Table 3 for accompanying lower bound and upper bound proof illustrations. Also see +Figure 9 for diagram of notions used within proofs of these reductions. +▶ Reduction 12. r1 (�X, �0, �0, �X) → (�1) + 0 +▶ Reduction 13. r2 (�X, �0, �2, �X) → (�1) + 1 +▶ Reduction 14. r3 (�X, �0, �0, �2, �X) → (�2) + 1 +▶ Reduction 15. r4 (�X, �2, �0, �2, �X) → (�2) + 2 +▶ Reduction 16. r5 (�X, �2, �0, �0, �2, �X) → (�2) + 2 +Now we proceed to show correctness of these reductions. First group consists of reductions +where a leaf cycle is reduced to �1 vertex u and its leaf v. The vertices of the expended leaf +cycle are denoted by u, u1, u2, . . . , un−1, u. +▶ Lemma 5.22. Let G′ be G after application of Reduction r1. +G is defended with the same +number of guards as G′. +Proof. Using Observation 4.3 to identify u2 with u1 then using Reduction m2 results in +lower bound of 0. +For the upper bound, we first expand {u, v} to multiedges e1 and e2 as per Reduction m2. +Then we take G′ and change it to G by splitting v into two vertices u1 and u2. We create β by +substituting all occurrences of v in P(β′) with u1, and create γ by substituting all occurrences +of v in P(γ′) with u2. The transition between them becomes T (β, γ) = {(u1, u2)}. The +strategy is interface equivalent as the strategy did not change states or transitions of the +interface. +We set Lu1,u2 = Nu,u1, Ru1,u2 = Nu,u1, and Nu1,u2 = Lu,u1 so the new edge {u1, u2} +holds Property 2 and the strategy for G holds Property 3. + +34 +Computing m-Eternal Domination Number of Cactus Graphs in Linear Time +Table 3 List of constant component reductions; thick red edges do not hold Property 2. +Reduction +Lower bound +Upper bound +r1 +−0 +u +u1 +u2 +u +v +u +v +u +v ++0 +α′ +β′ +γ′ +α′ +β′ +α +β +γ +r2 +u +v +u +v +−1 +u +u1 +u2 +u +v ++1 +α′ +β′ +γ′ +α′ +β′ +α +β +γ +Ω +r3 +−1 +u +u +u +u1 +u2 +u3 +R3 +β +α ++1 +Ω +Ω +γ +u +β +γ +α +Ω +u1 u2 +Ω +α′ β′ +r4 +−2 +u +u1 +u2 +u3 +R3 +R1 +u +u +α ++1 +Ω +Ω +Ω +β +α +Ω +Ω +γ +u +u1 u2 +β +γ ++1 +r5 +u1 +u4 +−2 +u +u +R4 +u2 +u3 +R1 +u +u1 u2 +γ +Ω +Ω +Ω +α γ1 +β +γ2 ++2 ++0 +α +β +No guard was added so γ∞ +m (G) ≤ γ∞ +m (G′) and by Lemma 4.2 we get that G is defended +with the same number of guards as G′. +◀ +We recall that by Ri we denote all leaves adjacent to ui. +▶ Lemma 5.23. Let G′ be G after application of Reduction r2. +G is defended with 1 more +guard than G′. +Proof. Using Observation 4.8 on {u2} ∪ R2 then using Reduction m2 results in lower bound +of 1. +We do the same expansion as in the proof of Lemma 5.22. After that, we use Lemma 4.33 +to add leaves R2 to u2 while using one extra guard to defend it. Graph holds Property 3 by +the same argument as in the proof of Lemma 5.22. We added one extra guard which results +in γ∞ +m (G) ≤ γ∞ +m (G′) + 1 and by Lemma 4.2 we get that G is defended with one more guard +than G′. +◀ +We now prove correctness of the other three cases. The reduced graph G′ now consists of +a single �2 vertex u (and its leaves). The partial labelled strategy on G′ has states α′ and +β′ that defend the two leaves adjacent to u. Also, let δ′ = Ω′ \ (α′ ∪ β′), which may be an +empty set. +▶ Lemma 5.24. Let G′ be G after application of Reduction r3. +G is defended with 1 more +guard than G′. +Proof. Using Observation 4.8 on u3 and one of its leaves, identifying u2 with u using +Observation 4.3, and using Reductions m1 and m2 to remove loops and multiedges results in +lower bound of 1. + +V. Blažej, J. M. Křišťan, and T. Valla +35 +For the upper bound, let u1 and u3 be the two leaves adjacent to u in G′. Let α′ = S′(u1) +and β′ = S′(u3). +We make SG′ = S′ +G′ □β′ {β, γ}. +Now we expand the graph G′ by +first applying Lemma 4.33 on u3, adding 2 new leaves to it using one additional guard. +Next, we add a vertex u2 while connecting it to u1 and u3 and we move γ from R3 to +u2 which is easy as u2 is a neighbor of u3. The only major change in transitions is that +T (α, γ) = {(u1, u2), (u, u), (u3, u3)} instead of {(u1, u), (u, u3), (u3, u2)}. No other transitions +change, and u behaves the same, so the exchanged graphs are interface equivalent. See +Figure 18 for strategy SG. +β +α +Ω +γ +δ +β +α +Ω +Ω +γ +α +γ +β +α +γ +β +δ +Figure 18 Strategy for (�X,�0,�0,�2,�X) leaf cycle +We note that each is traversed at some point and that α ∈ Nu2,u3, β ∈ Nu1,u2, and +γ ∈ Nu,u1 so these edges hold Property 2 and the strategy for G holds Property 3. The edge +{u, u3} does not need to hold the property as it is a special case tackled in Lemma 5.27. +We got that γ∞ +m (G) ≤ γ∞ +m (G′) + 1 and by Lemma 4.2 we get that G is defended with one +more guard than G′. +◀ +▶ Lemma 5.25. Let G′ be G after application of Reduction r4. G is defended with 2 more +guards than G′. +Proof. Using Observation 4.8 first on {u1, v1} where v1 ∈ R1, then again on {u3, v3} where +v3 ∈ R3, identifying u2 with u using Observation 4.3, and using Reductions m1 and m2 to +remove loops and multiedges results in lower bound of 2. +For upper bound, repeat exactly the expansion from Lemma 5.24 on G′ which uses +one extra guard. Continue by applying Lemma 4.33 on u1 which adds the leaves R1 using +one extra guard while returning the defending labelled strategy on G. The properties for +edges {u1, u2}, {u2, u3}, and interface equivalency still hold from Lemma 5.24. However, +we can split γ into two states γ1 and γ2 which dictates whether T (γi, δ) traverses through +{(u2, u1), (u1, u)} or {(u2, u3), (u3, u)}. This ensures Property 3 for {u, u1} and {u, u3} as +former cannot be traversed from γ2 and latter from γ1. Hence, we have γ∞ +m (G) ≤ γ∞ +m (G′) + 2 +and by Lemma 4.2 we get that G is defended with two more guards than G′. +◀ +▶ Lemma 5.26. Let G′ be G after application of Reduction r5. G is defended with 2 more +guards than G′. +Proof. Using Observation 4.8 first on {u1, v1} where v1 ∈ R1, then again on {u4, v4} where +v4 ∈ R4, and last identifying u2 and u3 with u using Observation 4.3 results in lower bound +of 2. +For upper bound, repeat exactly the expansion from Lemma 5.25 on G′ which uses two +extra guards (we do not use part of the proof which proved the property). Then we make +SG′ = S′ +G′ □S′(u2) {γ1, γ2}, i.e., splitting γ into γ1 and γ2. We expand G′ to G by splitting +u2 into u2 and u3 (while renaming u3 to u4). We preserve a guard of γ1 on u2 and γ2 on u3. +Transition between them will be T (γ1, γ2) = {(u2, u3), (u, u)}. This is interface equivalent. +See Figure 19 for strategy SG. +We have Property 3 as each edge is traversed and {u1, u2} cannot be traversed from γ2, +{u2, u3} from α, and {u3, u4} cannot be traversed from γ1. We note that the other two cycle + +36 +Computing m-Eternal Domination Number of Cactus Graphs in Linear Time +Ω +Ω +α γ1 +β +γ2 +δ +α +γ1 +β +γ2 +δ +Ω +Ω +Ω +α γ1 +β +γ2 +α +γ1 +β +γ2 +Ω +Ω +α γ1 +β +γ2 +α +γ1 +β +γ2 +Figure 19 Strategy for (�X,�2,�0,�0,�2,�X) leaf cycle +edges {u, u1} and {u, u4} are part of the exception which is tackled in Lemma 5.27. Hence, +we have γ∞ +m (G) ≤ γ∞ +m (G′) + 2 and by Lemma 4.2 we get that G is defended with two more +guards than G′. +◀ +Now we tackle the exception in Property 3 which influences Reductions r3 and r5. +▶ Lemma 5.27. The order of reductions can be changed so that in a (�X, �0, �0, �2, �X) or +(�X, �2, �0, �0, �2, �X) leaf cycle Property 2 is not required for edges that connect a �X and a �2 +vertex. +Proof. Let us label by e an edge which connects a �X and a �2 vertex. Reductions which +require the Property 2 on an edge are Reductions c1, c4, and c5. If e is not a result of any of +these reductions then there is no need for e to hold Property 2. Otherwise, let us analyze +the cases separately. +Reduction c1 resulted in e – before reduction we had (�X, �1, �2, . . . ) where we can use +Reduction c2 instead. This results in (�X, �2, . . . ) without needing the property for e. +Reduction c4 resulted in e – before reduction we had (�X, �0, �0, �0, �2, �0, �0, [�2, ] �X). Hence, we +may use Reduction c5 instead. This results in (�X, �0, �0, �0, [�2, ] �X) where e has the property. +Reduction c5 resulted in e – before reduction we had (�X, �0, �2, �0, �2, �0, �0, [�2, ] �X) so we may +use Reduction c5 on the second �2 vertex instead. This results in (�X, �0, �2, �0, [�2, ] �X) where e +has the property. +We used other reductions to avoid reaching these leaf components by reductions that would +require Property 2. The first described case can be used at any point. The last two described +cases are used on constant leaf components and as the result is different, it follows that their +edges hold the property. +◀ +This concludes the constant component reductions which together with cycle components +and approach described in Section 5.1 give us a polynomial algorithm to solve m-Eternal +Domination. +6 +Future Work +The presented tools could be useful in a future study of the m-Eternal Domination on +different graph classes. For instance, grids of size {3, 5} × n were extensively studied [16, 18]. +We believe it would interesting to see to which extent the tools could by applied in study of +grids of less restricted dimensions. +Another noteworthy class of graphs are the so called dually chordal graphs, for which +many domination related problems are polynomial time solvable. It would be interesting to +see whether m-Eternal Domination remains polynomial time solvable as well. +Furthermore, the computational complexity of the decision variant of the m-eternal +domination problem is still mostly unknown as mentioned in the introduction. It remains +open whether the problem is in PSPACE and whether it is PSPACE-hard. + +V. Blažej, J. M. Křišťan, and T. Valla +37 +References +1 +Václav Blažej, Jan Matyáš Křisťan, and Tomáš Valla. On the m-eternal domination number +of cactus graphs. In Reachability Problems - 13th International Conference, RP 2019, volume +11674 of Lecture Notes in Computer Science, pages 33–47. Springer, 2019. +2 +Andrei Braga, Cid C. de Souza, and Orlando Lee. The eternal dominating set problem for +proper interval graphs. Information Processing Letters, 115(6):582–587, 2015. +3 +Alewyn P. Burger, Ernest J. Cockayne, W. R. Gründlingh, Christina M. Mynhardt, Jan H. +van Vuuren, and Wynand Winterbach. Infinite order domination in graphs. 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Theo- +retical Computer Science, 794:27–46, November 2019. +16 +Margaret-Ellen Messinger. Closing the gap: Eternal domination on 3 x n grids. Contributions +to Discrete Mathematics, Vol 12:No 1 (2017), 2017. +17 +Martín Rinemberg and Francisco J. Soulignac. The eternal dominating set problem for interval +graphs. Information Processing Letters, 146:27–29, June 2019. +18 +Christopher M. van Bommel and Martin F. van Bommel. Eternal domination numbers of +5 × n grid graphs. Journal of Combinatorial Mathematics and Combinatorial Computing, +97:83–102, 2016. +A +Complete Strategies +We note that if the strategy SG was a complete graph, then strategy S′ +G created by the +application of Lemma 4.37 is also a complete graph. +▶ Property 4. A partial labelled strategy B = (G, SG, P, T , R) is complete if SG is a complete +graph, i.e., there is {α, β} ∈ F for every α, β ∈ Ω. + +38 +Computing m-Eternal Domination Number of Cactus Graphs in Linear Time +We note that there are graphs where every optimal strategy is not complete, see Ap- +pendix B for such an example. Complete strategies can be effectively pruned to contain at +most |V (G)| states in the following way. +▶ Lemma A.1. For any complete defending labelled strategy of cardinality k with the minimum +number of vertices of SG it holds |V (SG)| ≤ |V (G)| − k + 1. +Proof. Pick an arbitrary complete defending strategy SG which uses k guards. For each +v ∈ V (G) we shall pick one state αv ∈ V (SG) such that v ∈ P(α). First, pick any α ∈ V (SG) +and assign it as state to each v ∈ P(α). Then, for every v ∈ V (G) \ P(α) assign αv ∈ S(v) +as its state. We just picked |V (G)| − k + 1 states such that they form a strategy where every +pair of states is traversable and which is defending as it covers all the vertices of G. +◀ +Similar to completeness of a strategy we may talk about the graph class of SG to describe +its properties. +B +Non-complete Strategy +▶ Observation B.1. An optimal m-Eternal Domination strategy on 5 × 5 grid uses at least 7 +guards. +Proof. Let us denote vertices of the grid by ui,j where 1 ≤ i, j ≤ 5. +First, we show a lower bound of 7. Assume for a contradiction that there is a defending +strategy S6 with at most 6 guards. Any state of S6 needs to dominate all 25 vertices. There +must exist a state C where u2,2 is occupied. In C there also must be at least one guard +in the closed neighborhood of each corner (u1,1, u5,1, u1,5, and u5,5). In the grid a vertex +may dominate at most 5 vertices and a vertex on the side of the grid may dominate at most +4 vertices. All vertices in the closed neighborhood of corners are on the side of the grid. +Additionally, vertex which dominates u1,1 may dominate at most 2 new vertices, as u2,2 +already dominates many of vertices in its neighborhood. In total, the 6 guards of C may +dominate at most 2 · 5 + 3 · 4 + 2 = 24, a contradiction. +◀ +The upper bound can be shown by construction of a strategy, however, we have no good +tools to show that all the strategies are not complete graphs – we found this using a full +strategy-space search. A construction which uses 7 guards contains three states and majority +of their reflections and rotations, see them on Figure 20. In this case, we do not show the +strategy, as it contains roughly 20 states (depending on a slight optimization, it may be less) +that would contain 190 transitions. +Figure 20 The 5 × 5 grid has 19 m-eternal dominating sets. Each of the configurations can be +expressed as a combination of rotations and reflections of exactly one of these 3 basic configurations. +Each of the 19 configurations is necessary for the strategy and can move into at most 12 other states. + diff --git a/QtE4T4oBgHgl3EQflA0t/content/tmp_files/load_file.txt b/QtE4T4oBgHgl3EQflA0t/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e9b81ab5437a43d6d8ccb61cc4953153658d94e7 --- /dev/null +++ b/QtE4T4oBgHgl3EQflA0t/content/tmp_files/load_file.txt @@ -0,0 +1,1871 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf,len=1870 +page_content='Computing m-Eternal Domination Number of Cactus Graphs in Linear Time Václav Blažej Faculty of Information Technology, Czech Technical University in Prague, Prague, Czech Republic Jan Matyáš Křišťan Faculty of Information Technology, Czech Technical University in Prague, Prague, Czech Republic Tomáš Valla Faculty of Information Technology, Czech Technical University in Prague, Prague, Czech Republic Abstract In m-eternal domination attacker and defender play on a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Initially, the defender places guards on vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In each round, the attacker chooses a vertex to attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Then, the defender can move each guard to a neighboring vertex and must move a guard to the attacked vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The m-eternal domination number is the minimum number of guards such that the graph can be defended indefinitely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In this paper, we study the m-eternal domination number of cactus graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We consider two variants of the m-eternal domination number: one allows multiple guards to occupy a single vertex, the second variant requires the guards to occupy distinct vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We develop several tools for obtaining lower and upper bounds on these problems and we use them to obtain an algorithm which computes the minimum number of required guards of cactus graphs for both variants of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 2012 ACM Subject Classification Graph Theory Keywords and phrases Graphs, Algorithms, Eternal domination 1 Introduction Consider the following game, played by an attacker and a defender on graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The defender controls a set of guards, which he initially places on the vertices of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Each vertex can be occupied by at most one guard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In each round, the attacker first chooses one vertex, which he attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The defender then must defend against the attack by moving some or all of his guards along their adjacent edges, so that one of the guards moves to the attacked vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' If the attacked vertex is not occupied by a guard after the attack, the attacker wins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The defender wins if he can defend indefinitely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Defending a graph from attacks using guards for an infinite number of steps was introduced by Burger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In this paper, we study the concept of m-eternal domination, which was introduced by Goddard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' [6] (eternal domination was originally called eternal security).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Here, the notion of the letter “m” emphasizes that multiple guards may move during each round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' There is also a variant of the problem studied by Goddard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' [6] where only one guard may move during each round, which is not considered in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The m-eternal domination number γ∞ m (G) is the minimum number of guards which defend against all attacks indefinitely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Goddard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' [6] established γ∞ m exactly for paths, cycles, complete graphs and complete bipartite graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Since then, several results have focused on The authors acknowledge the support of the OP VVV MEYS funded project CZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='01/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='0/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='0/16_019/0000765 “Research Center for Informatics”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This work was supported by the Grant Agency of the Czech Technical University in Prague, grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' SGS20/208/OHK3/3T/18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Supported by the grant 22-19557S of the Czech Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='05155v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='DS] 12 Jan 2023 2 Computing m-Eternal Domination Number of Cactus Graphs in Linear Time finding bounds on γ∞ m under different conditions or graph classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Among the studied graph classes are trees [12, 8, 14], grids [4, 18, 16, 10, 5, 9, 15], and interval graphs [2, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' For a good survey of other related results and topics, see Klostermeyer and Mynhardt [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Very little is known regarding the algorithmic aspects of m-eternal domination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The decision problem (asking if γ∞ m (G) ≤ k) is NP-hard and belongs to EXPTIME, however, it is not known whether it lies in the class PSPACE [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='1 Original Results In this paper, we focus on the class of cactus graphs (connected graphs where each edge lies in at most one cycle) and provide an algorithm for computing γ∞ m in cactus graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In Section 4, we provide a set of tools with more general applications to proving upper and lower bounds of γ∞ m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Those tools are then used in Section 5 to describe a set of reductions, which allow us to compute γ∞ m of cactus graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This is a significant expansion of basic principles which were introduced by Klostermeyer and MacGillivray [11], in which they provide an algorithm for computing γ∞ m of trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Our main result is summarized in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let G be a cactus graph on n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Then there exists a polynomial algorithm which computes γ∞ m (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='2 Preliminaries Let us now review all the standard concepts formally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' A graph is a cactus if its every edge lies on at most one cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' For an undirected graph G let a configuration be a multiset of its vertices C = {c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' , cn | ci ∈ V (G)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We will refer to the elements of configurations as guards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' If a vertex is an element of a configuration, then it is occupied (by a guard).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Two configurations C1 and C2 of G are mutually traversable if there is some set of pairs T (C1, C2) = {(v1, u1), (v2, u2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' , (vn, un)} such that C1 = {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' , vn} and C2 = {u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' , un} and {vi, ui} ∈ E(G) for all i from 1 to n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We perceive the guards as tokens which move through the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The elements of T (C1, C2) are called movements and a single ordered pair among them is a move of a guard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' A guard that moves in T (C1, C2) to the same vertex where he started is called stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' A strategy in G is a graph SG = (C, F) where C is a set of configurations over V (G) such that all of the configurations have the same size and F ⊆ C2 describe possible transitions between the configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The order of a strategy is the number of guards in each of its configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In papers on this topic it is often assumed that the strategy edges are given implicitly as F = � {C1, C2} ∈ C2 | C1 and C2 are mutually traversable in G � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' For our purposes, we want to prescribe the strategy explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We introduce the notions for exact strategy prescription in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We call the strategy SG to be defending against vertex attacks if for any C ∈ C the configuration C and its neighbors in SG cover all vertices of G, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=', when a vertex v ∈ V (G) is “attacked” one can always respond by changing to a configuration which has a guard at the vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Formally, SG = (C, F) is defending if (∀C ∈ C) (∀v ∈ V (G)) � v ∈ C ∨ (∃C′ ∈ C)({C, C′} ∈ F ∧ v ∈ C′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Note that every configuration in a strategy which defends against vertex attacks induces a dominating set in G as otherwise, the attacker would win in the next round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We investigate two variants of the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The variants differ in whether they allow multiple guards to occupy the same vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let an m-Eternal Guard Strategy in G be a strategy defending against vertex attacks in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Blažej, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Křišťan, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Valla 3 Input: An undirected graph G = (V, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Question: What is the minimum number of guards γ∞ m such that there exists an m-Eternal Guard Strategy SG where each vertex is occupied by at most one guard that defends against vertex attacks in G?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' m-Eternal Domination Input: An undirected graph G = (V, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Question: What is the minimum number of guards Γ∞ m such that there exists an m-Eternal Guard Strategy SG that defends against vertex attacks in G?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' m-Eternal Guard Configuration The open neighborhood of u in G will be denoted as NG(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' By Pn we denote a path with n edges and n + 1 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' By G[U] we denote the subgraph of G induced by the set of vertices U ⊆ V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 2 High-level Overview of the Proof In order to solve the m-Eternal Domination and the m-Eternal Guard Configuration on cactus graphs, we use induction on the number of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Base cases will be presented in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In the induction step, we show how to reduce cactus graph G to a smaller cactus graph G′ while showing lower bound and upper bound in the following ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Reduction from G to G′ is done using Observations 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='8–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' These directly show a lower bound Γ∞ m (G) ≥ Γ∞ m (G′) + K for some constant K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Then, we show an expansion from G′ to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We assume that G′ has an optimal defending strategy that holds several nice properties from the induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We show that a part of the graph G′ along with its strategy can be exchanged for a different one by showing that Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='25 holds for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Such parts are then exchanged using Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='27 which expands G′ into G while showing that an upper bound devised by Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='29 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This gets us an upper bound γ∞ m (G) ≤ γ∞ m (G′) + K (the same K as in the lower bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Combining the lower and upper bound using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='2 gets us the optimal number of guards for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The used reduction depends on a leaf component that the cactus graph contains by Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' One case is that the subgraph is a tree and the second case is that there is a leaf cycle – a cycle with leaves which is connected to the rest of the graph via a single articulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We split the reductions into three groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The first group called leaf reductions shown in Section 3 has a few simple reductions of leaves which are not incident to a leaf cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' These were shown to be sufficient to determine the γ∞ m for any tree by Goddard, Hedetniemi, and Hedetniemi [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We reintroduce these reductions in our framework and show more general results so that the reductions can be used over tree subgraphs of non-tree graph classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' They also serve as an introductory example of how to use the tools from Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Further reductions are more involved and require non-trivial manipulation with strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' It is beneficial to establish strategies with nice properties in the induction to allow a stronger induction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='2, we show the properties which are used in the two other groups of reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The last two groups called cycle reductions and constant component reductions are shown in Sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Cycle reductions concern substructures that appear on leaf cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We fix a leaf cycle and use these reductions repeatedly on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Each reduction shortens the leaf cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Eventually, the cycle is very short and is reduced by constant component reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 4 Computing m-Eternal Domination Number of Cactus Graphs in Linear Time After these reductions, the leaf cycle is removed entirely and only zero, one, or two leaves are left in its place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Such leaves are then processed either as tree leaves or leaf vertices adjacent to another leaf cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 3 Reducing Trees In this section, we intuitively present tools to achieve lower and upper bounds and which will be formally introduced in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We focus on tree reductions, which were first described by Goddard, Hedetniemi, and Hedetniemi [6] as a part of the linear algorithm for computing the m-eternal domination number γ∞ m on trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We now show this set of reductions along with the proofs of their correctness in Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='1–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' For graph G, let us have a vertex u ∈ V (G) which is adjacent to ℓ ≥ 1 leaves and has degree d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let v be one of the leaves adjacent to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We define three leaf reductions of G to G′ as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' See Table 1 for an illustration of respective bound proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Reduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' t1 If ℓ = 1 and d ≤ 2, let G′ = G \\ {u, v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Reduction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' t2 If ℓ > 2, let G′ = G \\ {v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Reduction 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' t3 If ℓ = 2 and d = 3, let G′ = G \\ {all leaves adjacent to u}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Reductions t2 and t3 can be joined to a single reduction which removes all leaves of a vertex with ℓ ≥ 2 and d = ℓ + 1 (used in [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' However, Reduction t2 may be used in a wider range of scenarios as it does not require a specific value of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Assume now, that we know the optimum number of guards for G′ (for both Γ∞ m and γ∞ m ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Our goal is to show two things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' By showing that G always uses at least K more guards than G′ we get a lower bound on the number of guards necessary for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' By showing that there is a strategy for G which uses at most K more guards than an optimum strategy on G′ we get an upper bound on the number of guards on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Together, these bounds give us an optimum number of guards for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This concept is formally introduced in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Table 1 Leaf reductions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Lower bound side depicts clique reductions (removal of marked vertices and joining its neighborhood with a clique);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Upper bound side labels vertices with Greek letters of states where they belong, and arrows show how one state transitions to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The marked groups of vertices are created with Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Reduction Lower bound Upper bound t1 −1 v u a a v α β u +1 a a t2 −0 u u v +0 u Ω′ α′ β′ w u Ω α β w vγ t3 u u −1 v a a u β u +1 a a α′ β′ α Ω v w V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Blažej, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Křišťan, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Valla 5 −2 −2 −1 +1 +2 +2 +1 Ω Ω α β α α β1 γ δ1 γ δ Ω Ω Ω Ω µ ν δ2 Ω β2 β1β2β3 β4 t2 t2 t1 t3 t3 t1 t3 Figure 1 An example application of leaf reductions on a tree graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Dotted lines signify that the strategy does not use that edge, and the strategies on subtrees are independent, which is caused by Reduction t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Note how Reduction t3 can be used even when there is no vertex a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Having the reductions in hand see Figure 1 for an example of how the reductions are used to construct a strategy for a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We now proceed to show the bounds obtained from these reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Generally, the proofs contain lower bound and upper bound portions, see Table 1 for accompanying illustrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Lower bounds can be shown quite easily – delimit a connected part of a graph which is guaranteed to contain K guards, remove it, and join its neighborhood with a clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Upper bounds are more tricky – we assume some optimal strategy on G′, which has nice properties, and then we expand it to G while preserving the properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The notation used in the following proofs is defined in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We say that graph G is defended with k guards if k = γ∞ m (G) = Γ∞ m (G) and the strategy using k guards is proper in the sense of Property 3, which is defined in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This allows us the state the lemmas concisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let us now see the proofs for the three leaf reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let G′ be G after application of Reduction t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' G is defended with 1 more guard than G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' As {u, v} is a leaf and its neighbor, there is always at least one guard so we may apply Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='8 on {u, v} to get a lower bound of Γ∞ m (G) ≥ Γ∞ m (G′) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' To get an upper bound γ∞ m (G) ≤ γ∞ m (G′) + 1, we dedicate one new guard to defend {u, v} independently on the rest of the strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Putting the lower bound and upper bound together using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='2 we get that G is defended with 1 more guard than G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ◀ Note, that the final strategy graph after Reduction t1 is a Cartesian product of the strategy graph on G′ and a graph with a single edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Cartesian product is a basis for Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='30 where we introduce an operation which joins strategies even if the strategies are not entirely independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We shall use this operation along with a property shown in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='37 – that a strategy can be altered so that a vertex adjacent to multiple leaves is always occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' To ease notation, we shall reserve the prime symbol (′) to denote structures of the reduced instance such as the graph G′, defending strategy B′, strategy graph S′ G′, its states (vertices) Ω′ and transitions (edges) F′, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let G′ be G after application of Reduction t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' G is defended with the same number of guards as G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Lower bound of 0 is obtained by using Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='3 to identify v with u so Γ∞ m (G) ≥ Γ∞ m (G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 6 Computing m-Eternal Domination Number of Cactus Graphs in Linear Time For upper bound, from induction we have a defending labelled strategy B′ of G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We wish to alter it so it defends vertex v as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='37 to alter B′ so that u is occupied in each state of Ω′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let w be a leaf adjacent to u distinct from v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We partition all states (vertices) of the strategy S′ G′ as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' A state α′ belongs to S′(w) if in α′ vertex w is occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Now we perform graph Cartesian product of S′ G′ with a single edge {α, β} over subset S′(w) (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Written in short as SG′ = S′ G′ □S′(w) {α, β}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This splits all vertices of the strategy where w is occupied into two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We denote the new sets as α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This got us a new strategy graph SG′ over the reduced graph G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Now we expand from G′ to G while altering the strategy slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In β we substitute the guard on w with a guard on v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The guards shall transition between states of α and β as T (α, β) = {(w, u), (u, v)} while the rest of them shall not move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' As w and v are siblings it follows that we can transition from any γ ∈ Ω to w the same way as to v if they were swapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This remains defending by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Hence, γ∞ m (G) ≤ γ∞ m (G′) and by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='2 we get that G is defended with the same number of guards as G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ◀ The shown strategy basically defends v in the “same way” it defends w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We can do this when one can transition from one to the other in a single step while the remaining guards remain stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' For a detailed explanation see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='30 and its lemmas that show its properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The previous reduction bounds were proven with an extensive explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In the following proofs, we just use the tools to arrive at the result directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Note that a very similar argument could be used to obtain an arbitrary number of leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let G′ be G after application of Reduction t3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' G is defended with 1 more guard than G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Lower bound of 1 is obtained by using Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='8 on vertices {u, v}, which results in a graph isomorphic to one that is created by removing all leaves adjacent to u which gets us Γ∞ m (G) ≥ Γ∞ m (G′) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' For upper bound, we apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='33 which adds the two leaves to u using one extra guard which directly results in γ∞ m (G) ≤ γ∞ m (G′) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='2 we get that G is defended with 1 more guard than G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ◀ Note that the bounds devised for Reductions t1, t2, and t3 do not require the graph to be a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We may use these reductions in any graph class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Hence, we may reduce any leaves in subtrees which appear as parts of other graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In particular, Reduction t2 may be also used to reduce the number of leaves adjacent to any vertex to 2 because connections of u to other vertices do not interfere with the reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Note that in that case, we obtain lower bounds for the m-Eternal Guard Configuration and upper bounds for the m-Eternal Domination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' It was previously shown by Goddard, Hedetniemi, and Hedetniemi [6] that these reductions (originally given in a slightly different form) are sufficient to solve any tree graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Note that this can be shown by rooting the tree and repeatedly applying Reductions t1, t2, and t3 on the parent of the deepest leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Also note, that the reductions t1 and t3 do not require the rest of the graph (signified by vertex a) to be there at all, hence, these solve base cases where only a single edge or a star remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' When the reductions are used on a tree we get a partitioning of vertices into subtrees which are defended independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' These components constitute a neo-colonization, a notion introduced by Goddard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' [6] and often used in contemporary papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Blažej, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Křišťan, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Valla 7 4 The m-eternal domination Toolbox This section gives tools to show lower and upper bounds for the m-eternal domination problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Before we present the approach in detail we show several key ideas and a detailed structure for the rest of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Throughout this paper, we reserve prime (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' G′ and α′) to denote structures of the reduced instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' γ(G) ≤ Γ∞ m (G) ≤ γ∞ m (G) ≤ 2 · γ(G) for any graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Every m-Eternal Domination strategy can be applied as an m-Eternal Guard Config- uration strategy so γ∞ m ≥ Γ∞ m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Every configuration in each of these strategies must induce a dominating set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Therefore, they are all lower bound by the domination number γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' It is also known that an m-Eternal Domination strategy can be constructed by defending neighborhood of each vertex in the dominating set independently of each other (with a simple strategy for stars) that uses at most 2 · γ(G) guards as shown by Klostermeyer and Mynhardt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ◀ We now show the lemma which sums up how the bounds of the optimal strategies are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let us assume that for graphs G, G′, and an integer constant k γ∞ m (G) ≤ γ∞ m (G′) + k, (1) Γ∞ m (G) ≥ Γ∞ m (G′) + k, (2) γ∞ m (G′) = Γ∞ m (G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' (3) Then γ∞ m (G) = Γ∞ m (G) = γ∞ m (G′) + k = Γ∞ m (G′) + k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Given the assumptions, we have γ∞ m (G) (1) ≤ γ∞ m (G′) + k (3)= Γ∞ m (G′) + k (2) ≤ Γ∞ m (G) Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='1 ≤ γ∞ m (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' As the first and the last term is identical all these values are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ◀ Hence, it suffices to prove that for G and its reduction G′ we have γ∞ m (G) ≤ γ∞ m (G′)+k and Γ∞ m (G′) ≤ Γ∞ m (G) − k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' If we already have the optimal strategy for G′, then our constructive upper bounds together with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='2 give us an optimal strategy for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We present the tools for obtaining lower bounds in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='1, terminology and new concepts for upper bounds in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='2, and tools which use the new concepts to obtain upper bounds in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' See Figure 2 for a detailed section overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='1 Lower Bounds We start this section with a few elementary observations about strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Then, we show a pair of lemmas which are the main tools in obtaining lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Last, using these lemmas, we obtain three lower bound observations which we use frequently in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We say that G′ is a result of identifying u with v in G if it is a result of removing u while adding the edges so that NG′(v) = NG(u) ∪ NG(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='3 (Vertex identification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let G be a graph and u and v be its two distinct vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Then for a graph G′, which is a result of identifying u with v in G, Γ∞ m (G′) ≤ Γ∞ m (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 8 Computing m-Eternal Domination Number of Cactus Graphs in Linear Time Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='1 Lower bounds Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='3 Vertex identification Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='4 Substitute guard Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='5 Clique reduction Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='6 Lower bound lemma Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='7 Ink lemma Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='8 Leaf lower bound Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='9 Star lower bound Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='10 Path lower bound Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='2 Upper bounds Definitions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='11–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='14, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='17, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='19: State, Movement, Interface, Transition, (Partial) Defended graph and subgraph Property 1 Symmetry Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='21 Compatible Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='22 Cutting Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='23 Composing Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='24 Composing compatible Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='25 Interface equivalent Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='26 Transferred compatibility Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='27 Expansion Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='28 Equivalency constant Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='29 Upper bound Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='3 Tools for altering strategies Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='30 Cartesian product over subset Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='33 Leaves addition Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='35 Group state Figure 2 Overview of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Boxes represent respective subsections;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Arrows on the left side show which notions are used to prove other notions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Right arrows show which notions are frequently used in Section 5 to obtain results for cactus graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let v′ ∈ V (G′) be the vertex created by identifying u with v in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let SG be an optimal strategy of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let SG′ be a strategy on G′ which is the same as SG except that in every configuration each u and v is substituted by v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Any pair of traversable configurations in SG is still traversable in SG′ as in every movement u and v can be replaced by v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Any attack on V (G′) \\ {v′} is defended by a configuration in SG′ which was created from a respective configuration of SG, and v′ is defended by a configuration which defended u in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ◀ ▶ Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' If a graph has a clique on distinct vertices u, v, w, then guard movements (u, v) and (v, w) can be substituted with movement (u, w) and a stationary guard on v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Along with vertex identification, the following reduction is the main tool for obtaining lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='5 (Clique reduction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let G be a graph and H be its non-empty induced connected subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' By clique reduction of H in G we mean the creation of a new graph G′ that is the result of removing H from G and mutually connecting all neighbors of H in G \\ H by an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' See Figure 3 for an illustration of a clique reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Using Observations 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='4 we now show that the clique reduction implies a lower bound on G which can be later used to show tight strategy lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='6 (Lower bound lemma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let G be a graph and H be its non-empty induced connected subgraph such that in at least one optimal m-eternal guard strategy, there are always at least k guards present on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let G′ be the result of a clique reduction of H in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Then Γ∞ m (G′) ≤ Γ∞ m (G) − k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' If there is no neighbor of H in G, then clique reduction removes H and adds no edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We can remove all the guards which were standing on H so G′ is clearly defended by Γ∞ m (G) − k guards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Otherwise, there is a neighbor of H in G \\ H, say v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let SG = (C, F) be an optimal strategy on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We use Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='3 to identify all vertices V (H) with v in G to obtain V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Blažej, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Křišťan, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Valla 9 a subgraph of G′ along with a strategy SG′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Note that in G′ each configuration of SG′ has at least k guards on v because before identification H always contained k guards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Also, configurations which defend v in SG′ have at least k + 1 guards on v by the same argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let S− G′ be a strategy which is the same as SG′, except it has k less guard on v in each configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We see that each configuration which defended v in SG had at least k + 1 guards on v so it defends v in S− G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Guards which defended V (G) \\ (V (H) ∪ {v}) remain unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' It remains to check whether configurations which were traversable in SG remain traversable in S− G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Our goal is to show that there exists a set of movements between each pair of configurations of SG′ which have k stationary guards on v which are not needed for defending G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Such guards then may be removed to obtain S− G′ and the remaining movements show that the respective configurations are traversable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Each movement (u, h) and (h, w) such that h ∈ V (H) in SG has its respective pair of movements (u, v), (v, w) in SG′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' As u and w are neighbors of V (H) then there is an edge {u, w} ∈ E(G′) added by the construction of G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' By Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='4 we may substitute movements (u, v), (v, w) with (u, w) and a stationary guard on v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Assume there are less than k stationary guards on v in SG′ after applying the substitution exhaustively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Then there must be at most k − 1 stationary guards and at least one guard which leaves v or at least one guard which arrives to v, but there may not be both (one leaving and one arriving) as they would form (u, v), (v, w) pair and the substitution could be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' When the guard is leaving or arriving there are at most k − 1 guards in the final or starting configuration, respectively, which is a contradiction because there are at least k guards on v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Removing k guards from v in SG′ yields S− G′ where each configuration pair remains traversable, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ◀ To use Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='6 we need to show that an induced subgraph H of G is always occupied by at least k guards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' To do that we have the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='7 (Ink lemma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let H be an induced subgraph of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let (v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' , vk) be a sequence of vertices in H such that it holds d(u, vi) > i for every i and for every u ∈ V (G) \\ V (H), and also for every j < i it holds that d(vj, vi) > i − j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Then there are at least k guards on H in every defending m-eternal guard configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let C be any fixed configuration of a defending strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We show that C contains at least k guards on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Assume that the attacker performed a sequence of attacks (v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' , vk) one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' At the i-th step of the attack sequence, the following is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Guards who were standing on V (G) \\ V (H) at the beginning of the attack sequence are more than i edges far from vi so they cannot reach vi in time to defend it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Similarly, any guard that defended vj with j < i can not defend the attack on vi as their distance from vi is more than i − j at the time they defended vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Therefore, none of the guards can reach vi in time and we need an additional guard placed on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In total, we need k guards on H in a configuration to be able to defend the attack sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This is true for any configuration so every defending strategy must have k guards on H in every configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ◀ The operation in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='6 together with the lower bound obtained from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='7 allows us to make a graph smaller while showing that the removed part required some minimum number of guards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' See an example usage of Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='7 in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 10 Computing m-Eternal Domination Number of Cactus Graphs in Linear Time G H G′ Figure 3 A graph G with an induced subgraph H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Graph G′ is obtained by a clique reduction from Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='7 we have that every configuration contains at least 1 guard on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Hence, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='6 we have Γ∞ m (G) ≥ Γ∞ m (G′) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In graph G, let v be a leaf vertex and let u be its neighbor, and let H = G[{u, v}].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='7 with a sequence (v) we obtain that 1 guard is on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In other words, the closed neighborhood of every leaf must contain at least one guard otherwise an attack on the leaf could not be defended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let G′ be a graph obtained by using the operation of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='6 on H, this gives us Γ∞ m (G) ≥ Γ∞ m (G′) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In graph G, let u be a vertex which is adjacent to at least two leaves {v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' }, and let H = N[u] denote the closed neighborhood of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='7 with a sequence (v1, v2) we obtain that 2 guards are on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In other words, the closed neighborhood of u must contain at least 2 guards otherwise two consecutive attacks on different leaves adjacent to u could not be defended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let G′ be a graph obtained by using the operation of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='6 on H, this gives us Γ∞ m (G) ≥ Γ∞ m (G′) + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let us have graph G and its induced subgraph H that is isomorphic to a path on three vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We label these three vertices of H as u1, u2, u3 (in order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='7 with a sequence (u2) we have the lower bound of 1 on the number of guards on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let G′ be a graph obtained by using the operation of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='6 on {u1, u2, u3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This gives us Γ∞ m (G) ≥ Γ∞ m (G′) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='2 Upper Bounds This section introduces notation to describe strategies which are used to achieve upper bounds for γ∞ m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We assume that we have a graph G and its reduced copy G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The main idea is that a strategy for G′ can be locally changed to obtain a strategy for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' To accommodate this local change, we show how to cut and compose parts of the graph while preserving its strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' At the end of this section, we present a set of sufficient rules that allow such a local change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Then, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='3, we present tools which we use to obtain upper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In our constructions, we need to have control over the movements of the guards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We also need a way to represent only part of the strategy over an induced subgraph of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' To do so, we introduce states (labelled configurations) and labelled strategy that prescribes the guard movements on state transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='11 (States).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let states be a set of labels Ω and let state vertex mapping P of Ω to V (G) be P : Ω → 2V (G), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=', a state α ∈ Ω represents a subset of vertices P(α) ⊆ V (G) (also called guards) of a graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let S(v) = {β | β ∈ Ω, v ∈ P(β)} (states that contain v) for every v ∈ V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We will use Greek letters such as α, β, γ, δ, φ to signify states or sets of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Move of a guard is still an ordered pair of vertices (u, v) such that {u, v} ∈ E(G) or u = v (stationary guard).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Blažej, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Křišťan, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Valla 11 Building towards a comprehensive definition of a labelled strategy, we first build a more general concept – partial labelled strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This will allow us to do cutting and composing with a well-defined strategy over a subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='12 (Interface).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let an interface R of a graph G with respect to its supergraph H be a subset of vertices such that R = � u | (∃v) u, v ∈ V (H), {u, v} ∈ E(H), u ∈ V (G), v ̸∈ V (G) � , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=', those vertices of G which have a neighbor in V (H) \\ V (G) in H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' H G R G R α α α β β β β α αβ α α β u v w Figure 4 Left: An interface R of G with respect to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Bold edges signify the cut between V (G) and V (H) \\ V (G) that is responsible for the vertices in R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Right: Transition T (α, β) from α to β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Arrows signify movements of the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' All vertices in states α and β must be paired up with a movement if they are not in the interface R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In the interface, a guard moves from u to the rest of the graph outside of G so a movement is missing for u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Note the difference between v and w: in w the same guard stays on the vertex, in v (as there is no (v, v) movement) the guard on v moves out and a different one moves to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' See Figure 4 for an example of an interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The interface marks the vertices where the strategy may be incomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The transitions between states incorporate the interface by allowing the moves to be incomplete in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='13 (Transition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' For states α and β and a graph G with an interface R, let a transition (from α to β) denoted by T (α, β) be a set of moves such that T (α, β) ⊆ � (u, v) | u ∈ P(α), v ∈ P(β), {u, v} ∈ E(G) ∨ u = v � , for each u ∈ P(α) \\ R there exists exactly one (u, v) ∈ T (α, β), for each v ∈ P(β) \\ R there exists exactly one (u, v) ∈ T (α, β), for each u ∈ P(α) ∩ R there exists at most one (u, v) ∈ T (α, β), and for each v ∈ P(β) ∩ R there exists at most one (u, v) ∈ T (α, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' See Figure 4 for an example of a transition and how it interacts with an interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Note that if the interface R is empty, then the transition yields a bijection between guards of the states, which gives an exact prescription on how they move between the two states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Transition gives us that each guard can be in relation with at most one other guard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In our case, there is at most one guard on each vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Hence, we may use the standard relation terminology for the set of pairs defined by a transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='14 (Partial labelled strategy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' A partial labelled strategy is (G, SG, P, T , R) where G is a graph, SG = (Ω, F) is a strategy graph such that Ω is a set of vertices (states) and F is a set of edges, P is a state vertex mapping of Ω to V (G), R ⊆ V (G) is an interface of G, and T maps orientations (α, β) and (β, α) of every edge {α, β} ∈ F to transitions T (α, β) and T (β, α), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 12 Computing m-Eternal Domination Number of Cactus Graphs in Linear Time For various purposes, it may be beneficial to think of the strategy graph SG as an oriented graph (allowing non-symmetric transitions, see Property 1), or even multigraph (allowing multiple different transitions between the same set of states).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The labelled strategy may defend against attacks indefinitely if it is in accordance with the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='15 (Defending).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' A partial labelled strategy (G, (Ω, F), P, T , R) is defending G if for every state α ∈ Ω and each vertex v ∈ V (G) there is β ∈ Ω such that {α, β} ∈ F and v ∈ P(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In other words, Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='15 says that for each vertex of the graph every state is either occupying it or a state which occupies it is reachable with only one transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This directly leads to the following observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Recall that S(u) denotes a set of states that contain u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' A strategy is defending a graph if for every u ∈ V (G) set S(u) is a dominating set of the strategy graph SG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='17 (Labelled strategy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' A labelled strategy is D = (G, SG, P, T ) such that (G, SG, P, T , ∅) is a partial labelled strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Note, that all the states in the labelled strategy must contain the same number of guards because the transitions are bijections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' When the strategy is optimal the number of guards corresponds to γ∞ m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Partial labelled strategies can have several nice properties, which we present now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' When the strategy graph is unoriented it is natural to require symmetry of transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Property 1 (Symmetry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' A partial labelled strategy B = (G, (Ω, F), P, T , R) is symmetrical if and only if T (α, β) is a converse relation to T (β, α) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=', T (α, β) = {(a, b) | (b, a) ∈ T (β, α)}) for every {α, β} ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' For each partial labelled strategy B = (G, (Ω, F), P, T , R) there exists a symmetrical partial labelled strategy B′ = (G, (Ω, F), P, T ′, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' For each pair of states {α, β} ∈ F fix an arbitrary orientation (α, β) and take the T (α, β) with an interface R which gives us T (α, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Note that by swapping u with v and α with β in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='13 we obtain the same definition but for T (β, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We substitute the transition T (β, α) for this newly found transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Performing this substitution for every pair of states in F gives us the desired T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ◀ By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='18, we will always assume that the partial labelled strategy is symmetrical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We also use this property to infer transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We show only one direction of the transition mapping and let the other direction be the converse transition given by symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' It is not easy to grasp the labelled strategy description only from the formal notation so we shall draw many auxiliary pictures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Vertices and edges shall be depicted by small circles (or squares) and line segments, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' vertices may be labelled by their letter name;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' by Greek letters we signify the states which contain respective vertices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' guard moves in transitions are depicted by differently styled arrows on edges which point between the state labels (Note that the arrows are always shown only in one direction because we assume Property 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' the interface vertices are marked by gray-filled areas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' see Figure 5 for an example of a labelled strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We will need to cut part of the labelled strategy and put something slightly different in its place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' To tackle that we put forward the following notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Blažej, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Křišťan, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Valla 13 α β γ a b c u v a b c u v + u B A C cutting composing γ β α α β γ γ β γ β α Figure 5 Example of a labelled strategy B and two partial labelled strategies A and C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' the full formal description of the labelled strategy B = (G, (Ω, F), P, T ) is G = (V, E), V = {a, b, c, u, v}, E = � {a, b}, {b, u}, {u, v}, {a, c}, {c, u}� , Ω = {α, β, γ}, F = � {α, β}, {α, γ}, {β, γ}� , P(α) = {v, a}, P(β) = {u, b}, P(γ) = {u, c}, (R = ∅), T (α, β) = {(a, b), (v, u)}, T (α, γ) = {(a, c), (v, u)}, T (β, γ) = {(b, u), (u, c)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' the formal descriptions of partial labelled strategies A and C are similar while restricted to their subgraph and they contain an interface R = {u}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='19 (Partial labelled substrategy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The partial labelled substrategy B′ of a labelled strategy B = (G, (Ω, F), P, T ) for some induced subgraph G′ of G is a partial labelled strategy B′ = (G′, (Ω, F), P ′, T ′, R) where R is an interface of G′ with respect to G, for all α ∈ Ω it holds P ′(α) = P(α) ∩ V (G′), and for all β, γ ∈ Ω it holds T ′(β, γ) = {(a, b) | (a, b) ∈ T (β, γ) ∧ a, b ∈ V (G′)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' It is not immediately obvious that the Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='19 made a partial labelled substrategy in a way that it constitutes a partial labelled strategy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' so we show that next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' A partial labelled substrategy B′ = (G′, (Ω, F), P ′, T ′, R) of a labelled strategy B = (G, (Ω, F), P, T ) is a partial labelled strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' G′ is a graph, Ω is a set of labels, and R is a subset of V (G′) which is in accordance to Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' P ′ was created by restricting P to the vertices of V (G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We only removed some guards from the mapping so this is okay by Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Last, the only guards which are not included in T ′ are those whose moves in T went outside of G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Assume such guard on vertex u with a move (u, v) where v ̸∈ G′, hence, u ∈ R by Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' As stated in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='13 any guard in R does not have to be included in a move so any partial labelled substrategy is a partial labelled strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ◀ Now we present conditions which are necessary to be able to combine two partial labelled strategies into one labelled strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Based on that, we show how to split a labelled strategy into two partial labelled strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Figure 5 shows an example of the following operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='21 (Compatible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Two partial labelled strategies B1 and B2 (denoted as Bi = (Gi, (Ωi, Fi), Pi, Mi, Ri)) are called compatible if the following conditions hold true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' R1 = R2 = V (G1) ∩ V (G2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=', their graphs overlap exactly in the interface, (Ω1, F1) = (Ω2, F2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=', the strategy graphs are the same, M1(α, β) ∪ M2(α, β) is a bijection between P1(α) ∪ P2(α) and P1(β) ∪ P2(β) for every α, β ∈ Ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The conditions for compatible partial labelled strategies ensure that the interfaces overlap in a way that a composed function will be a bijection which allows us to cut and compose them in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='22 (Cut).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let us have a labelled strategy B and a vertex cut R which partitions the vertices of G(B) into R, A and C in such a way that there are no edges between A and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We say B is cut along R into two partial labelled substrategies A and C where A is a 14 Computing m-Eternal Domination Number of Cactus Graphs in Linear Time partial labelled substrategy induced by V (G(A)) = R∪A and C is a partial labelled substrategy induced by V (G(C)) = R ∪ C such that A and C are compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='23 (Composing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' By composing two partial labelled strategies B1 and B2 (Bi = (Gi, (Ωi, Fi), Pi, Mi, Ri)) we mean getting (G∗, (Ω, F), P ∗, T ∗) where G∗ = � V (G1) ∪ V (G2), E(G1) ∪ E(G2) � , Ω = Ω1 ∪ Ω2, F = F1 ∪ F2, ∀γ ∈ Ω we have P ∗(γ) = P1(γ) ∪ P2(γ), and T ∗(α, β) = M1(α, β) ∪ M2(α, β) for every α, β ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Composing two compatible partial labelled strategies yields a labelled strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let us use the notation of Definitions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='21 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We need to check whether (G∗, (Ω, F), P ∗, T ∗, ∅) is a partial labelled strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' First, G∗ is a graph where we unite vertices and edges, while only the interface vertices are overlapping;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' this constitutes a well- defined graph without multiedges and loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The states Ω1 are the same for the compatible B1 and B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Next, mapping of the states to vertices is done by uniting the individual sets P1(γ) ∪ P2(γ) for each γ ∈ Ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Each vertex is now guarded in the union of states it was guarded before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Last, we check whether the union of M1 and M2 always maps to a well- defined transition, however, this is ensured by compatibility conditions over Pi and Mi in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ◀ We define an equivalency relation (reflexive, symmetric, and transitive) with respect to the interfaces as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='25 (Interface equivalent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Two partial labelled strategies B1 and B2 (Bi = (Gi, (Ωi, Fi), Pi, Mi, Ri)) are interface equivalent if G[R1] = G[R2], Ω1 = Ω2, F1 = F2, for all α ∈ Ω1 we have P1(α) ∩ R1 = P2(α) ∩ R2, and we have (a, b) ∈ M1(β, γ) ⇔ (a, b) ∈ M2(β, γ) for all u such that a = u ∨ b = u for all β, γ ∈ Ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Interface equivalent partial labelled strategies have the same states with respect to the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This allows us to infer compatibility as stated in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' For three partial labelled strategies B1, B2, and B3 if B1 is compatible with B2, B2 is interface equivalent with B3, and V (G(B1)) ∩ V (G(B3)) = R(B3), then B1 is compatible with B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let Bi = (Gi, (Ωi, Fi), Pi, Mi, Ri).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We will check the conditions stated in Defini- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' As V (G1) ∩ V (G3) = R3 and R3 = R2 by interface equivalency, and R2 = R1 by compatibility, the first condition holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' As G3[R3] = G2[R2] and V (G1) ∩ V (G3) = R3 there are no possible edges which would be shared by G1 and G3 outside of R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Ω1 = Ω2 by their compatibility, Ω2 = Ω3 by interface equivalency, so Ω1 = Ω3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The B3 is a partial labelled strategy so each guard on a vertex in V (G3) \\ R3 is covered by M3 exactly once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The guards on R3 are covered exactly when they were covered on R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' As B1 and B2 are compatible the guards on R2 were covered by M1 exactly when they were not covered by M2 and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Hence, this property still holds for B1 and B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ◀ The culmination of the previous notions and lemmas is the following procedure which we use as one major part for proving upper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='27 (Expansion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let us have a labelled strategy B with a partial labelled substrategy C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let us also have a partial labelled strategy C′ which is interface equivalent with C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' An expansion of B from C to C′ is the following sequence of operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Cutting B along R(C) into C and D (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='22), composing D with C′ into a labelled strategy R (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Blažej, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Křišťan, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Valla 15 Partial labelled strategies D and C′ are compatible due to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The result R is a labelled strategy due to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' To establish the difference in the number of guards used to defend B and R we have the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' For two interface equivalent partial labelled strategy B1 and B2 (as in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='25) there is some constant K(P1, P2) ∈ Z such that for all α ∈ Ω1 we have K(P1, P2) = |P2(α)| − |P1(α)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Suppose we have arbitrary states α, β ∈ Ω1 and let Kα = |P2(α)| − |P1(α)| and Kβ = |P2(β)| − |P1(β)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let M1(α, β) be part of B1 and M2(α, β) part of B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Each defines a pairing of guards in respective states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' However, the guards on the interface are not required to participate in the pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' So we have |P1(α)| + g1(α, β) = |P1(β)| + g1(β, α) where gi is the number of guards that do not participate in the pairing of respective Mi (we assume symmetric moves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Similarly for B2 we have |P2(α)| + g2(α, β) = |P2(β)| + g2(β, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' As the partial labelled strategies are interface equivalent, the sets of guards which do not participate in the pairings is the same, so g1(γ, δ) = g2(γ, δ) for all γ, δ ∈ Ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We get Kα = |P2(α)| − |P1(α)| = |P2(β)| + g1(β, α) − g1(α, β) − (|P1(β)| + g2(β, α) − g2(α, β)) = |P2(β)| − |P1(β)| + (g1(β, α) − g2(β, α)) + (g2(α, β) − g1(α, β)) = |P2(β)| − |P1(β)| = Kβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We set K(P1, P2) = Kα as we showed that this value is the same irrespective of the chosen α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ◀ To be able to use an expansion we need to select a partial labelled substrategy C of B and then show that C is interface equivalent with C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The expansion then proceeds as in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='27 and an upper bound is obtained from the following observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let us have an expansion of B from C to C′ (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='27) which results in a labelled strategy R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The expansion increases the number of used guards by K(P(C), P(C′)) due to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Assuming that B is an optimal strategy we obtain γ∞ m (G(R)) ≤ γ∞ m (G(B)) + K(P(C), P(C′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We showed a way to describe a labelled strategy and how we can exchange the underlying defended graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' However, to be able to do this we need the strategy to be the same for the original and expanded graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' So before we start expansion we alter the strategy on the original graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This is discussed in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='3 Tools for Altering Strategies In this section, we introduce further notions useful for working with strategies when building upper bound constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The typical upper bound proof uses tools introduced in this section to alter the strategy and then applies expansion (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='27) which gives the upper bound by Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' First, let us note that all the notions can be thought of as “up to isomorphism” because we can relabel graph or strategy vertices and relabeling does not fundamentally change them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We skipped this in definitions for the sake of readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let us also set from now on 16 Computing m-Eternal Domination Number of Cactus Graphs in Linear Time B = (G, SG, P, T , R) and SG = (Ω, F), and similarly for B′ and B∗ have respective graphs G′ and G∗, strategies, mappings, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Now we present the main operation for altering strategy graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='30 (Graph Cartesian product over subset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let us have graphs G1 and G2 while A ⊆ V (G1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The graph Cartesian product over subset A denoted as G1 □A G2 is a graph H such that V (H) = {(u, ∅) | u ∈ V (G1) \\ A} ∪ {(u, v) | u ∈ A, v ∈ V (G2)}, {(a, b), (c, d)} ∈ E(H) ⇔ � (a = c) ∧ (a ∈ A) ∧ ({b, d} ∈ E(G2)) � ∨ ∨ � {a, c} ∈ E(G1) ∧ ((b = d) ∨ (b = ∅) ∨ (d = ∅)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The operation in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='30 can be thought of as a Cartesian product where the sets of vertices created from G1 which are not present in A are identified to a single vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Equivalently, H can be constructed by taking the graph Cartesian product of G[A] and H, adding G[V (G) \\ A], relabeling each new vertex u as (u, ∅) and connecting each such (u, ∅) ∈ V (G) \\ A to all (v, x) ∈ A × V (H) such that v ∈ NG1(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This operations will prove very useful when altering strategy graphs – we will see it used soon in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='37 and many times in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The aim of this operation is to defend parts of the graph almost independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The edges created from G1 represent changes of guard positions within one part of the graph and edges from G2 represent changes in another part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' While guards move within one part of the graph then the guards in the other part will remain stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The necessity of the set A comes from the fact that the strategy in one part assumes that a guard occupies vertex (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' u) so then the altered part is restricted to vertices where the guard is present on the vertex (A = S(u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' See Figure 6 for an example application of the Cartesian product over subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' A = A G1 G2 H Figure 6 Example of a graph Cartesian product of G1 and G2 over subset A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We shall use the Cartesian product of G′ and complete graph over subset very often so we will use short notation that allows us to focus on what happens in the created strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='31 (Short notation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let G □A {α1, α2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' , αn} = G □A Kn where V (Kn) = {β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' , βn} and αi denotes sets of states created from βi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=', αi = {(a, βi) | a ∈ A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The Cartesian product over subset will be used to first alter the strategy graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The multiplied states shall defend the same set of vertices as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Then, during expansion, the guards shall be moved in order to defend new parts of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' There, we need to ensure that the strategy remains defending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' For this, we have the following lemma that tackles the unchanged and changed states in separate cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let us have H = G1 □A G2 with vertices labelled as in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Blažej, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Křišťan, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Valla 17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' If C is a dominating set of G1, then {(c, b) | (c, b) ∈ V (H), c ∈ C, b = ∅ ∨ b ∈ A} is a dominating set of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' If A is a dominating set of G1 and B is a dominating set of G2, then A × B (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=', {(a, b) | a ∈ A, b ∈ B}) is a dominating set of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let us have a vertex (x, y) in V (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In Case 1, there is a c ∈ C that dominates x in G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Therefore, (c, y′) with y′ = ∅ if c ̸∈ A or with y′ = y if c ∈ A dominates (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In Case 2, if x ̸∈ A (so y = ∅), then there is some a ∈ A that dominates x in G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' As |B| ≥ 1 there is (a, b) ∈ A×B that dominates (x, y) in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Otherwise x ∈ A so there is b ∈ B that dominates y in G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Hence, (x, y) is dominated by (x, b) ∈ A × B in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ◀ When changing the strategy, we want to keep the properties of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='32 to ensure that labelled strategy is defending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The following lemma shows the second major operation for changing strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' It allows us to add leaves to arbitrary vertex and defend the new graph with one more guard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='33 (Leaves addition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let us have a graph G and let u ∈ V (G) such that it has ℓ ≥ 1 adjacent leaf vertices v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' , vℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let G′ be a graph G with vertices v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' , vℓ removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' For any defending labelled strategy B′ there is a defending labelled strategy B with strategy graph SG = S′ G′ □S(u) Kℓ that uses one more guard than B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' First, let SG′ = S′ G′ □S(u) {α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' , αℓ} (see short notation Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' As u must be defended S(u) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' By its construction, all guards of strategy SG′ are stationary on T (αi, αj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let δ′ = Ω′ \\ {α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' , αℓ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We expand the strategy over SG′ to G by adding u to δ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=', P(δ) = P ′(δ′) ∪ {u}) and adding vi to αi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We set T (αi, αj) = {(vi, u), (u, vj)} and we extend T (δ, αi) with (u, vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' As αi dominates the clique and S′(u) dominates S′ G′ we have by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='32 that B is a defending labelled strategy for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ◀ ▶ Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='33 the construction works the same for any number of leaves, hence, we may add additional leaves after its use retroactively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We shall build strategies where vast majority of leaves are defended with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This gives a merit to treat all such states in the same way as their transitions with respect to the rest of the graph are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' To do this we put forward the following notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='35 (Group state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let a group defense be a set of states which were created by Cartesian product of G and a clique Kn over a subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We shall use group defense only to describe groups of leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' By Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='34 we will be able to add new leaves to such group at any point of the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In group states, but also in general strategies we investigated, it seems that vertices which are adjacent to multiple leaves are often permanently occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' To get a concrete result from this observation let us show how to alter an m-Eternal Domination strategy such that such vertices are permanently occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='36 (Permanently defended).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' A vertex u is permanently defended (permanently occupied) in B if S(u) = Ω, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=', u ∈ P(α) for every α ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' For a graph G and its arbitrary defending labelled strategy B′ we may create a defending labelled strategy B which uses the same number of guards and where each vertex adjacent to at least 2 leaves is permanently defended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 18 Computing m-Eternal Domination Number of Cactus Graphs in Linear Time Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let vertex u be a vertex with at least 2 adjacent leaves such that u is not permanently occupied in in strategy B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let α′ be a state where no guard occupies u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In P ′(α′), there must be a guard on each leaf adjacent to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let v and w be two of the leaves adjacent to u in G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let SG′ = S′ G′ □Ω′\\S′(u) {αv, αw} (see short notation Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let P ′(αv) = P(α′) ∪ {u} \\ {w} and P ′(αw) = P(α′) ∪ {u} \\ {v}, so P(αv) occupies v and P(αw) occupies w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' To create transitions T , we keep all transitions between states within S′(u) the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' For all β ∈ Ω \\ S′(u), we set T (αv, β) as T (α′, β) with w in each movement substituted by u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Such substitution still constitutes movements as N[w] ⊆ N[u].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Similarly, we create the transitions for T (αw, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Furthermore, we set T (αw, αv) = {(v, u), (u, w)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Note, that the transitions in reverse direction are derived from symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This shows that there are valid transitions for all edges of the created strategy graph SG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In the obtained strategy B vertex u is permanently defended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' As we use Cartesian product with a complete graph over a dominating subset it follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='32 that the strategy is still defending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We repeat the above procedure for each vertex adjacent to at least 2 leaves until all such vertices are permanently defended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ◀ 5 Reducing Cactus Graphs In this section, we prove that Γ∞ m (G) = γ∞ m (G) for cactus graphs by showing optimal strategies and unconditional lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The main idea is to repeatedly use reductions on the cactus graph G to produce smaller cactus graph G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Then we prove that a strategy for G uses a fixed number of guards more than an optimal strategy for G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Respective lower bound then shows that the strategy for G is indeed also optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We will describe precise way we get such results in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='1 but before that, we show the overall structure of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The proof uses an induction on the number of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The base case is a small graph (1 or 2 vertices) where the optimal strategy is elementary (see Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The induction step is described in detail later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Now we show several structural properties of cactus graphs which allow us to do the induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='1 (Leaf cycle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Leaf cycle is a cycle which has at most one vertex (called connecting vertex) which has neighbor such that it is not a vertex of the cycle nor a leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' See a leaf cycle on Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='2 (Leaf component).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' By a leaf component we mean either a leaf cycle or a leaf vertex which is not adjacent to a leaf cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Every cactus graph with at least 3 vertices contains a leaf component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let us obtain the block-cut tree T representation of the cactus graph and root it in an arbitrary block node (see [7, block-cutpoint trees, page 36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Each block node of T either represents a single edge or a cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We observe the deepest nodes of T to get the following three cases, see Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' A There is a deepest node which represents a cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' B A deepest node’s grandparent block is a single edge block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' C A deepest node’s grandparent block is a cycle block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' D No deepest node has a grandparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Blažej, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Křišťan, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Valla 19 A B C D or Figure 7 Example subtrees for structures which always appear in the block-cut tree of a cactus graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Big squares represent cycle nodes, small full circles represent articulations, and small empty circles represent single edge nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Note that in cases A and C the graph contains a leaf cycle, in case B it contains a leaf vertex which is not adjacent to a leaf cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The case D is trivial and the graph is either a cycle or a single edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Hence, a cactus graph always contains a leaf component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ◀ ▶ Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='4 (Vertex colors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' A non-connecting vertex v of a leaf cycle C is labeled with a color col(v) which depends on the number of adjacent leaves in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' col(v) is \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 �0 if v is adjacent to 0 leaves �1 if v is adjacent to 1 leaf �2 if v is adjacent to at least 2 leaves We shall label v as col(v) = �X if v is the connecting vertex of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' When a vertex can have different colors (to cover several cases at once) we list them by set of colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' For that purpose, we may write �0 instead of {�0}, and similarly for �1, �2, and �X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Also, we use a shortcut to denote all colors �⋆ = {�0, �1, �2, �X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' See Figure 8 for an example of �0, �1, and �2 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' a b c d G e �0 �1 �2 �X �⋆ Figure 8 A leaf cycle of a graph G with a partial labelled strategy B containing vertices a, b, c, d, and e with colors �0, �1, �2, �X, and �⋆, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The original graph was bigger and its rest was connected to d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' To look at the leaf cycle in isolation, the graph was cut in vertex d that now constitutes interface of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This cycle would be denoted by a leaf sequence (�X,�⋆,�0,�1,�2,�X) (or reversed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' To describe reductions over leaf components we will use a concise notation for the leaf cycles which just lists the colors of consecutive vertices of the cycle as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' See Figure 8 for an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='5 (Leaf sequence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' , vn) be n consecutive vertices of a leaf cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The leaf sequence of vertices (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' , vn) is (col(v1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' , col(vn)) where col(vi) ⊆ {�0, �1, �2, �X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Moreover, given two leaf sequences A and B and a graph G which contains a leaf cycle with a leaf sequence A, let A → B denote a reduction of subgraph with leaf sequence A to one with leaf sequence B in G to obtain G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Note that if the leaf sequence starts and ends with a connecting vertex and contains no �⋆, then it describes the whole cycle because colors correspond to the number of leaves and there is only one connecting vertex in a leaf cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Now, we show the base case and the overview of the induction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 20 Computing m-Eternal Domination Number of Cactus Graphs in Linear Time ▶ Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='6 (Base cases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let the base cases be the following graphs along with their optimal defending labelled strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' A single isolated vertex with no edges defended by labelled strategy � ({u}, ∅), ({α}, ∅), {α → {u}}, ∅ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' A single isolated edge is defended by labelled strategy �� {u, v}, {{u, v}} � , � {α, β}, {{α, β}} � , � α → {u}, β → {v} � , � (α, β) → {(u, v)} �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='1 Technique and Overview For the induction step, every reduction takes the cactus graph G and changes it to G′ which has smaller number of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Reductions will be performed on a leaf component which by Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='3 is always present in a cactus graph on at least three vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The two cactus graphs which have at most two vertices are covered by base cases from Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' More precisely, every reduction shows lower bound and upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Lower bound is shown for the m-Eternal Guard Configuration and involves using Observations 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='8–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' These tools make the graph smaller and show that in any defending strategy the removed parts required some minimum number of guards;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' they give us lower bound Γ∞ m (G) ≥ Γ∞ m (G′) + K for some constant K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Upper bounds are shown for the m-Eternal Domination and usually involve two separate steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' First step takes the reduced graph G′ and its optimal strategy S′ G′ and shows how to alter the strategy by tools shown in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This does not change the number of guards, but only structure of the defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Second step uses the framework shown in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' It takes part of the graph we intend to expand (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='27), cuts it, and replaces with an interface equivalent (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='25) partial labelled strategy, as described in Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' During the expansion, the strategy graph SG does not change (so SG = SG′), however the graph and mapping does change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The labelled strategy now maps strategy graph states so that there are new guards and some states have guards moved to other vertices of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We also show how the transitions change between states that were altered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' When the defense of the graph is managed with K additional guards, this gives us an upper bound γ∞ m (G) ≤ γ∞ m (G′) + K (the same K as in the lower bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Combining the lower and upper bound using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='2 results in an optimal number of guards for G for m-Eternal Domination and m-Eternal Guard Configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The used reduction depends on a leaf component that the cactus graph contains by Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' If the deepest node is not adjacent to a leaf cycle, then we use leaf reductions shown in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Using these reduction exhaustively results in having a leaf cycle (or a base case) – we will show this soon in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' To reduce leaf cycles we will need additional properties on edges of the leaf cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This involves being able to forbid movement along an edge, and forcing move along an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We achieve this by partitioning all states of the strategy graph into tree groups which ensure these properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The properties are established in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Having the properties we take the leaf cycle and look at its vertex colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' If a color pattern is listed among reductions then we have a way to remove it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The reductions are split into two groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' First recognizes just a small part of the cycle, making it shorter – these are called cycle reductions Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The second recognizes the whole cycle and removes it entirely, leaving just a few leaves in its place – these are constant component reductions shown in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We show that one of these reductions may always be used V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Blažej, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Křišťan, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Valla 21 by exhaustive search of all possibilities in depicted in Figure 14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' and doubling this function, we show a slightly different proof in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We end this section with the aforementioned proof of the cactus graph structure after application of leaf reductions in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='7 and a diagram overview of the remaining sections in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Exhaustive application of Reductions t1, t2, and t3 on a cactus graph G results in reaching the base case or it results in a cactus graph with a leaf cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We saw in Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='3 that in every cactus there either is a leaf cycle or there is a set of ℓ ≥ 1 leaves with a common parent which is connected to the rest of the graph with a single edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The number ℓ directly implies which tree reduction may be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' If ℓ = 1, then we may use Reduction t1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' if ℓ = 2, then we use Reduction t3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' and last, if ℓ > 2, then we use Reduction t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' After exhaustive application we either reach the base case or the other case applies – we have a leaf cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ◀ Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='1 Lower bounds Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='3 Vertex identification Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='8 Leaf lower bound Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='9 Star lower bound Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='10 Path lower bound Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='2 Properties of cycle edges Property 3 Proper labelled strategy Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='3 Tools for altering strategies Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='30 Cartesian product over subset Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='33 Leaves addition Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='35 Group state Section 3 Leaf reductions Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='1 Reduction t1 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='2 Reduction t2 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='3 Reduction t3 Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='3 Cycle reductions Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='10 Reduction c1 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='11 Reductions c2 and c3 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='12 Reduction c4 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='13 Reduction c5 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='16 Reduction c6 Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='4 Constant component reductions Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='19 Cactus multigraph Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='20 Reduction m1 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='21 Reduction m2 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='22 Reduction r1 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='23 Reduction r2 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='24 Reduction r3 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='25 Reduction r4 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='26 Reduction r5 Figure 9 Overview of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Left boxes represent tools obtained in Section 4 (see Figure 2) and properties we introduce in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Right box shows structure of Section 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Left-to-right arrows show which tools are used for which results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Right-to-right (green) arrows show that the reduction is partially based on or uses another reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='2 Properties of Cycle Edges We shall assume that the built strategy over the graph holds some properties which allow us to make stronger induction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' More precisely, these properties shall be necessary to show Reductions c1, c4, and c5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='8 (Edge states).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' By edge states of (u, v) in Ω (where {u, v} ∈ E(G)) we mean 22 Computing m-Eternal Domination Number of Cactus Graphs in Linear Time creating sets Lu,v, Ru,v, and Nu,v such that α ∈ Lu,v if ∃β ∈ Ω, (u, v) ∈ T (α, β), α ∈ Ru,v if ∃β ∈ Ω, (v, u) ∈ T (α, β), Nu,v = Ω \\ (Lu,v ∪ Ru,v) Note that because the orientation of the edge plays a role in these definitions, we have La,b = Rb,a, Ra,b = Lb,a, and Na,b = Nb,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The names of the sets reflect from which side a guard can traverse the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Also note, that if we assume symmetry then when moving to and from Nu,v the edge {u, v} cannot be traversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We propose the following edge property which is somewhat similar to Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Property 2 (Proper edge states).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' For a strategy B over graph G an edge {u, v} holds Property 2 if and only if its edge states Lu,v, Ru,v, and Nu,v are all non-empty, Lu,v∩Ru,v = ∅, and each of them is a dominating set over SG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' There are several ramifications of an edge having Property 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Because of Lu,v ∩ Ru,v = ∅ there is no state where we may choose to move over (u, v) or (v, u), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=', at most one of these movements is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' At the same time, as each of these sets is dominating SG, it follows that we may get into any of these sets in one transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Last, as each set is non-empty we may force the strategy to forbid to move over {u, v} in the current and one future transition by moving to Nu,v at any point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Additionally, we may force a movement over (u, v) by moving first to some α ∈ Lu,v and then to β ∈ Ru,v such that (u, v) ∈ T (α, β) as per Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' All the properties that proper edge states additionally have compared to non-proper edge states are true irrespective of permutations of Lu,v, Ru,v, and Nu,v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Hence, we may use the same sets on different edges by permuting them and checking that they constitute edge states of the new edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' For edges {u, v} and {a, b} if we map proper edge states Lu,v, Ru,v, and Nu,v to new sets La,b, Ra,b, and Na,b (with possibly permuting them) then these constitute proper edge states of {a, b} if and only if they constitute edge states of {a, b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' If the new edge states La,b, Ra,b, and Na,b do not constitute edge states then they trivially cannot be proper edge states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' When Lu,v, Ru,v, and Nu,v are proper edge states then they are disjoint and nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' These properties do not depend on their order so as long as the new states are edge states they will be proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ◀ Our goal will be to have Property 2 on all edges that lie on a leaf cycle that are incident to at least one �0 or �X vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We shall also show that it holds in some special cases to make several reductions easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In reductions, we will check that an edge has Property 2, however, the intuition about it is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We need to check whether each cycle edge is traversed at least once and whether it is not traversed at all by at least one state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Also, it is usually trivial, but we should check that the edge cannot be traversed in both directions from some state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Property 3 (Proper labelled strategy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' A partial labelled strategy B over a cactus graph G has Property 3 if and only if Property 2 holds for each edge that lie on a cycle and is incident to a �0 or a �X vertex, or is on a leaf cycle (�X, �2, �2, �X), or is incident to a �X vertex while not being a edge which lies between �X and a �2 vertex on leaf cycle (�X, �2, �0, �0, �2, �X) or (�X, �0, �0, �2, �X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Blažej, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Křišťan, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Valla 23 Our goal is to keep our cactus graph proper (as per Property 3) in all steps of reducing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' For simplicity, we shall work with reductions as if all edges on cycles which are incident to �0 or �X vertex have Property 2 and we shall tackle the exceptions to this rule separately in Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='18 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='3 Cycle Reductions Due to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='7 we know that applying tree reductions may result in either solving the instance entirely or we obtain a leaf cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In this section, we will tackle leaf cycles with cycle reductions which results in a leaf cycle of constant size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Constant-sized leaf cycles are then resolved in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let C denote a leaf cycle where vertices are labeled with colors according to Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Cycle reductions consist of the following reductions (see notation in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=', Reduction c1 describes that a graph G with a leaf cycle that contains consecutive vertices U with colors (�⋆, �1,�⋆) may be changed to G′ by substituting U with a vertices of colors (�⋆,�⋆) (so just �1 was removed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' At the same time, it claims that γ∞ m (G) ≤ γ∞ m (G′) + 1 and Γ∞ m (G) ≥ Γ∞ m (G′) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' All of this is concisely written as (�⋆, �1,�⋆) → (�⋆,�⋆) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Reduction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' c1 (�⋆, �1,�⋆) → (�⋆,�⋆) + 1 where (�⋆,�⋆) has Property 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Reduction 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' c2 (�2, �1,�⋆) → (�2,�⋆) + 1 ▶ Reduction 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' c3 (�2, �2,�⋆) → (�2,�⋆) + 1 ▶ Reduction 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' c4 (�⋆, �0, �0, �0,�⋆) → (�⋆,�⋆) + 1 where (�⋆,�⋆) has Property 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Reduction 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' c5 (�⋆, �0, �2, �0,�⋆) → (�⋆,�⋆) + 2 where (�⋆,�⋆) has Property 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Reduction 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' c6 (�X, �2, [�0, �2]2k, �X) → (�1) + 3k + 1 and (�X, �2, [�0, �2]2k+1, �X) → (�2) + 3k + 2 Let a and b be the first and the last vertex of the leaf cycle in G′, respectively, that are described by the reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' It is clear that these reductions may be used in cases where a and b are non-connected disjoint vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We note that the reductions will be used when {a, b} ∈ E(G′) though the result contains a pair of multiedges between a and b in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Moreover, these reductions may be used even in case where a = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Applying the reduction in such a case results in a loop in a within G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Though loops and multiedges may be created by the process they will be immediately removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' These cases will be addressed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Reductions c1, c4, and c5 require the edge that is being expended (edge {a, b} in G′) holds Property 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We shall ensure this by keeping Property 3 for G′ while ensuring that during every expansion this property is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let G′ be G after application of Reduction c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' G is defended with 1 more guard than G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let us label the vertices of colors (�⋆, �1,�⋆) by a, u, b, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let v be the leaf adjacent to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' By using Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='8 on vertices {u, v} we get lower bound Γ∞ m (G) ≥ Γ∞ m (G′) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' For upper bound, let La,b, Ra,b, and Na,b be edge states of the edge a, b in the strategy of G′ obtained as stated in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We extend all states of La,b and Ra,b by adding u to them, and we add v to Na,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We substitute movements (a, b) with {(a, u), (u, b)} in T (La,b, Ra,b) and we add (u, v) to T (La,b ∪ Ra,b, Na,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The new vertices are defended as La,b and Na,b are dominating the strategy graph because Property 2 holds for {a, b} in G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 24 Computing m-Eternal Domination Number of Cactus Graphs in Linear Time Table 2 List of the cycle reductions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' notation is the same as in Table 1 Reduction Lower bound Upper bound c1 a b −1 u v a b v u a b α β γ γ a b α′ β′ +1 c2 −1 u a b a b a u a b γ′ γ +1 β′ β Ω Ω b c3 −1 u a b a b a u a b γ′ γ +1 β′ β Ω Ω b c4 a b −1 u a b a b α β α β u γ +1 c d a b α′ β′ c5 a b −2 u b a a b α β α β u Ω +2 c d γ a b α′ β′ c6 See Figures 10 and 11 The edge states for the new edges {a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' u} and {u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' b} in G remain the same as for {a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' b} in G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Therefore, these edges now hold Property 2 in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' By extending all states with one guard we got a defending labelled strategy, so γ∞ m (G) ≤ γ∞ m (G′) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='2 we get that G is defended with one more guard than G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ◀ Reductions c2 and c3 merge a group of consecutive red and pink vertices and defend leaves adjacent to them by a group state (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let G′ be G after application of Reduction c2 or c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' G is defended with 1 more guard than G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The reductions are separate for the sake of future argument but they are proven in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let us label the vertices of colors (�2, {�1, �2},�⋆) by a, u, b, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let R1 denote all leaves adjacent to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' By applying Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='8 on R1 ∪ {u} we get lower bound Γ∞ m (G) ≥ Γ∞ m (G′) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' For upper bound, let γ′ be the group state for leaves adjacent to a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We add to leaves R1 two new vertices which are defended by γ′ as pointed out in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Now we split u from a, taking its leaves with it that we now label by R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Transitions between leaves is extended to T (R1, R2) = {(R1, a), (a, u), (u, R2)} and similarly, we extend all transitions which used a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The transitions that interacted with a and b are preserved, so the reduction expands interface equivalent partial labelled strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Though we did not need Property 2 the graph still has Property 3 because the new edge {u, b} takes on exact transitions that {a, b} had.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' So if {a, b} held the property in G′, then {b, u} holds it in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We added one guard so γ∞ m (G) ≤ γ∞ m (G′)+1 and by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='2 we get that G is defended with one more guard than G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ◀ ▶ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let G′ be G after application of Reduction c4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' G is defended with 1 more guard than G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Blažej, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Křišťan, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Valla 25 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let us label the vertices of colors (�⋆, �0, �0, �0,�⋆) by a, c, u, d, b, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Using Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='10 on {c, u, d} we get lower bound Γ∞ m (G) ≥ Γ∞ m (G′) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' For upper bound, let L′ a,b, R′ a,b, and N ′ a,b be edge states of the edge a, b in the strategy of G′ obtained from Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' These are proper edge states as {a, b} holds Property 2 in G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We extend the states by adding d to all states of L′ a,b, c to R′ a,b, and u to N ′ a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' this creates sets La,b, Ra,b, and Na,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We substitute movements along (a, b) with {(a, c), (d, b)} in T (La,b, Ra,b), hence, the exchanged parts of the graph are interface equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We add (c, u) to T (Ra,b, Na,b) and (d, u) to T (La,b, Na,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The new {c, u, d} vertices are defended by the nonempty sets Ra,b, Na,b, and La,b, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The edge states for the new edges are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' La,b = La,c = Ld,b = Nc,u = Ld,u Ra,b = Ra,c = Rd,b = Lc,u = Nd,u (4) Na,b = Na,c = Nd,b = Rc,u = Rd,u As these are only permutations of the edge sets by Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='9 they hold Property 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We get γ∞ m (G) ≤ γ∞ m (G′) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='2 we get that G is defended with one more guard than G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ◀ ▶ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let G′ be G after application of Reduction c5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' G is defended with 2 more guards than G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The proof goes very similarly as the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='12, but all states Na,b shall group defend leaves adjacent to u while u will be permanently occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We label the vertices of colors (�⋆, �0, �2, �0,�⋆) by a, c, u, d, b, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let R be the leaves neighboring u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Using Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='9 on u and its neighborhood we get lower bound Γ∞ m (G) ≥ Γ∞ m (G′) + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Repeat the same sequence of steps as in the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='12 which uses one guard and then add leaves adjacent to u by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='33 using one extra guard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This does not change transitions over the edges which are not incident to the leaves so by Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='9 they still hold Property 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We get γ∞ m (G) ≤ γ∞ m (G′) + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='2 we get that G is defended with two more guards than G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ◀ We remark that using reductions t1, t2, t3, c1, c4, and a small set of constant component reductions is sufficient to solve so-called Christmas cactus graphs (graphs where each edge is in at most one cycle and each vertex is in at most two 2-connected components) for which the optimal strategy we presented in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The remaining reductions tackle vertices of color �2, which are not present in the class of Christmas cactus graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The last cycle reduction is a curious special case, let us recall it first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Reduction 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' c6 (�X, �2, [�0, �2]2k, �X) → (�1) + 3k + 1 and (�X, �2, [�0, �2]2k+1, �X) → (�2) + 3k + 2 We shall use all the other cycle reductions first and if none of them can be used, then we use Reduction c6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This allows us to assume a particular structure which we define and prove now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The reason behind this structure may also be well understood from decision diagram of reduction application in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='14 (RW-cycle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' A leaf cycle C is a RW-cycle if it consists of vertices with alternating �2 and �0 colors such that the first and last is �2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=', C = (�X, �2, �0, �2, �0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ,�2, �0, �2, �X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Assume a leaf cycle C where Reductions c1, c2, c3, c4, and c5 cannot be applied anywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Then, |C| ≤ 6 or C is a RW-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 26 Computing m-Eternal Domination Number of Cactus Graphs in Linear Time Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' First, note that if the leaf cycle contains �1 vertices, then there always is a �1 vertex which neighbors �0 or �X, in that case we use Reduction c1, or it neighbors �2, we use Reduction c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Hence, all �1 vertices are removed if Reductions c1 and c2 were exhaustively used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Next, as only �2, �0, and �X vertices remain, exhaustively using Reduction c3 ensures that there are no two adjacent �2 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' (Note that if only red vertices remained, than we end up with (�X, �2, �X) which removes the multiedge by Reduction m2 and then uses Reduction t3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=') Last, if there is a (�2, �0, �0) part of a leaf cycle then the �2 vertex is either also adjacent to �X or Reduction c5 can be used as would (�⋆, �0, �2, �0, �0) necessarily occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Exhaustively using Reduction c4 ensures that such cases do not occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' So whenever there are two �0 adjacent vertices the �X vertex is at distance at most 2 from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This means that either the cycle has at most 5 vertices or the vertices of colors �2 and �0 alternate and constitute a RW-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ◀ The lower bound for an RW-cycle will not be much harder than for other reductions, however, for the upper bound we will need a strategy made just right for such a cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Reduction c6 is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let us label the vertices along the cycle as u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' , un with u1 being the connecting vertex (so that �2 vertices are even).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let un+1 = u1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let R2, R4, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' be the leaves adjacent to the red vertices u2, u4, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' First, we use Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='8 on {u6} ∪ R6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' second, we use Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='9 on N[u4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We shortened the cycle by 4 and got lower bound of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' By repeating the argument k times we end up with a cycle G′ of constant size 2 or 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The cycle of size 2 gets reduced by m2 and then t3 which results in a lower bound of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The RW-cycle of size 4 has form (�X, �2, �0, �2, �X) and its lower bound is shown in Reduction r4 to be 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Putting the 3k for every 4 vertices together with 1 and 2 lower bound for respective sizes of the cycle, we get the desired lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' See Figure 10 for an illustration of shortening the cycle by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' −3 u2 u4 R2 u8 R6 R8 u7 u6 u1 u2 R1 u8 R8 u7 u1 R4 u3 u5 Figure 10 Part of the lower bound proof for Reduction c6 Strategy for the upper bound is quite tricky to describe so let us define a few new notions just for its description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let red-parity of an even number i be the parity of i/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This divides red vertices of the RW-cycle into red-odd and red-even, based on the red-parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let reverse labeling be the labeling of the RW-cycle in opposite ordering, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=', if u′ 1 = un+1, u′ 2 = un, u′ 3 = un−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' , u′ n = u2, and u′ n+1 = u1, then u′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' , u′ n is the reverse labeling with respect to labeling u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' , un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' For the upper bound, we distinguish two cases depending on the size of the RW-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' First case is that the size is n = 4k + 2, the second has size n = 4k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We now focus on the first case, where the RW-cycle has size n = 4k + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Note that in RW-cycle of this size reverse labeling does not change the red-parity of red vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We alter the strategy by gradually expanding the states as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' S∗ G′ = S′ G′ □S′(u1) {α1, α2, β4, β8, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' , β4k} and SG′ = S∗ G′ □Ω\\S′(u1) {β2, β6, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' , β4k+2} V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Blažej, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Křišťan, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Valla 27 Now we perform expansion from G′ to G and set the states Ω of SG as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' P(α1) = k� x=0 {u4x+3} P(α2) = k� x=0 {u4x+1} P(β4x) = � (P(α2) ∩ {uj}ji) � ∪ {R4x} (5) P(β4x+2) = � (P(α1) ∩ {uj}ji) � ∪ {R4x+2} \\ {u1} Notice that u1 ∈ P(α2) as u4x+1 = u1 for x = 0 and u1 ∈ P(α1) as u4x+3 = u4k+3 = un+1 = u1 for x = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' For P(β4x+2) the intersections imply that u1 is not contained, but we mention it explicitly for clarity (as we do not consider un+1 to be uj for j < i even though it equals u1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The state β2i group defends leaves adjacent to u2i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Note that they behave differently based on their their red-parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Now for the transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' To make the notation concise let us shorten consecutive move- ments through red vertices as Fi = {(ui, ui+1), (ui+1, ui+2)} and Bi = {(ui, ui−1), (ui−1, ui−2)} (as forward and backward).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Note that we use Fi and Bi only for odd values of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let us have integers x and y and assume, without loss of generality, that x ≤ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We set the movements of transitions as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' If 2x and 2y have the same red-parity, then T (β2x, β2y) = {(R2x, u2x), (u2x, u2x+1)} ∪ y−4 � i=x F2i+3 ∪ {(u2y−1, u2y), (u2y, R2y)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' (6) Otherwise, 2x and 2y have different red-parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' If x is odd (so y is even), then their transition is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' T (β2x, β2y) = {(R2x, u2x), (u2x, u2x−1)}∪ x� i=2 B2i−3 ∪ 2k−1 � i=y+1 B2i+3 ∪{(u2y+1, u2y), (u2y, R2y)} If the red-parity is different and x is even, then in reverse labeling and swapping x with y we end up in the case where the red-parity is still different, but x is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This case was already solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The other direction of these transitions is filled in by symmetry (Property 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' It remains to describe transitions with α1 and α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' T (α1, α2) = {(u1, u1)} ∪ k−1 � i=0 F4i+3 T (β4x, α1) = {(R4x, u4x), (u4x, u4x−1)} ∪ x−1 � i=1 B4i+1 (7) T (β4x+2, α2) = {(R4x+2, u4x+2), (u4x+2, u4x+1)} ∪ x� i=1 B4i−1 (8) Note that α1 is α2 in reverse labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Hence, the case T (β4x, α2) is equivalent to T (β4x, α1) in reverse labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Similarly, the case T (β4x+2, α1) is equivalent to T (β4x+2, α2) in reverse labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' See a part of this strategy on Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' For the interface equivalency, note that in α1, α2, and β4k occupy u1 and states β4k+2 do not occupy u1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We showed how to transition between every pair of states, so the strategies are interface equivalent with a single pink vertex with expanded states as in SG′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Now for the second case, where the RW-cycle has size n = 4k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Note that in RW-cycle of this size reverse labeling changes the red-parity of red vertices, which was not true in the first case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The difference in the construction of the strategy is that now we expand from a 28 Computing m-Eternal Domination Number of Cactus Graphs in Linear Time u1 u2 u3 u4 u5 u6 u7 u8 u9 u10u11u12u13 u14u15 u16u17 u18 u19 = u1 β2 β4 β8 β10 β12 β14 β16 β18 β6 α1 α1 α1 α1 α1,2, β4k α2 α2 α2 α2 β6 β12 α1 α2 Figure 11 Part of a strategy on a RW-cycle of red-odd size 18 with guards (shown purple) placed on P(β6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' A few selected transitions are shown as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' red vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let u′ 1 and u′ 2 be the two leaves of the red vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let δ′ = Ω′ \\ (S(u′ 1) ∪ S(u′ 2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We gradually alter the strategy in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' S1 G′ = S′ G′ □S(u′ 1) {γ, β4, β8, β12, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' , βn} S2 G′ = S1 G′ □S(u′ 2) {α1, β2, β6, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' , βn−2} SG′ = S2 G′ □δ′ {α2} We perform the expansion to get SG such that all the states have exactly the same definitions as in the first case, see Equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We note a major difference: in the second case, u1 is not an element of P(α1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We added one extra state γ which has P(γ) = P(α1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' There will be a major significance for this state when proving edge properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Now we describe the transitions for the the strategy on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' For 2x and 2y of the same red-parity, the transition Equation (6) still holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In case 2x and 2y (with x < y) have different red-parity, then we consider two separate cases based on red-parity of 2x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' T (β4x, β4y+2) = {(R4x, u4x), (u4x, u4x−1)} ∪ x−1 � i=1 B4i+1 ∪ k� i=y+2 B4i−1 ∪ ∪ {(u4y+1, y4y+2, (u4y+2, R4y+2)} T (β4x+2, β4y) = {(R4x+2, u4x+2), (u4x+2, u4x+1)} ∪ x−1 � i=0 B4i+3 ∪ k+1 � i=y+2 B4i−3 ∪ ∪ {(u4y−1, y4y, (u4y, R4y)} Notice the difference in u1 – transition T (β4x, β4y+2) does not move through u1 so there u1 is stationary during it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' in T (β4x+2, β4y) movements {(u2, u1), (u1, un)} happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' To fill all possibilities of mutual transitions among β2x we add transitions obtained by reversed labeling and symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Now we show the transitions with α1 and α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Note that reversed labeling does not change these two states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' For β4x+2 we can apply Equation (8) to get T (β4x+2, α2), and by reversing the labeling this gives us also T (β4x, α2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Note that after this transition there is one less guard on G as it leaves through the interface {u1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In particular, T (β2x, α2) moves to u1 via (u2, u1) if 2x is red-odd, and via (un, un+1) if 2x is red-even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Similarly, for β4x we can apply Equation (7) to get T (β4x, α1), and by reversing the labeling we get T (β4x+2, α1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This transition did not interact with the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The transition among the two states is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' T (α1, α2) = k−1 � i=0 F4i+3 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Blažej, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Křišťan, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Valla 29 Note that this again results in a move (un, un+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Last, we introduce the new state γ which has the same guard configuration as α1, but differs in one transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' So T (γ, β2x) = T (α1, β2x), and T (γ, α2) = ∅ (all guards are stationary), but T (γ, α2) shall be T (α1, α2) in reverse labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' More precisely, T (γ, α2) = k−1 � i=0 B4i+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This contains a move (u2, u1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' See how movements interact with the interface in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' u1 u1 red-odd red-even Figure 12 Movements through the �X vertex in red-odd and red-even RW-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We discussed the interface impact of all transitions and note that they are equivalent to those in SG′, hence, the exchanged strategy is interface equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' It remains to show that SG is a proper strategy in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The strategy started S′ G′ was a clique and by Cartesian product over single vertices it remained a clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Thus, it suffices to say that there is at least one state in Lui,ui+1, Rui,ui+1, and Nui,ui+1, and that Lui,ui+1 ∩ Rui,ui+1 = ∅ for every i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' , n}, as any non-empty subset of vertices of the clique is dominating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Now we show the partitioning of the states into Lux,ux+1, Rux,ux+1, and Nux,ux+1 for each x ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' , n}, see Figure 13 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' First, observe that all closed neighborhoods of u2x for 4 ≤ 2x ≤ n − 2 contain exactly 2 guards in all the states we defined for this strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let x be an even integer such that 4 ≤ x ≤ n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let e = {ux+1, ux+2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' N R L R L R L R L R L e1 e2 Figure 13 The states of β2x that belong to Le, Re, Ne for e equal to the edges e1 and e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We show that βx ∈ Ne by a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Assume that in some transition from βx a guard moved through e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' As in βx vertex ux+1 is not occupied the guard must have moved from ux+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' However, then N[ux] would have 3 guards after the transition which cannot happen as we observed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' For edges that could not be addressed in the argument because they are too close to the �X vertex – e1 = {u1, u2} and e2 = {u3, u4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We observe that for red-even RW-cycles α1 ∈ Ne1 and γ ∈ Ne2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' For red-odd RW-cycles β2 ∈ Ne2 and βn ∈ Ne1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Now we claim that for any even x such that 2 ≤ x ≤ n, e = {ux+1, ux+2}, the states βy where y ̸= x are in Le if and only if ux+1 ∈ P(βy), and they are in Re otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We shall prove this more intuitively, as otherwise the claim can be proved by exhaustively listing all edges in all the transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' First notice, that all movements from βy which are not incident to leaves are performed over a continuous part of the cycle which starts in uy, and that they move “away” from uy towards the other end of the part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The movements always move an 30 Computing m-Eternal Domination Number of Cactus Graphs in Linear Time occupied �0 to �2 and if the part continues then moves the �2 to the adjacent �0 (this is true even when moving through the �X vertex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Hence, if e (that is not incident to �X) is included in the part of the movement, then we move (ux+1, ux+2) if and only if xx+1 is occupied, and we move (ux+2, ux+1) if and only if ux+1 was unoccupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Remainder of the edges which start at even positions and their N, L, and R sets can be obtained by the same argument on reversed labelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ◀ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='4 Constant Component Reductions The following lemma shows that considering the constant component cases completes the list of all necessary reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let us have a cactus graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' After an exhaustive application of leaf and cycle reductions the reduced cactus graph G′ is either a base case or it contains a leaf cycle of constant size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' First, by Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='3 the cactus always contains a leaf component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' If we exhaustively apply tree reductions, then by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='7 we are either done or there is a leaf cycle C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='15 we saw that an exhaustive application of the cycle rules results either in a base case or a cycle with alternating �2 and �0 vertices, which gets tackled by Reduction c6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The cases that remain are cycles of constant sizes where none of the reductions may be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ◀ We obtain the list of constant leaf cycles by the following procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' First, we apply Reductions c1 and c2 exhaustively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This removes all pink vertices from the leaf cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Now, let us scan over the vertices of the leaf cycle in a linear order of vertices along the cycle, starting from the connecting vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' On the one hand, whenever there is a cycle reduction applicable on the vertices which were scanned so far, then we can apply it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Hence, such a leaf cycle does not belong to constant leaf cycle cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' On the other hand, when the cycle returns back to the connecting vertex and still no cycle reduction may be used, then this cycle constitutes a constant leaf cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We present a full search diagram in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Again, we shall denote the reductions concisely as defined by Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' However, in constant component reductions the leaf sequence describes the whole leaf cycle and the connecting vertex is listed as the first and the last vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The vertices of the leaf cycle will be denoted by u, u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' , un−1, u where u is the connecting vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' , Rn−1 denote sets of all leaves adjacent to vertices u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' , un−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Note that the size 0 ≤ |Ri| ≤ 2 and directly coincides with color of respective vertex ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' See Figure 15 for an example of a leaf sequence of constant leaf cycle and notation of its vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Recall that the cycle reductions may be used even when the result does not create a simple graph, which is resolved in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' A strategy for a leaf cycle (u, u1, u2, u) with colors (�X, �2, �2, �X) is built in such a way that the edge (u1, u2) holds Property 2 (even though it is not incident to a �0 vertex) which makes an expansion of Reductions c4 or c5 over this edge possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This leaf cycle gets reduced by Reduction c3, then m2, and last with tree reduction t3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We show that in the strategy resulting for expansions holds Property 2 on edges {u, u1} and {u, u2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' See Figure 15 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Checking the exact movements of this strategy, we have that (u1, u2) ̸∈ T (Lu,u1, Ru,u1), (u1, u2) ̸∈ T (Lu,u1, Nu,u1), (u1, u2) ∈ T (Ru,u1, Nu,u1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Blažej, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Křišťan, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Valla 31 m1 m2 r1 c4 r3 c5 c3 r2 c5 c3 r2 r3 c4 c∗ 4 c5 c3 r5 c5 c3 r4 c5 c∗ 5 c5 c3 c6 c3 c3 c3 �X �X �X �X �0 �X �0 �2 �2 �0 �X �0 �2 �2 �0 �X �X �X�0 �X �0 �2 �2 �0 �X �0 �2 �2 �0 �X �X�0 �X �X�0 �2 �0 �2 �2 �0 �2 �2 �0 �2 �2 �0 �X �2 Figure 14 Case analysis of applied reductions on a leaf cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Vertices �1 were removed first by exhaustively applying their reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Scanning over vertices of a leaf cycle in order from the connecting vertex we identify these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The leaves show which reduction should be used for the scanned leaf cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Labels c1 up to c5 (yellow leaves) signify cycle reductions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' labels mi and ri (red leaves) signify constant component reductions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' nodes with a star ∗ require Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We can check that all the cases are covered by seeing that all inner (empty) nodes have outgoing edges labelled �0, �2, and �X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In particular, we may set Lu,u1 = Nu1,u2, Ru,u1 = Lu1,u2, and Nu,u1 = Ru1,u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' As the edge move sets for {u, u1} holds the properties which require all these sets to be non-empty, we have that they hold for {u1, u2} as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ◀ β γ Ω α Ω α β γ u1 u0 =un u2 γ α β Ω α β Ω α β t3 m2 c3 R1 R2 Figure 15 Left: Building the strategy for a (�X,�2,�2,�X) leaf cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The states α, β, and γ are representants of sets Lu,a, Ru,a, and Nu,a, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Right: Example guard configurations for states α, β, and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' From Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='18 we know that the cases (�X, �2, �0, �0, �0, �2, �X) and (�X, �2, �0, �2, �0, �2, �X) can be reduced by Reductions c4 and c5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' See these cases in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='1 Loops and Multiedges Similarly to Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='18, for the constant cases where we need to show that the properties hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' By allowing cycle reductions to apply in cases where the vertices a and b are adjacent, or even identical, we allowed the result of the reduction to contain multiedges or loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This intermediate form of the graph can be thought of as a generalized cactus graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='19 (Cactus multigraph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let the cactus multigraph be a multigraph (possibly with loops) that is connected and its every edge lies on at most one cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' A cactus multigraph differs from a cactus graph by allowing loops on arbitrary vertices (cycles of size 1) and allowing 2 multiedges between some vertices (cycles of size 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The 32 Computing m-Eternal Domination Number of Cactus Graphs in Linear Time cactus multigraph may be changed to a cactus graph by removing multiedges and loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The following two reductions take care of that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Reduction 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' m1 Let G′ be G with one loop removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Reduction 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' m2 Let G′ be G with a multiedge {u, v} (2 edges) where v has degree 2 (1 neighbor) changed to a single edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' u u u v u v Figure 16 Left: loop reduction m1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Right: multiedge reduction m2 Observe these reductions on Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We now prove that they do not need any additional guards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let G′ be G after application of Reduction m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' G is defended with the same number of guards as G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The strategy on G can be easily adapted to G′ by replacing any guard movement along the loop of u by not moving the guard on u, thus Γ∞ m (G′) ≤ Γ∞ m (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' At the same time, any strategy on G′ is applicable on G, so γ∞ m (G) ≤ γ∞ m (G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The equality follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' However, we would like the loop in u to have Property 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Intuitively, to keep the properties, we could say that at any configuration where u is occupied the guard can be moved along the loop in any direction or to be forbidden from moving along it while the configuration stays the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Formally, we can achieve the same by setting SG = S′ G′ □S′u {α, β, γ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We now set that T (α, β) = {(u, u)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This creates Lu,u = α, Ru,u = β, and Qu,u = Ω \\ {α, β}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This altered strategy holds Property 2 for the loop of u as the sets Lu,u, Ru,u, and Nu,u are non-empty and dominating SG because S′(u) dominates S′ G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ◀ In our case, Reduction m1 gets used after Reduction c4 is used on (�X, �0, �0, �0, �X) or after Reduction c5 is used on (�X, �0, �2, �0, �X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' It could also be used on (�X, �1, �X) after Reduction c1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' but in that case we can remove the multiedge first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let G′ be G after application of Reduction m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' G is defended with the same number of guards as G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let e1, e2 be the two different edges {u, v} oriented as (u, v) in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We assume that G′ is G with e2 removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Lower and upper bound are clear as every move along e2 can be changed to a move along e1 and the strategy on G′ is applicable to G without change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The challenge is, again, to show that Property 2 holds for e1 and e2 in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let β′ = S′(v) and α′ = Ω′ \\ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' To prove the property on e1 and e2, we will modify the strategy on G′ in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' If β′ ̸= Ω′, then there is a move along e1 in G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In that case, we set SG′ = S′ G′ □β′ {β, γ} while we alter the movements T (α, γ) to move along e2 instead of e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The edge states have α ∈ Le1, β ∈ Re1, and γ ∈ Ne1, and similarly for e2 (with swapped β and γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Second case is that β′ = Ω′ while α′ ̸= Ω′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Here, we alter the strategy such that for all states where u is not occupied, we move the guard from v to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This makes it so that v is occupied in states α which we now split into α1 and α2 in the same way as in the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Blažej, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Křišťan, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Valla 33 The last case is β′ = Ω′ while α′ ̸= Ω′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Here we set SG′ = S′ G′ □Ω′ {α1, α2, α3} and setting T ({α1, α2}) = {(u, v), (v, u)}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=', transitioning along e1 and e2 in opposite directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Also T (α1, α3) and T (α2, α3) have all guards stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This makes edge states as α1 ∈ Le1, α2 ∈ Re1, and α3 ∈ Ne1 while the exact same edge states work for e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In all the cases the edge states are non-empty, hence, Property 2 holds for e1 and e2 after Reduction m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ◀ α α′ β′ m2 β γ α1 Ω Ω Ω Ω α2 Ω Ω e1 e1 e2 α′ u v Figure 17 Cases of Reduction m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Left: There is a movement along the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Middle: Leaf is permanently occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Right: Leaf and its neighbor are permanently occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We note that in our strategy the case where v is permanently defended shall not occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' If we did not use Reduction m2 the number of constant size leaf cycle reductions would be significantly bigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' It gets used after reduction of (�X, �0, �1, �X) or (�X, �1, �1, �X) by c1, (�X, �2, �1, �X) by c2, (�X, �2, �2, �X) by c3, (�X, �0, �0, �0, �0, �X) or (�X, �0, �0, �0, �2, �X) by c4, (�X, �0, �0, �2, �0, �X) or (�X, �0, �2, �0, �2, �X) by c5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Without Reduction m2 each of these cases would have to be analyzed separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='2 Constant Size Leaf Cycle Reductions By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='17 the last cases that have to be resolved are covered by the following reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' See Table 3 for accompanying lower bound and upper bound proof illustrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Also see Figure 9 for diagram of notions used within proofs of these reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Reduction 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' r1 (�X, �0, �0, �X) → (�1) + 0 ▶ Reduction 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' r2 (�X, �0, �2, �X) → (�1) + 1 ▶ Reduction 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' r3 (�X, �0, �0, �2, �X) → (�2) + 1 ▶ Reduction 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' r4 (�X, �2, �0, �2, �X) → (�2) + 2 ▶ Reduction 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' r5 (�X, �2, �0, �0, �2, �X) → (�2) + 2 Now we proceed to show correctness of these reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' First group consists of reductions where a leaf cycle is reduced to �1 vertex u and its leaf v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The vertices of the expended leaf cycle are denoted by u, u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' , un−1, u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let G′ be G after application of Reduction r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' G is defended with the same number of guards as G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Using Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='3 to identify u2 with u1 then using Reduction m2 results in lower bound of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' For the upper bound, we first expand {u, v} to multiedges e1 and e2 as per Reduction m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Then we take G′ and change it to G by splitting v into two vertices u1 and u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We create β by substituting all occurrences of v in P(β′) with u1, and create γ by substituting all occurrences of v in P(γ′) with u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The transition between them becomes T (β, γ) = {(u1, u2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The strategy is interface equivalent as the strategy did not change states or transitions of the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We set Lu1,u2 = Nu,u1, Ru1,u2 = Nu,u1, and Nu1,u2 = Lu,u1 so the new edge {u1, u2} holds Property 2 and the strategy for G holds Property 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 34 Computing m-Eternal Domination Number of Cactus Graphs in Linear Time Table 3 List of constant component reductions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' thick red edges do not hold Property 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='Reduction ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='r5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='u1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='u4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='R4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='u2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='u3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='R1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='u1 u2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='α γ1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='+2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='+0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='No guard was added so γ∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='m (G) ≤ γ∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='m (G′) and by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='2 we get that G is defended with the same number of guards as G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ◀ We recall that by Ri we denote all leaves adjacent to ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let G′ be G after application of Reduction r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' G is defended with 1 more guard than G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Using Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='8 on {u2} ∪ R2 then using Reduction m2 results in lower bound of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We do the same expansion as in the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' After that, we use Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='33 to add leaves R2 to u2 while using one extra guard to defend it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Graph holds Property 3 by the same argument as in the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We added one extra guard which results in γ∞ m (G) ≤ γ∞ m (G′) + 1 and by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='2 we get that G is defended with one more guard than G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ◀ We now prove correctness of the other three cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The reduced graph G′ now consists of a single �2 vertex u (and its leaves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The partial labelled strategy on G′ has states α′ and β′ that defend the two leaves adjacent to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Also, let δ′ = Ω′ \\ (α′ ∪ β′), which may be an empty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let G′ be G after application of Reduction r3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' G is defended with 1 more guard than G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Using Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='8 on u3 and one of its leaves, identifying u2 with u using Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='3, and using Reductions m1 and m2 to remove loops and multiedges results in lower bound of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Blažej, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Křišťan, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Valla 35 For the upper bound, let u1 and u3 be the two leaves adjacent to u in G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let α′ = S′(u1) and β′ = S′(u3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We make SG′ = S′ G′ □β′ {β, γ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Now we expand the graph G′ by first applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='33 on u3, adding 2 new leaves to it using one additional guard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Next, we add a vertex u2 while connecting it to u1 and u3 and we move γ from R3 to u2 which is easy as u2 is a neighbor of u3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The only major change in transitions is that T (α, γ) = {(u1, u2), (u, u), (u3, u3)} instead of {(u1, u), (u, u3), (u3, u2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' No other transitions change, and u behaves the same, so the exchanged graphs are interface equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' See Figure 18 for strategy SG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' β α Ω γ δ β α Ω Ω γ α γ β α γ β δ Figure 18 Strategy for (�X,�0,�0,�2,�X) leaf cycle We note that each is traversed at some point and that α ∈ Nu2,u3, β ∈ Nu1,u2, and γ ∈ Nu,u1 so these edges hold Property 2 and the strategy for G holds Property 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The edge {u, u3} does not need to hold the property as it is a special case tackled in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We got that γ∞ m (G) ≤ γ∞ m (G′) + 1 and by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='2 we get that G is defended with one more guard than G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ◀ ▶ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let G′ be G after application of Reduction r4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' G is defended with 2 more guards than G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Using Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='8 first on {u1, v1} where v1 ∈ R1, then again on {u3, v3} where v3 ∈ R3, identifying u2 with u using Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='3, and using Reductions m1 and m2 to remove loops and multiedges results in lower bound of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' For upper bound, repeat exactly the expansion from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='24 on G′ which uses one extra guard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Continue by applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='33 on u1 which adds the leaves R1 using one extra guard while returning the defending labelled strategy on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The properties for edges {u1, u2}, {u2, u3}, and interface equivalency still hold from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' However, we can split γ into two states γ1 and γ2 which dictates whether T (γi, δ) traverses through {(u2, u1), (u1, u)} or {(u2, u3), (u3, u)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This ensures Property 3 for {u, u1} and {u, u3} as former cannot be traversed from γ2 and latter from γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Hence, we have γ∞ m (G) ≤ γ∞ m (G′) + 2 and by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='2 we get that G is defended with two more guards than G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ◀ ▶ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let G′ be G after application of Reduction r5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' G is defended with 2 more guards than G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Using Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='8 first on {u1, v1} where v1 ∈ R1, then again on {u4, v4} where v4 ∈ R4, and last identifying u2 and u3 with u using Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='3 results in lower bound of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' For upper bound, repeat exactly the expansion from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='25 on G′ which uses two extra guards (we do not use part of the proof which proved the property).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Then we make SG′ = S′ G′ □S′(u2) {γ1, γ2}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=', splitting γ into γ1 and γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We expand G′ to G by splitting u2 into u2 and u3 (while renaming u3 to u4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We preserve a guard of γ1 on u2 and γ2 on u3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Transition between them will be T (γ1, γ2) = {(u2, u3), (u, u)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This is interface equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' See Figure 19 for strategy SG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We have Property 3 as each edge is traversed and {u1, u2} cannot be traversed from γ2, {u2, u3} from α, and {u3, u4} cannot be traversed from γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We note that the other two cycle 36 Computing m-Eternal Domination Number of Cactus Graphs in Linear Time Ω Ω α γ1 β γ2 δ α γ1 β γ2 δ Ω Ω Ω α γ1 β γ2 α γ1 β γ2 Ω Ω α γ1 β γ2 α γ1 β γ2 Figure 19 Strategy for (�X,�2,�0,�0,�2,�X) leaf cycle edges {u, u1} and {u, u4} are part of the exception which is tackled in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Hence, we have γ∞ m (G) ≤ γ∞ m (G′) + 2 and by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='2 we get that G is defended with two more guards than G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ◀ Now we tackle the exception in Property 3 which influences Reductions r3 and r5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The order of reductions can be changed so that in a (�X, �0, �0, �2, �X) or (�X, �2, �0, �0, �2, �X) leaf cycle Property 2 is not required for edges that connect a �X and a �2 vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let us label by e an edge which connects a �X and a �2 vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Reductions which require the Property 2 on an edge are Reductions c1, c4, and c5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' If e is not a result of any of these reductions then there is no need for e to hold Property 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Otherwise, let us analyze the cases separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Reduction c1 resulted in e – before reduction we had (�X, �1, �2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ) where we can use Reduction c2 instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This results in (�X, �2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ) without needing the property for e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Reduction c4 resulted in e – before reduction we had (�X, �0, �0, �0, �2, �0, �0, [�2, ] �X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Hence, we may use Reduction c5 instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This results in (�X, �0, �0, �0, [�2, ] �X) where e has the property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Reduction c5 resulted in e – before reduction we had (�X, �0, �2, �0, �2, �0, �0, [�2, ] �X) so we may use Reduction c5 on the second �2 vertex instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' This results in (�X, �0, �2, �0, [�2, ] �X) where e has the property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We used other reductions to avoid reaching these leaf components by reductions that would require Property 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The first described case can be used at any point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The last two described cases are used on constant leaf components and as the result is different, it follows that their edges hold the property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ◀ This concludes the constant component reductions which together with cycle components and approach described in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='1 give us a polynomial algorithm to solve m-Eternal Domination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 6 Future Work The presented tools could be useful in a future study of the m-Eternal Domination on different graph classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' For instance, grids of size {3, 5} × n were extensively studied [16, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We believe it would interesting to see to which extent the tools could by applied in study of grids of less restricted dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Another noteworthy class of graphs are the so called dually chordal graphs, for which many domination related problems are polynomial time solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' It would be interesting to see whether m-Eternal Domination remains polynomial time solvable as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Furthermore, the computational complexity of the decision variant of the m-eternal domination problem is still mostly unknown as mentioned in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' It remains open whether the problem is in PSPACE and whether it is PSPACE-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Blažej, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Křišťan, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Valla 37 References 1 Václav Blažej, Jan Matyáš Křisťan, and Tomáš Valla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' On the m-eternal domination number of cactus graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In Reachability Problems - 13th International Conference, RP 2019, volume 11674 of Lecture Notes in Computer Science, pages 33–47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Springer, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 2 Andrei Braga, Cid C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' de Souza, and Orlando Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The eternal dominating set problem for proper interval graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Information Processing Letters, 115(6):582–587, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 3 Alewyn P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Burger, Ernest J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Cockayne, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Gründlingh, Christina M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Mynhardt, Jan H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' van Vuuren, and Wynand Winterbach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Infinite order domination in graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Journal of Combinatorial Mathematics and Combinatorial Computing, 50:179–194, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 4 Stephen Finbow, Margaret-Ellen Messinger, and Martin F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' van Bommel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Eternal domination on 3 × n grid graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Australasian Journal of Combinatorics, 61:156–174, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 5 Stephen Finbow and Martin F van Bommel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The eternal domination number for 3× n grid graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Australas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' J Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=', 76:1–23, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 6 Wayne Goddard, Sandra M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Hedetniemi, and Stephen T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Hedetniemi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Eternal security in graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Journal of Combinatorial Mathematics and Combinatorial Computing, 52:169–180, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 7 Frank Harary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Graph Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Addison-Wesley Publishing Company, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=', 1969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 8 Michael A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Henning and William F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Klostermeyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Trees with large m-eternal domination number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Discrete Applied Mathematics, 211:79–85, October 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 9 Fionn Mc Inerney, Nicolas Nisse, and Stéphane Pérennes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Eternal domination in grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In Lecture Notes in Computer Science, pages 311–322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Springer International Publishing, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 10 Fionn Mc Inerney, Nicolas Nisse, and Stéphane Pérennes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Eternal domination: D-dimensional cartesian and strong grids and everything in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Algorithmica, 83(5):1459–1492, February 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 11 William F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Klostermeyer and Gary MacGillivray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Eternal dominating sets in graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Journal of Combinatorial Mathematics and Combinatorial Computing, 68:97–111, February 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 12 William F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Klostermeyer and Gary MacGillivray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Eternal domination in trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' CoRR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' arXiv preprint arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='03107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 13 William F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Klostermeyer and Christina M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Mynhardt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Protecting a graph with mobile guards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Applicable Analysis and Discrete Mathematics, 10, July 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 14 William F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Klostermeyer and Christina M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Mynhardt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Domination, eternal domination and clique covering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Discussiones Mathematicae Graph Theory, 35(2):283, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 15 Ioannis Lamprou, Russell Martin, and Sven Schewe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Eternally dominating large grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Theo- retical Computer Science, 794:27–46, November 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 16 Margaret-Ellen Messinger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Closing the gap: Eternal domination on 3 x n grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Contributions to Discrete Mathematics, Vol 12:No 1 (2017), 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 17 Martín Rinemberg and Francisco J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Soulignac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' The eternal dominating set problem for interval graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Information Processing Letters, 146:27–29, June 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 18 Christopher M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' van Bommel and Martin F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' van Bommel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Eternal domination numbers of 5 × n grid graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Journal of Combinatorial Mathematics and Combinatorial Computing, 97:83–102, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' A Complete Strategies We note that if the strategy SG was a complete graph, then strategy S′ G created by the application of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='37 is also a complete graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Property 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' A partial labelled strategy B = (G, SG, P, T , R) is complete if SG is a complete graph, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=', there is {α, β} ∈ F for every α, β ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' 38 Computing m-Eternal Domination Number of Cactus Graphs in Linear Time We note that there are graphs where every optimal strategy is not complete, see Ap- pendix B for such an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Complete strategies can be effectively pruned to contain at most |V (G)| states in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ▶ Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' For any complete defending labelled strategy of cardinality k with the minimum number of vertices of SG it holds |V (SG)| ≤ |V (G)| − k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Pick an arbitrary complete defending strategy SG which uses k guards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' For each v ∈ V (G) we shall pick one state αv ∈ V (SG) such that v ∈ P(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' First, pick any α ∈ V (SG) and assign it as state to each v ∈ P(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Then, for every v ∈ V (G) \\ P(α) assign αv ∈ S(v) as its state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' We just picked |V (G)| − k + 1 states such that they form a strategy where every pair of states is traversable and which is defending as it covers all the vertices of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ◀ Similar to completeness of a strategy we may talk about the graph class of SG to describe its properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' B Non-complete Strategy ▶ Observation B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' An optimal m-Eternal Domination strategy on 5 × 5 grid uses at least 7 guards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Let us denote vertices of the grid by ui,j where 1 ≤ i, j ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' First, we show a lower bound of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Assume for a contradiction that there is a defending strategy S6 with at most 6 guards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Any state of S6 needs to dominate all 25 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' There must exist a state C where u2,2 is occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In C there also must be at least one guard in the closed neighborhood of each corner (u1,1, u5,1, u1,5, and u5,5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In the grid a vertex may dominate at most 5 vertices and a vertex on the side of the grid may dominate at most 4 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' All vertices in the closed neighborhood of corners are on the side of the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Additionally, vertex which dominates u1,1 may dominate at most 2 new vertices, as u2,2 already dominates many of vertices in its neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In total, the 6 guards of C may dominate at most 2 · 5 + 3 · 4 + 2 = 24, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' ◀ The upper bound can be shown by construction of a strategy, however, we have no good tools to show that all the strategies are not complete graphs – we found this using a full strategy-space search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' A construction which uses 7 guards contains three states and majority of their reflections and rotations, see them on Figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' In this case, we do not show the strategy, as it contains roughly 20 states (depending on a slight optimization, it may be less) that would contain 190 transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Figure 20 The 5 × 5 grid has 19 m-eternal dominating sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Each of the configurations can be expressed as a combination of rotations and reflections of exactly one of these 3 basic configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} +page_content=' Each of the 19 configurations is necessary for the strategy and can move into at most 12 other states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQflA0t/content/2301.05155v1.pdf'} diff --git a/R9FRT4oBgHgl3EQf8jhZ/content/2301.13684v1.pdf b/R9FRT4oBgHgl3EQf8jhZ/content/2301.13684v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c8b8b0f23055176d88d3bf55e0be9de337ef21a0 --- /dev/null +++ b/R9FRT4oBgHgl3EQf8jhZ/content/2301.13684v1.pdf @@ -0,0 +1,3 @@ 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Timea Csengeri,5 Orsolya F´eher,6, 7 +Rolf G¨usten,8 and Thomas Henning3 +1Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, E¨otv¨os Lor´and Research Network (ELKH), +Konkoly-Thege Mikl´os ´ut 15–17, 1121 Budapest, Hungary +2CSFK, MTA Centre of Excellence, Konkoly Thege Mikl´os ´ut 15–17, 1121, Budapest, Hungary +3Max Planck Institute for Astronomy, K¨onigstuhl 17, 69117 Heidelberg, Germany +4ELTE E¨otv¨os Lor´and University, Institute of Physics, P´azm´any P´eter s´et´any 1/A, 1117 Budapest, Hungary +5Laboratoire d’astrophysique de Bordeaux, Univ. Bordeaux, CNRS, B18N, all´ee Geoffroy Saint-Hilaire, 33615 Pessac, France +6School of Physics and Astronomy, Cardiff University, Queen’s Buildings, The Parade, Cardiff CF24 3AA, UK +7IRAM, 300 Rue de la piscine, 38406 Saint-Martin-d’H`eres, France +8Max Planck Institute for Radioastronomy, Auf dem H¨ugel 69, 53121 Bonn, Germany +(Received January 10, 2023; Revised January 10, 2023; Accepted January 10, 2023) +Submitted to ApJ +ABSTRACT +FU Orionis-type objects (FUors) are low-mass pre-main-sequence objects which go through a short- +lived phase (∼100 years) of increased mass accretion rate (from 10−8 to 10−4 M⊙ yr−1). +These +eruptive young stars are in the early stages of stellar evolution and, thus, still deeply embedded in +a massive envelope that feeds material to the circumstellar disk that is then accreted onto the star. +Some FUors drive molecular outflows, i.e. low-velocity wide-angle magneto-hydrodynamical winds, +that inject energy and momentum back to the surrounding envelopes, and help clear the material +surrounding the young star. Here we present a 12CO (3–2), 13CO (3–2) and 12CO (4–3) survey of 20 +FUor-type eruptive young stars observed with APEX. We use our 13CO (3–2) observations to measure +the masses of the envelopes surrounding each FUor and find an agreement with the FUor evolutionary +trend found from the 10 µm silicate feature. We find outflows in 11 FUors, calculate their masses and +other kinematic properties, and compare these with those of outflows found around quiescent young +stellar objects gathered from the literature. This comparison indicates that outflows in FUors are more +massive than outflows in quiescent sources, and that FUor outflows have a higher ratio outflow mass +with respect to the envelope than the quiescent sample, indicating that the eruptive young stars have +lower star-forming efficiencies. Finally, we found that the outflow forces in FUors are similar to those +of quiescent young stellar objects, indicating that their accretion histories are similar or that the FUor +outflows have lower velocities. +1. INTRODUCTION +Jets and molecular outflows are a ubiquitous phe- +nomenon in the process of star formation. +The for- +mer are highly collimated gas streams at high veloci- +ties (≥100 km s−1), and the latter have wider opening +angles and velocities between 1 km s−1 and 50 km s−1 +in the case of low-mass stars. +Jets are detected with +optical, near-infrared, radio molecular lines, and radio +Corresponding author: Fernando Cruz-S´aenz de Miera +cruzsaenz.fernando-at-csfk.org +continuum, while the slower outflows are typically de- +tected with molecular line tracers (Frank et al. 2014; +Bally 2016). +Both types of mass ejection events are driven by accre- +tion, thus the physical properties of the outflows depend +on the accretion history of the star. Indeed, evidence has +shown that Class 0 objects (i.e. younger protostars with +higher mass accretion rates) have elevated outflow mass +loss rates and higher outflow forces compared to more +evolved Class I or Class II objects (Mottram et al. 2017). +The mass accretion rates from protostellar disks to pro- +tostars are expected to undergo episodic variations. De- +arXiv:2301.03387v1 [astro-ph.SR] 9 Jan 2023 + +2 +Cruz-S´aenz de Miera et al. +tailed analysis of jet knots (e.g. Ellerbroek et al. 2014; +Lee et al. 2017; Garufi et al. 2019) and molecular out- +flow shells (Plunkett et al. 2015; Zhang et al. 2019; Nony +et al. 2020; Vazzano et al. 2021) show how the study of +outflows can shed light on the accretion history of the +protostars that drive them. +FU Orionis-type objects (FUors) are examples of +the episodic nature of accretion (Hartmann & Kenyon +1996; Audard et al. 2014; Fischer et al. 2022). These +eruptive young stars are low-mass protostars charac- +terized by a sudden increase in their mass accretion +rate, going from typical values of ∼10−8 M⊙ yr−1 up +to ∼10−4 M⊙ yr−1. These events are typically detected +as a 3 − 5 magnitude brightening at optical and near- +infrared wavelengths, and are expected to last up to +a century, meaning that these events increase the final +stellar mass by a significant amount. FUor-type events +generally occur in Class I objects. Accretion outbursts +have been detected in earlier stages, e.g. the Class 0 +HOPS 383 (Safron et al. 2015), and in later stages, +e.g. the Class II Gaia20eae (Cruz-S´aenz de Miera et al. +2022), however, these are not considered FUors. This +differentiation is because to classify an object as a FUor, +the near-infrared spectrum of the protostar must also +present the spectral signatures found in the prototyp- +ical FUors (Connelley & Reipurth 2018). In the case +a protostar shows these signatures and the photomet- +ric outburst was not detected, the source is considered +FUor-like. And if a Class I protostar shows an outburst +and none, or a minimal amount, of the spectral signa- +tures, then it is considered as Peculiar. +Outflows play an important role in the star forma- +tion process as they remove angular momentum from +the accretion disk, inject mass and energy into their sur- +roundings, and clear material from the envelope (Arce & +Sargent 2006). The circumstellar envelopes are the re- +mains of the parent molecular cloud core that surround +the protostar, and their properties (i.e. , mass and ex- +tension) are deeply connected with how evolved a young +star is, with younger objects having more massive and +larger envelopes than their evolved counterparts (Andre +& Montmerle 1994). Therefore, if the elevated accre- +tion rates during the outbursts can inject more momen- +tum into the envelopes via outflows, then these episodic +events must play an important role in the evolution of +their protostellar system. Indeed, it is expected that af- +ter an eruption, the inner circumstellar disk becomes de- +pleted and will be replenished by the surrounding enve- +lope (Vorobyov & Basu 2006) until the system can erupt +again (Bell & Lin 1994; Takami et al. 2018). Eventu- +ally, the repetitive outbursts will clear out the envelope +and the young system will move to its next evolutionary +phase, from Class I to Class II (Green et al. 2006; Quanz +et al. 2007; Green et al. 2013). +Previous observations of CO rotational transitions +have shown the presence of outflows in some known +FUors: V1057 Cyg (Rodriguez et al. 1990), V1735 Cyg +(Evans et al. 1994), L1551 IRS 5 (Wu et al. 2009, and +references therein), V883 Ori (Ru´ız-Rodr´ıguez et al. +2017a), Reipurth 50 (Ru´ız-Rodr´ıguez et al. 2017b), +FU Ori (Hales et al. 2015), V1647 Ori (Principe et al. +2018), V2775 Ori (Zurlo et al. 2017), V346 Nor (K´osp´al +et al. 2017b) and V900 Mon (Takami et al. 2019). In +other cases, optical and near-infrared spectroscopy have +shown indication of high-velocity jets: Z CMa (Poetzel +et al. 1989), V899 Mon (Ninan et al. 2015), iPTF 15afq +(Hillenbrand 2019), and V346 Nor (K´osp´al et al. 2020b). +Each of the aforementioned studies focused on a sin- +gle FUor-type object or on a few of them, preventing a +statistical analysis of their properties. In this paper we +present a systematic study of the envelopes surrounding +FUors, and we search for outflows among our full sam- +ple. We then compare our results with outflows found +in YSOs that are currently quiescent and for which it is +unknown whether they experienced an outburst or not. +As outflows found at thousands of astronomical units are +an indication of the accretion history of a protostar, this +comparison allows us to examine how comparable is the +histories of the FUors with those of the quiescent sam- +ple. The triggering mechanisms behind the FUor-type +outbursts is still not understood, however, it is possi- +ble that an examination of the differences between the +two samples might hint that FUors are protostar with +intrinsic differences that caused the outburst. Alterna- +tively, it could show the samples are similar and, thus, +we cannot rule out that quiescent sources experienced +FUor-type outbursts in the past. +The structure of the paper is as follows. The observed +sample is briefly introduced in Section 2, while in Sec- +tion 3 we describe the observations and the data reduc- +tion. +In Section 4 we present the distribution of the +gas in the environment surrounding the FUors, and the +properties of the integrated line profiles. The main goal +of this paper is to study the properties of the circum- +stellar gas, this analysis is found in Section 5, including +the characterization of the molecular outflows where de- +tected. In Section 6 we compare the outflows found in +the FUor sample with non-outbursting sources and draw +our conclusions. Finally, in Section 7 we summarize our +work and present our main findings. +2. SAMPLE +Our sample is composed of 20 eruptive young stars, +including most of the known FUors accessible from the + +Molecular outflows in FUors with APEX +3 +APEX site (Audard et al. 2014; Connelley & Reipurth +2018). +We note that not all the targets in our list +are considered FUors as some objects are cataloged as +FUor-like objects. This subclassification is used when +the photometric outburst was not detected but their +near-infrared spectrum shows features similar to those +of the prototypical FUors (e.g. BBW 76, Connelley & +Reipurth 2018). +The target list also includes erup- +tive young stars with peculiar accretion histories (e.g. +V1647 Ori) and a massive star with a powerful accre- +tion outburst (V723 Car). The non-FUor objects were +included because of their sudden increases of their mass +accretion rate, and thus the properties of their outflows +could be affected. The first part of the sample, com- +posed of eight targets, was analyzed by K´osp´al et al. +(2017a), where they found outflows in three objects: +HBC 494, Haro 5a IRS and V346 Nor. +Here we will +analyze the full sample, including a re-processing of the +target list presented in K´osp´al et al. (2017a). The sam- +ple presented in this paper includes ∼50% of the cur- +rently known FUors and FUor-like objects (Connelley +& Reipurth 2018). The full target list is presented in +Table 1. +3. OBSERVATIONS AND DATA REDUCTION +We carried out two programs with the FLASH+ re- +ceiver (Klein et al. 2014) at the APEX telescope (G¨usten +et al. 2006) to measure the 12CO (3–2), 13CO (3–2), +and 12CO (4–3) lines towards our targets. +Program +094.F-9508 was observed between 2014 August 23–28 +and program 098.F-9505 between 2016 August 25 and +2016 September 10. Both programs used the same tech- +nical setup and reduction process. The lower frequency +channel was tuned to 344.2 GHz in USB to cover the +13CO (3–2) at 330.588 GHz, and the 12CO (3–2) at +345.796 GHz, respectively. The higher frequency chan- +nel was tuned to the 12CO (4–3) line at 461.041 GHz in +USB. We used the XFFTS backends providing a nomi- +nal 38 kHz spectral resolution for the J = 3–2 lines and +76 kHz for the J = 4–3 line, these resulted in spectral +resolutions of ∼34.5 m s−1, ∼32.9 m s−1 and ∼49.4 m s−1 +for the 13CO (3–2), CO (3–2) and CO (4–3) lines, respec- +tively.. +For each target, 120′′×120′′ on-the-fly (OTF) +maps were obtained at 6 ′′ s−1, using a relative reference +off position 1000′′ away in right ascension. +We removed a first order baseline from the spectra, +and calibrated the data using a main beam efficiency of +0.73 and 0.60 at 352 GHz and 464 GHz, respectively, and +the values were converted to Jansky using 41 Jy K−1 and +48 Jy K−1 at 352 GHz and 464 GHz, respectively. +We +calculated the noise levels of each CO line by first se- +lecting the first and the last 100 channels of each cube +(individually confirmed to be free of line emission), cal- +culated the noise levels for each FUor using these chan- +nels, and then we calculated the median noise level of +all FUors to obtain representative values. The rms noise +levels, at the native spectral resolution mentioned ear- +lier, are 3.6 Jy for 13CO (3–2), 3.7 Jy for 12CO (3–2), +and 9.4 Jy for 12CO (4–3). The telescope’s half-power +beam-width is 19.′′2, and 15.′′3 at the corresponding fre- +quencies. As mentioned earlier, the first half of the sur- +vey has already been published by K´osp´al et al. (2017a), +and here we use their calibrated data for our analyses. +4. RESULTS +4.1. Distribution of gas +We constructed velocity integrated emission maps +(Moment 0) for 12CO (3–2) using all channels in our +data cubes. The resulting maps are presented in Fig- +ure 1. Some young eruptive stars are still deeply embed- +ded, therefore, it is possible that the observed CO emis- +sion originates from the remaining material in their sur- +rounding envelope. In order to verify that our CO detec- +tions come from the FUors, we compared our Moment 0 +maps with the dust continuum emission. We searched +for 250 µm continuum maps taken with Herschel/SPIRE +maps in the Herschel Science Archive1 and found data +at an angular resolution of 17.6′′ for 18 sources. +For +the two remaining sources (V900 Mon and Z CMa), +we searched the Canadian Astronomy Data Centre2 for +archival 850 µm observations taken with the James Clerk +Maxwell Telescope3 (JCMT) with an angular resolution +of 14.5′′. +In the cases of L1551 IRS 5, Haro 5a IRS, +V883 Ori, Reipurth 50, V899 Mon, V960 Mon, Z CMa, +V346 Nor, GM Cha, and HBC 687, the peaks of both the +dust emission and the gas emission are located at the po- +sition of the protostar. For five of our targets (V582 Aur, +AR 6A, iPTF 15afq, V723 Car, and OO Ser) the brighter +peaks of both gas and dust emission are offset from the +position of the protostar. The continuum peaks to these +five sources are 27′′ to the Southeast, 12′′ to the East, +28′′ to the Northwest, 22′′ to the West, and 6′′ to the +Southwest, respectively. In the case of OO Ser, there +is continuum emission toward the position of the FUor, +1 http://archives.esac.esa.int/hsa/whsa/ +2 https://www.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/en/ +3 The James Clerk Maxwell Telescope has historically been oper- +ated by the Joint Astronomy Centre on behalf of the Science +and Technology Facilities Council of the United Kingdom, the +National Research Council of Canada and the Netherlands Or- +ganisation for Scientific Research. + +4 +Cruz-S´aenz de Miera et al. +Table 1. Objects observed. +Name +Coordinates +Classa +FUor classification +vLSRb +Distancec +Lbold +[km s−1] +[pc] +[L⊙] +L1551 IRS 5 +04:31:34.07 +18:08:04.9 +I +FUor-like +6.46 +147 +25 +V582 Aur +05:25:51.97 +34:52:30.0 +F +Bona fide FUor +−10.85 +1320 +146 +Haro 5a IRS +05:35:26.75 −05:03:55.1 +I +FUor-like +10.90 +391 +50 +V883 Ori +05:38:18.09 −07:02:25.9 +F +Bona fide FUor +4.10 +388 +400 +Reipurth 50e +05:40:27.45 −07:27:30.0 +I +Peculiar +3.76 +460 +300 +FU Ori +05:45:22.36 +09:04:12.2 +I/II +Bona fide FUor +11.96 +402 +420 +V1647 Ori +05:46:13.13 −00:06:04.8 +I/II +Peculiar +10.06 +388 +39 +V2775 Ori +05:42:48.48 −08:16:34.7 +I +Bona fide Fuor +3.08 +428 +25 +V899 Mon +06:09:19.24 −06:41:55.8 +F/II +Peculiar +9.57 +785 +419 +AR 6Af +06:40:59.30 +09:35:52.3 +I +Peculiar +5.02 +890 +450 +V900 Mon +06:57:22.22 −08:23:17.6 +I +Bona fide FUor +13.77 +1130 +106 +V960 Mon +06:59:31.58 −04:05:27.7 +II +Bona fide FUor +23.81 +2068 +Z CMa +07:03:43.15 −11:33:06.2 +I +FUor-like +13.91 +1150 +500 +iPTF 15afqg +07:09:21.39 −10:29:34.4 +I/F +Peculiar +14.04 +920 +7.2 +BBW 76e +07:50:35.59 −33:06:23.9 +I +FUor-like +17.64 +1040 +287 +V723 Car +10:43:23.44 −59:33:55.3 +I +Peculiar +−19.58 +2500 +4000 +GM Cha +11:09:28.55 −76:33:28.1 +I/II +Peculiar +4.86 +160 +1.5 +V346 Nor +16:32:32.19 −44:55:30.7 +0/I +Peculiar +−3.08 +700 +135 +OO Ser +18:29:49.13 +01:16:20.6 +I +Peculiar +8.36 +311 +31 +HBC 687h +19:29:00.87 +09:38:42.9 +II +FUor-like +16.98 +400 +10 +aHere we refer as Class to the classification based on the shape of its SED (Lada 1987; Andre et al. 1993; +Greene et al. 1994). +b See Section 4.2. +c See Section 5.1. +dObtained from the literature (e.g. Audard et al. 2014; Connelley & Reipurth 2018, and references +therein). +eReipurth 50 and BBW 76 are labeled as HBC 494 and Bran 76 in K´osp´al et al. (2017a). +fAlso known as V912 Mon. +gAlso known as Gaia19fct. +hAlso known as Parsamian 21. +however the brightest peak is the one previously men- +tioned. We find that for FU Ori, V1647 Ori, V2775 Ori, +and V900 Mon, the dust emission is located at the po- +sition of the protostar, however, the peak of the gas +emission is offset. Finally, in the case of BBW 76, the +dust emission peaks at the position of the source, how- +ever, the CO map shows that the emission is weak and +extended. +4.2. Systemic velocity +As it is shown below, to estimate the kinematic prop- +erties of the outflows, we need a reliable estimate of the +systemic velocity for each target so that we can mea- +sure the velocity of the outflow relative to the proto- +star. The systemic velocities of our targets were esti- +mated by fitting a Gaussian function to the line profile +of 13CO, extracted using a circular aperture with ra- +dius of 10 000 au, and using the center of the best-fitting +Gaussian as the systemic velocity. Due to the proximity +of L1551 IRS 5 and the field-of-view of our observations, +we had to use a smaller aperture of 8000 au for this FUor. + +Molecular outflows in FUors with APEX +5 +50" +0" +-50" +Dec +L1551 IRS 5 +V582 Aur +Haro 5a IRS +V883 Ori +Reipurth 50 +50" +0" +-50" +Dec +FU Ori +V1647 Ori +V2775 Ori +V899 Mon +Ar 6a +50" +0" +-50" +Dec +V900 Mon +V960 Mon +Z CMa +IPTF15AFQ +BBW 76 +50" +0" +-50" +50" +0" +-50" +RA +Dec +V723 Car +50" +0" +-50" +RA +GM Cha +50" +0" +-50" +RA +V346 Nor +50" +0" +-50" +RA +OO Ser +50" +0" +-50" +RA +HBC 687 +Figure 1. Integrated intensity (Moment 0) maps of our targets for the 12CO (3–2) line observed with APEX (orange contours). +The Moment 0 maps were generated by integrating the full spectral cube in order to produce an unbiased map. The purple +contours are the 250 µm continuum emission from Herschel for most of our targets, the two exceptions are V900 Mon and Z CMa +where we show contours of the 850 µm continuum emission from the JCMT (see Section 4.1). The CO and the dust contours +are plotted with levels at 0.3, 0.4, . . . , 0.9 of the peak intensity, and are meant to be representative. The star symbols indicate +the nominal positions of the protostars. The bars at the bottom of each panel represent 10 000 au. +The line profiles and the best-fit Gaussians are shown in +Figure 2. +A number of sources have complicated line profiles +that could not be fitted by a single Gaussian. +Some +of these show asymmetric dips around the peak of the +emission (e.g. V1647 Ori, GM Cha, and OO Ser), which +can be due to the self-absorption of the envelope or due +to the rotation of the gas. The former scenario is more +likely based on the inspection of the channel maps of +13CO, thus, for these sources, we discarded the velocity +range of the dip and fitted the Gaussian function using +the remaining velocities. The line profiles of other ob- +jects show asymmetric shapes out to the wings of the line +profiles (e.g. V582 Aur, Reipurth 50, and V346 Nor), an +indication of multiple components (e.g. envelope, out- +flows, Keplerian disk, or unrelated gas in the same line- +of-sight) showing emission at 13CO. For these objects, +we fitted a combination of two or three Gaussian func- +tions to the line profile, and used the best-fit mean of +the Gaussian with the highest amplitude to estimate the +systemic velocity. +To +verify +our +estimated +systemic +velocities, +we +searched for the velocity around which the line profile +is most symmetrical. As expected, we found that the +sources with asymmetrical line profiles have the largest +differences, but still less than 1 km s−1. For the symmet- +rical sources, the differences are less than 0.3 km s−1. In +addition, we examined the channel maps of each target +to confirm our systemic velocity estimates. + +6 +Cruz-S´aenz de Miera et al. +5 +10 +0 +1000 +2000 +3000 +Flux [Jy] +L1551 IRS 5 +12.5 +10.0 +7.5 +0 +10 +20 +V582 Aur +7.5 +10.0 +12.5 +0 +2000 +4000 +Haro 5a IRS +2.5 +5.0 +7.5 +0 +250 +500 +750 +V883 Ori +0 +5 +0 +500 +1000 +Reipurth 50 +10 +15 +0 +200 +400 +Flux [Jy] +FU Ori +7.5 +10.0 +12.5 +0 +500 +1000 +1500 +V1647 Ori +0 +5 +0 +500 +V2775 Ori +7.5 +10.0 +12.5 +0 +500 +V899 Mon +2.5 +5.0 +7.5 +0 +100 +200 +Ar 6a +10 +15 +0 +100 +200 +Flux [Jy] +V900 Mon +20 +25 +0 +20 +40 +V960 Mon +10 +15 +0 +1000 +2000 +Z CMa +12.5 +15.0 +17.5 +0 +500 +1000 +1500 +IPTF15AFQ +15 +20 +0 +10 +BBW 76 +22.5 +20.0 +17.5 +vLSR [km s +1] +0 +50 +100 +Flux [Jy] +V723 Car +2.5 +5.0 +7.5 +vLSR [km s +1] +0 +500 +1000 +1500 +GM Cha +5 +0 +vLSR [km s +1] +0 +200 +400 +600 +V346 Nor +5 +10 +vLSR [km s +1] +0 +500 +1000 +1500 +OO Ser +15 +20 +vLSR [km s +1] +0 +25 +50 +HBC 687 +Figure 2. Line profiles of 13CO (black lines) extracted using a circular aperture with radius of 10 000 au to determine the +systemic velocities. The blue lines indicate the best-fit Gaussian when using the full velocity range, and the green lines where +the fit was done without including velocities close to the peak. The gray horizontal line at 0 Jy indicates the range of velocities +excluded from this second fit. The red lines show the best-fit when using two or three Gaussians. The vertical dashed line +indicates the systemic velocity of each FUor. In the cases of V883 Ori, V2775 Ori and V723 Car, i.e. FUors with known emission +from other sources in the same line of sight, we did not use additional Gaussians to fit the additional components because they +can be easily separate from the single Gaussian fit. See Section 4.2 for details. +As a final step, we compared our estimates with +those from the literature. +The differences between +our estimates and those obtained from previous ob- +servations +are +0.04 km s−1 +for +L1551 +IRS +5 +(Wu +et al. 2009), 0.02 km s−1 for V2775 Ori (Zurlo et al. +2017), +0.04 km s−1 +for +GM +Cha +(Mottram +et +al. +2017), 0.20 km s−1 for V883 Ori (Ru´ız-Rodr´ıguez et al. +2017a), 0.06 km s−1 for V1647 Ori (Principe et al. +2018), +0.35 km s−1 for V582 Aur (´Abrah´am et al. +2018), 0.27 km s−1 for V900 Mon (Takami et al. 2019), +0.56 km s−1 for FU Ori North (P´erez et al. 2020, who +resolved the binary system with an angular resolution +of 0.05” using ALMA). In the cases of Haro 5a IRS, +AR 6A, BBW 76, OO Ser, and HBC 687 the esti- +mates by (K´osp´al et al. 2017a) are in agreement within +0.30 km s−1. +For V899 Mon, V960 Mon, Z CMa, +iPTF 15afq, and V723 Car these are the first estimates +of their systemic velocity. +Two of our measurements +deviate from those determined by interferometric obser- +vations of C18O: V346 Nor and Reipurth 50. +In the +case of the former, K´osp´al et al. (2017b) found the line +profile peaks at −3.55 km s−1, indicating a difference of +0.47 km s−1 from our estimate, and in the case of the +latter FUor, Ru´ız-Rodr´ıguez et al. (2017b) determined +a systemic velocity of 4.6 km s−1, a value 0.77 km s−1 dif- +ferent from ours. It is likely that the larger differences +are due to the interferometric observations resolving out + +Molecular outflows in FUors with APEX +7 +emission from the extended envelopes. The final values +for the systemic velocities are presented in Table 1. +4.3. Line Profiles +In order to examine the outflows using their line pro- +files, we must select apertures that cover the gas emis- +sion. We began by exploring the channel maps of the +two 12CO transitions and checking which channels and +which regions show emission above the 3σ contour level. +The channels with wide extended emission that showed +little variations from channel to channel were considered +as envelope emission. Then we inspected the blue- and +red-shifted channel maps for emission similar to what is +found in outflows, i.e. emission whose red-shifted chan- +nels is in the opposite side from the blue-shifted channels +with respect to the expected position of the star, and +emission that is generally more extended in the chan- +nels with velocities closer to the systemic velocity and +more compact towards higher velocities. Finally, we cre- +ated a polygon whose shape would cover this emission in +both transitions. For the targets where the CO emission +does not the morphology described above, the spectra +were extracted using a 10 000 au aperture centered on +the nominal position of the protostar. The only excep- +tion is L1551 IRS 5, where we used a circular aperture +with a radius of 8000 au, due to the proximity of this +source (see below) and the size of our CO map. +For +each target, we used the same aperture in the three CO +maps. The aperture used for each target can be seen +in their channel maps in Appendix A, and the spectral +line profiles integrated over these apertures for all three +observed CO lines are presented in Figure 3. +The line profiles and the channel maps show con- +tamination caused by faint extended emission in four +of the FUors: V582 Aur (at ∼−9 km s−1), V883 Ori +(at ∼5.5 km s−1), V2775 Ori (at ∼5.9 km s−1), and +V723 Car (∼−24 km s−1). The peaks in the V883 Ori +profiles were reported by White et al. (2019), and the +blueshifted broad feature in V582 Aur was discussed by +´Abrah´am et al. (2018). +HBC 687, BBW 76, FU Ori, and V883 Ori show +the narrowest lines in our sample. The line profiles of +V1647 Ori, V900 Mon, and Z CMa are slightly wider +and do not show obvious indications of wings caused by +high velocity outflows. +The remainder of the sources +exhibit much wider profiles with clear indication of line +wings and possible outflows, mainly in the 12CO (4–3) +and 12CO (3–2) transitions. The 12CO (4–3) line is the +strongest line for most sources, except for AR 6A and +V960 Mon where both transitions are equally strong. +Indeed, for most FUors, the ratio between line profiles, +(J=4–3)/(J=3–2), is <1.5 at the systemic velocity of +each object. +The two exceptions are FU Ori, where +the J=3–2 transition almost reaches 0 due to strong +self-absorption, and V582 Aur, where there is a ratio of +∼4.5. For the latter, this ratio suggests different excita- +tion conditions, which can be explained by the intense +radiation from two early B type stars within 30 pc of +V582 Aur that are exciting the region surrounding the +FUor (Kun et al. 2017). +Some of our line profiles are different from those pre- +sented by K´osp´al et al. (2017a). These discrepancies are +because of differences in the distance to the FUors and +in the shape of the apertures. An example of the for- +mer is BBW 76, for which they used a distance 660 pc +larger than ours (see below), thus their aperture covered +fewer pixels, causing a difference in the integrated flux +of a factor of ∼3. A similar scenario applies to AR 6A. +Concerning the different shapes of the apertures, K´osp´al +et al. (2017a) used a 10,000 au circular aperture for all +targets while we tailored the shape of our apertures. +Haro 5a IRS, Reipurth 50 and V346 Nor are examples +of this, where our apertures produced higher integrated +fluxes by a factor of ∼3. +Unsurprisingly, we find that the 13CO (3–2) transition +produces the faintest line in all targets. Its line profiles +are single-peaked for most FUors with the maximum +at velocities close to the systemic velocity (see below). +GM Cha, iPTF 15afq, and OO Ser are double-peaked +with slightly less emission at the systemic velocity, a +possible indication that 13CO (3–2) is optically thick at +the line center. +5. ANALYSIS +5.1. Distances +The estimation of gas masses is dependent on the dis- +tance to the target. For FU Ori, V899 Mon, AR 6A, +V900 Mon, V960 Mon, and BBW 76 we used photoge- +ometric distances from the Gaia Early Data Release 3 +(Bailer-Jones et al. 2021). In the case of the more em- +bedded objects (i.e. undetected by Gaia), L1551 IRS 5, +V883 Ori, V1647 Ori, and V2775 Ori, we used the dis- +tances estimated from the distance to their molecular +clouds and the positions of the FUors within them (Con- +nelley & Reipurth 2018, and references therein). +For +V582 Aur we used the distance estimated by Kun et al. +(2017) under the assumption the FUor is related to the +Aur OB1 association. We followed Tapia et al. (2015) +and used the mean distance to the Great Carina Nebula +(NGC 3372) for V723 Car. The distance to iPTF 15afq +was estimated by Park et al. (2022) after comparing dif- +ferent distance estimates based from kinematics, Gaia +parallax and the distance to the CMa OB1 association +to which this object belongs. For Reipurth 50, Z CMa, + +8 +Cruz-S´aenz de Miera et al. +0 +5 +10 +15 +0 +2500 +5000 +7500 +Flux [Jy] +L1551 IRS 5 +20 +15 +10 +5 +0 +100 +200 +300 +400 +V582 Aur +5 +10 +15 +20 +0 +5000 +10000 +Haro 5a IRS +12CO (J = 4 +3) +12CO (J = 3 +2) +13CO (J = 3 +2) +5 +0 +5 +10 +0 +1000 +2000 +3000 +V883 Ori +5 +0 +5 +10 +0 +1000 +2000 +3000 +4000 +Reipurth 50 +5 +10 +15 +20 +0 +500 +1000 +Flux [Jy] +FU Ori +5 +10 +15 +20 +0 +1000 +2000 +3000 +V1647 Ori +5 +0 +5 +10 +0 +1000 +2000 +V2775 Ori +0 +5 +10 +15 +0 +500 +1000 +1500 +2000 +V899 Mon +0 +5 +10 +15 +0 +200 +400 +600 +Ar 6a +5 +10 +15 +20 +0 +200 +400 +600 +Flux [Jy] +V900 Mon +20 +30 +0 +25 +50 +75 +V960 Mon +5 +10 +15 +20 +0 +2000 +4000 +6000 +8000 +Z CMa +5 +10 +15 +20 +0 +1000 +2000 +IPTF15AFQ +10 +15 +20 +25 +25 +0 +25 +50 +75 +BBW 76 +25 +20 +15 +10 +vLSR [km s +1] +0 +100 +200 +300 +Flux [Jy] +V723 Car +5 +0 +5 +10 +vLSR [km s +1] +0 +1000 +2000 +3000 +4000 +GM Cha +10 +5 +0 +5 +vLSR [km s +1] +0 +500 +1000 +1500 +2000 +V346 Nor +0 +5 +10 +15 +vLSR [km s +1] +0 +2000 +4000 +OO Ser +10 +15 +20 +25 +vLSR [km s +1] +0 +100 +200 +HBC 687 +Figure 3. CO line profiles of our targets observed with APEX. The vertical dotted line is the systemic velocity. The vertical +dashed lines are the range of velocities of the CO (3–2) outflows. The line profiles have been smoothed for presentation purposes. +GM Cha, V346 Nor, and OO Ser we used distances com- +piled from the literature (Audard et al. 2014, and ref- +erences therein). We compared our distances to those +used by K´osp´al et al. (2017a) and found that four FUors +have different distance estimates: Haro 5a IRS, AR 6A, +V900 Mon and BBW 76. The difference between our +and their estimates are −79 pc, 90 pc, 30 pc and −660 pc, +respectively. If we had used the same apertures and ve- +locity integration ranges as K´osp´al et al. (2017a), these +differences in distance would translate to a difference in +masses of factors of 0.69, 1.24, 1.06 and 0.37, respec- +tively. +5.2. Envelope masses +We used the 13CO (3–2) emission to calculate the +masses of the envelopes surrounding the FUors. +To +calculate the integrated fluxes, we used the line pro- +files defined in Section 4.3, and integrated the channels +that had emission above 3σ. We assumed local thermo- +dynamical equilibrium, used an excitation temperature +of 20 K, a 13CO/12CO abundance ratio of 69 (Wilson +1999) and a 12CO/H2 abundance ratio of 10−4 (Bolatto +et al. 2013). The velocity range used to calculate the line +fluxes, the resulting line fluxes and the envelope mass es- +timates are presented in Table 2. To test the impact of +our choice on gas temperature, we did the calculations +using 10 K or 50 K, and found our estimated envelope +masses would change by a factor of ∼0.94 or ∼2.55, re- +spectively. As we show later, some FUors have optically +thick emission at velocities close to the systemic veloc- +ity, therefore, the estimated masses are lower limits for +these sources. + +Molecular outflows in FUors with APEX +9 +Table 2. Envelope masses of the FUors based on 13CO (3–2). vmin and vmax indicate the velocity +range used to integrate and calculate the line fluxes. The emission or absorption of the 10 µm +silicate feature is indicated, when known, and the reference used for each target. +Name +vmin +vmax +Int. Flux +Menv +10 µm feature +Si reference +[km s−1] +[km s−1] +[Jy km s−1] +[M⊙] +L1551 IRS 5 +3.11 +9.03 +4920.70 +0.193 +Absorption +Quanz et al. (2007) +V582 Aur +−11.73 +−8.34 +43.35 +0.137 +Emission +K´osp´al et al. (2020a) +Haro 5a IRS +8.92 +13.21 +3714.50 +1.032 +Absorption +Postel et al. (2019) +V883 Ori +3.00 +4.80 +506.04 +0.138 +Absorption +Quanz et al. (2007) +Reipurth 50 +1.28 +7.61 +1567.74 +0.603 +Absorption +Quanz et al. (2007) +FU Ori +11.20 +12.73 +323.91 +0.095 +Emission +Quanz et al. (2007) +V1647 Ori +7.95 +11.79 +1903.05 +0.521 +Absorption +Quanz et al. (2007) +V2775 Ori +0.72 +4.53 +1363.13 +0.454 +Absorption +Kim et al. (2016) +V899 Mon +7.92 +11.27 +196.46 +0.220 +Emission +K´osp´al et al. (2020a) +AR 6a +0.24 +8.40 +899.39 +1.295 +Unknown +V900 Mon +12.26 +14.50 +58.73 +0.136 +Emission +K´osp´al et al. (2020a) +V960 Mon +21.82 +25.70 +80.29 +0.624 +Emission +K´osp´al et al. (2020a) +Z CMa +11.96 +15.73 +499.94 +1.201 +Absorption +Quanz et al. (2007) +iPTF 15afq +12.27 +14.63 +73.41 +0.113 +Unknown +BBW 76 +17.36 +17.98 +8.39 +0.016 +Emission +Quanz et al. (2007) +V723 Car +−21.70 +−16.13 +294.20 +3.341 +Absorption +K´osp´al et al. (2020a) +GM Cha +3.31 +6.39 +4099.17 +0.191 +Absorption +Manoj et al. (2011) +V346 Nor +−6.60 +−0.86 +443.84 +0.395 +Absorption +Quanz et al. (2007) +OO Ser +5.22 +12.38 +3071.60 +0.540 +Absorption +Quanz et al. (2007) +HBC 687 +16.71 +17.40 +34.74 +0.010 +Emission +Quanz et al. (2007) +5.3. Outflow detection +Here we explain the process we followed to determine +whether a FUor had an outflow detection. We inspected +all the sources in our sample, including the ones that +K´osp´al et al. (2017a) considered as not having an out- +flow. +As mentioned earlier, high-velocity wings in the line +profiles of 12CO are a common indicator of outflows, +and these are present in some of our FUors. However, +this feature by itself is not enough. Thus, we examined +the 12CO channel maps for each target (found in Ap- +pendix A) to verify the existence of the outflows via a +visual inspection of the different distribution of the gas +between the channels close to the systemic velocity and +outwards to higher velocities. +The envelope emission +dominates the velocities closest to the systemic, which +are approximately the same velocities covered by the +13CO emission (Table 2), so we focused on velocities +beyond these. If there emission is not detected at ve- +locities higher the ones overran by the envelope then we +consider the FUor as not having an outflow. As out- +flows originate from the protostars, it is expected that +at lower velocities (with respect to the systemic) the out- +flow is extended and its position is closer to the FUor. +So in the case there is emission in the maps beyond the +envelope velocities, we checked the separation of this +gas with respect the position of the FUor. Should the +emission be close to the FUor we consider them to be +an outflow, and in the case they are separated we do +not. When compared to the outflow detections of K´osp´al +et al. (2017a), our methodology resulted in almost the +same detections and non-detections with the only dif- +ference begin V900 Mon. In this case, the line profiles +do not show high velocity wings and thus resulted in a +non-detection for them. +After we detected an outflow, we determined the ve- +locity ranges at which it was present in the blueshifted +and redshifted sides. First, we found the velocity chan- +nel on which the envelope is not dominant and consid- +ered it as the “inner” velocity, vin, of the outflow. Next, +we located the velocity channel where a 3σ detection +was not found and considered it as the “outer” velocity, +vout. We then calculated the maximum velocity of each +lobe of the outflow with respect to the systemic velocity + +10 +Cruz-S´aenz de Miera et al. +as vmax = vout − vsys. The list of FUors with outflows +and these three velocities are presented in Table 3. In +Figure 3 we marked with vertical dotted lines the veloc- +ity ranges where outflows are detected. For two FUors, +V900 Mon and iPTF 15afq, the 12CO (4–3) emission at +velocities close to the systemic does not appear to be +dominated by the envelope, thus, we used the systemic +as vin for both lobes. +Finally, we used these velocities to produce blueshifted +and redshifted integrated emission maps of the J=3–2 +and J=4–3 transitions of 12CO, and their contour maps +are presented in Figure 4 and Figure 5. We used these +maps to estimate the position angle of the outflow, re- +ported in Table 3, and to estimate the extension of each +lobe (see below). +5.4. Outflow properties +One of the goals of this work is to compare the out- +flows emanating from eruptive young stars to those from +quiescent young stellar objects. We carried out our cal- +culations for the blueshifted and redshifted parts of the +spectra separately and here we describe how we carried +out these calculations. The results for the J=3–2 and +J=4–3 transitions are presented in Table 4 and Table 5, +respectively. +5.4.1. Mass, momenta, and energy +The outflow masses were calculated assuming the wind +emission is in local thermodynamical equilibrium with +an excitation temperature of 75 K (van Kempen et al. +2009; Yıldız et al. 2015) and assuming a CO abundance +of 10−4 with respect to H2 (Bolatto et al. 2013). We +calculated the mass (Mv) for each velocity channel (v) +for all pixels above 3σ. Afterwards, we calculated the +momentum and kinematic energy for each channel with +Pv = Mv ×v and Ev = 0.5Mv ×v2, respectively. Finally, +we integrated the three properties over the same velocity +range to obtain the total values (Mof, Pof, Eof) for the +blueshifted and redshifted lobes of the outflow. +5.4.2. Force and luminosity +The outflow force and luminosity are calculated as +Fof = Pof/τd and Lof = Eof/τd, respectively, where τd +is the dynamical time of the outflow. The dynamical +time is defined as τd = Rlobe/vmax , where Rlobe is the +projected extension of the outflow lobe and vmax is the +maximum velocity of the outflow. +For the projected extension of the outflows, we used +the integrated emission maps (Figure 4 and Figure 5) to +measure the separation between the position of the star +and the maximum length at which the outflow is above +3σ for each transition separately. However, the outflows +around some of our targets extend beyond the field of +view of our observations so our extension measurements +are only a lower limit. The lobes for which this is the +case are indicated with an asterisk in Table 3. We esti- +mated the maximum outflow velocity for each lobe in- +dependently by calculating the difference between the +systemic velocity and the minimum/maximum velocity +where there is blueshifted/redshifted emission. In both +Rlobe and vmax , the sensitivity and spatial resolution of +the observations directly affect their measured values. +5.4.3. Caveats +Due to the nature of our observations and methodol- +ogy, it is important to understand the limitations of our +estimated values. +First, the large uncertainties in the estimations of the +outflow masses due to the presence of envelopes. This +surrounding material dominates emission at velocities +close to the systemic velocity, thus we have calculated +the outflow properties using only channels where the +outflow is the predominant flux contributor. Therefore, +we are knowingly underestimating the outflow masses by +not integrating the emission at low-velocities to prevent +this contamination. However, it is likely that some of the +envelope emission is still included into our calculations. +Indeed, in a couple of cases (V883 Ori and V1647 Ori) +we are not able to separate their known outflows due to +their emission being at velocities comparable to those of +the surrounding cloud. +Second, the sensitivity of the observations put strong +constraints on the maximum velocities where outflows +are detected. For example, in the case of L1551 IRS 5, +Yıldız et al. (2015) reported a sum of maximum ve- +locities of 21.5 km s−1, while we obtained ∼13.1 km s−1 +(see Table 3). +Their JCMT observations had typical +noise values ≲0.1 K, while our APEX observations have +a noise value of ∼0.38 K. +Finally, the extension of the outflow can reach beyond +the field of view of our observations. An extreme exam- +ple is that L1551 IRS 5 whose outflow extends out to +∼20′ (e.g. Stojimirovi´c et al. 2006) and our field of view +is less than 1′ (see Figure 4 and Figure 5). +Therefore, both the outflow masses and their dynam- +ical ages should be considered as lower limits, while the +outflow forces and luminosities must be considered as +highly uncertain. +5.5. Optical depth correction + +Molecular outflows in FUors with APEX +11 +Table 3. Position angles, velocities, extensions and dynamical times of outflows +CO (3–2) +CO (4–3) +Target +Inc. +PA +Side +vin +vout +|vmax | +Rlobe +τd +vin +vout +|vmax | +Rlobe +τd +[◦] +[◦] +[ km +s ] +[ km +s ] +[ km +s ] +[103 au] +[103 yr] +[ km +s ] +[ km +s ] +[ km +s ] +[103 au] +[103 yr] +L1551 IRS 5 +70 +65 +Blue* +5.03 +−0.27 +6.73 +11.7 +8.2 +5.64 +0.93 +5.54 +11.7 +10.0 +Red* +7.57 +12.80 +6.33 +11.7 +8.7 +7.67 +12.24 +5.77 +11.7 +9.6 +Haro 5a IRS +50 +70 +Blue* +9.20 +5.50 +5.35 +27.2 +24.1 +9.53 +5.30 +5.55 +25.3 +21.6 +Red* +13.40 +16.63 +5.78 +26.6 +21.8 +12.80 +15.50 +4.65 +22.9 +23.3 +Reipurth 50 +70 +150 +Blue* +1.03 +−0.98 +4.81 +18.8 +18.5 +2.60 +1.01 +2.82 +9.2 +15.5 +Red* +7.91 +8.97 +5.14 +32.0 +29.5 +5.18 +8.01 +4.18 +23.6 +26.7 +V2775 Ori +10 +− +Blue +1.36 +−5.58 +8.66 +6.3 +3.4 +1.61 +−1.62 +4.70 +10.7 +10.8 +Red +4.57 +8.24 +5.16 +4.8 +4.4 +4.34 +7.01 +3.93 +7.1 +8.5 +V899 Mon +50 +60 +Blue* +8.60 +6.92 +2.62 +46.1 +83.3 +8.93 +7.99 +1.55 +24.4 +74.6 +Red* +10.85 +12.47 +2.93 +23.0 +37.2 +10.71 +11.66 +2.12 +18.4 +41.2 +V900 Mon† +30 +80 +Blue +12.02 +11.22 +2.53 +21.5 +40.4 +13.44 +12.49 +1.26 +21.5 +80.9 +Red +14.46 +14.73 +0.98 +25.9 +125.7 +13.44 +14.23 +0.48 +25.9 +256.2 +V960 Mon +10 +− +Blue +21.42 +17.32 +6.48 +14.5 +10.6 +22.12 +20.93 +2.87 +14.5 +23.9 +Red +27.14 +28.80 +5.00 +14.5 +13.7 +24.61 +27.04 +3.24 +14.5 +21.2 +Z CMa +30 +45 +Blue +12.13 +10.71 +3.15 +29.9 +45.0 +12.44 +11.25 +2.61 +21.6 +39.2 +Red +16.00 +19.34 +5.48 +47.0 +40.6 +15.56 +19.18 +5.32 +40.8 +36.3 +iPTF 15afq† +50 +135 +Blue* +11.16 +9.84 +4.20 +28.1 +31.7 +13.00 +11.36 +2.89 +28.1 +49.7 +Red* +17.97 +20.65 +6.61 +60.4 +43.3 +13.00 +16.22 +1.97 +60.4 +131.3 +GM Cha +70 +100 +Blue +3.48 +2.82 +2.02 +2.8 +6.5 +3.65 +3.00 +1.84 +1.3 +3.3 +Red +6.59 +10.00 +5.16 +6.8 +6.2 +6.08 +9.80 +4.96 +6.4 +6.1 +V346 Nor +30 +45 +Blue +−5.41 +−11.16 +8.18 +22.7 +13.1 +−4.58 +−9.59 +6.61 +18.0 +12.9 +Red* +−0.25 +5.94 +8.92 +39.6 +21.1 +−1.26 +4.38 +7.36 +30.0 +19.3 +Note—The inclination angles here are the values used for the inclination correction in Section 6.5, see text for details. The +position angles were estimated by hand using the CO (3–2) integrated emission maps. The asterisk (*) indicates the lobes +that extend beyond the field-of-view of our observations, thus their values of Rlobe and τd are lower limits. The † labels the +two FUors with tentative outflow detections. +We calculated the outflow parameters assuming the +12CO lines are optically thin, however, this isotopologue +is typically optically thick. One way to correct for this +optical depth issue is by using the 13CO emission, under +the assumption that that isotopologue is optically thin, +and use it to correct the fluxes of 12CO. In this section +we present our methodology to correct the emission of +the J=3–2 transition of the 12CO. +This correction was done following the procedure pre- +sented by Dunham et al. (2014). We assumed that both +CO isotopologues are in local thermodynamical equi- +librium at the same excitation temperature, and with +identical beam filling factors. Under these conditions, +the brightness temperature ratio between the two iso- +topologues is given by +Tmb,12 +Tmb,13 += 1 − e−τ12 +1 − e−τ13 , +(1) +where Tmb,12 and Tmb,13 are the brightness temperatures +of 12CO and 13CO, respectively, and τ12 and τ13 are their +respective opacities. Assuming that 13CO is optically +thin, Equation 1 can be re-written as +Tmb,12 +Tmb,13 += [12CO] +[13CO] +1 − e−τ12 +τ12 +, +(2) +where [12CO]/[13CO] is the abundance ratio, for which +we use a value of 69 (Wilson 1999). +We began the estimation of the correction factor +(1 − exp (−τ12))/τ12 by calculating Tmb,12/Tmb,13 for +each channel where both isotopologues were detected +above 6σ. In some low-velocity channels for a few FUors, +the 13CO appears to be optically thick, therefore, we +dropped these points from the fitting. + +12 +Cruz-S´aenz de Miera et al. +Table 4. Outflow properties from 12CO (3–2) observations assuming they are optically thin. +Target +Side +Mof +Pof +Eof +Fof +Lof +[M⊙] +[M⊙ km s−1] +[erg] +[M⊙ yr−1 km s−1] +[L⊙] +L1551 IRS 5 +Blue +6.3 × 10−3 +1.9 × 10−2 +6.8 × 1041 +2.4 × 10−6 +6.9 × 10−4 +Red +6.8 × 10−3 +1.7 × 10−2 +4.9 × 1041 +1.9 × 10−6 +4.7 × 10−4 +Haro 5a IRS +Blue +1.6 × 10−2 +3.6 × 10−2 +8.7 × 1041 +1.5 × 10−6 +3.0 × 10−4 +Red +8.6 × 10−3 +2.7 × 10−2 +8.6 × 1041 +1.2 × 10−6 +3.2 × 10−4 +Reipurth 50 +Blue +3.5 × 10−3 +1.1 × 10−2 +3.6 × 1041 +6.0 × 10−7 +1.6 × 10−4 +Red +7.2 × 10−3 +3.3 × 10−2 +1.5 × 1042 +1.1 × 10−6 +4.4 × 10−4 +V2775 Ori +Blue +4.6 × 10−2 +1.7 × 10−1 +7.4 × 1042 +4.9 × 10−5 +1.8 × 10−2 +Red +3.5 × 10−2 +9.5 × 10−2 +2.8 × 1042 +2.2 × 10−5 +5.3 × 10−3 +V899 Mon +Blue +1.3 × 10−2 +2.0 × 10−2 +3.2 × 1041 +2.4 × 10−7 +3.2 × 10−5 +Red +2.2 × 10−2 +3.8 × 10−2 +6.8 × 1041 +1.0 × 10−6 +1.5 × 10−4 +V900 Mon† +Blue +6.0 × 10−3 +1.2 × 10−2 +2.5 × 1041 +3.1 × 10−7 +5.3 × 10−5 +Red +1.3 × 10−2 +1.0 × 10−2 +8.0 × 1040 +7.9 × 10−8 +5.2 × 10−6 +V960 Mon +Blue +1.1 × 10−1 +4.2 × 10−1 +1.7 × 1043 +3.9 × 10−5 +1.4 × 10−2 +Red +5.9 × 10−2 +2.3 × 10−1 +8.7 × 1042 +1.6 × 10−5 +5.2 × 10−3 +Z CMa +Blue +1.9 × 10−2 +4.2 × 10−2 +9.3 × 1041 +9.4 × 10−7 +1.7 × 10−4 +Red +4.7 × 10−2 +1.5 × 10−1 +4.8 × 1042 +3.5 × 10−6 +9.7 × 10−4 +iPTF 15afq† +Blue +1.8 × 10−2 +5.8 × 10−2 +1.9 × 1042 +1.8 × 10−6 +5.0 × 10−4 +Red +3.8 × 10−2 +1.8 × 10−1 +8.9 × 1042 +4.2 × 10−6 +1.7 × 10−3 +GM Cha +Blue +7.4 × 10−4 +1.2 × 10−3 +1.8 × 1040 +1.8 × 10−7 +2.3 × 10−5 +Red +1.2 × 10−3 +3.3 × 10−3 +9.8 × 1040 +5.3 × 10−7 +1.3 × 10−4 +V346 Nor +Blue +1.7 × 10−2 +6.4 × 10−2 +2.7 × 1042 +4.8 × 10−6 +1.7 × 10−3 +Red +4.6 × 10−2 +2.0 × 10−1 +9.8 × 1042 +9.8 × 10−6 +3.9 × 10−3 +Note—The † labels the two FUors with tentative outflow detections. +We then fitted a parabola +Tmb,12 +Tmb,13 += A + B (v − vsys) + C (v − vsys)2, +(3) +which will allow us to correct for the velocity channels +where the 13CO emission was not detected. We fixed +B = 0 to keep the parabola symmetric with respect to +the systemic velocity and prevent over-correcting one +side of the outflow. +Finally, the correction factor se- +lected for each channel was the lower value between the +fitted parabola and the expected abundance ratio of 69. +The plots and the values of the fitted parabolas for each +target are presented in Appendix B. We note that in +the case of iPTF 15afq, due to the complex emission of +13CO, we only used blueshifted points to fit the parabola +(see Appendix B). +6. DISCUSSION +Here we present our discussion about the envelope +masses and their relationship with the FUor evolution- +ary scheme presented by Quanz et al. (2007). Then we +discuss the FUors for which we detected outflows, and +we comment on the sources for which an outflow was +not detected. Finally, we make a statistical comparison +between the properties of the outflows in our FUor sam- +ple and those from other works in the literature focused +on quiescent protostars. +6.1. Envelope masses +Quanz et al. (2007) targeted 14 FUor-type objects and +obtained mid-infrared spectra. They found that the sil- +icate feature at 10 µm could be present in either absorp- +tion or emission, and suggested that when the feature +is in absorption, it is an indication of higher content of +mass in the envelope surrounding the FUor and, thus, +an indication of the object being younger. In Figure 6 +we compare our estimations of envelope masses to the +emission/absorption of the silicate feature based on the +references for each object listed in Table 2. + +Molecular outflows in FUors with APEX +13 +Table 5. Outflow properties estimated from the 12CO (4–3) observations. +Target +Side +Mof +Pof +Eof +Fof +Lof +[M⊙] +[M⊙ km s−1] +[erg] +[M⊙ yr−1 km s−1] +[L⊙] +L1551 IRS 5 +Blue +3.8 × 10−3 +8.4 × 10−3 +2.3 × 1041 +8.4 × 10−7 +1.9 × 10−4 +Red +2.9 × 10−3 +6.9 × 10−3 +1.9 × 1041 +7.2 × 10−7 +1.6 × 10−4 +Haro 5a IRS +Blue +1.8 × 10−2 +3.3 × 10−2 +6.7 × 1041 +1.6 × 10−6 +2.6 × 10−4 +Red +1.3 × 10−2 +2.9 × 10−2 +6.9 × 1041 +1.2 × 10−6 +2.4 × 10−4 +Reipurth 50 +Blue +4.3 × 10−3 +7.3 × 10−3 +1.3 × 1041 +4.6 × 10−7 +6.8 × 10−5 +Red +1.9 × 10−2 +4.5 × 10−2 +1.2 × 1042 +1.7 × 10−6 +3.7 × 10−4 +V2775 Ori +Blue +9.9 × 10−3 +2.4 × 10−2 +6.4 × 1041 +2.2 × 10−6 +4.9 × 10−4 +Red +1.0 × 10−2 +2.3 × 10−2 +5.7 × 1041 +2.7 × 10−6 +5.5 × 10−4 +V899 Mon +Blue +3.2 × 10−3 +3.3 × 10−3 +3.7 × 1040 +4.5 × 10−8 +4.2 × 10−6 +Red +2.9 × 10−3 +4.5 × 10−3 +7.3 × 1040 +1.1 × 10−7 +1.4 × 10−5 +V900 Mon† +Blue +1.8 × 10−2 +1.3 × 10−2 +1.1 × 1041 +1.7 × 10−7 +1.2 × 10−5 +Red +1.3 × 10−2 +2.8 × 10−3 +8.1 × 1039 +1.1 × 10−8 +2.5 × 10−7 +V960 Mon +Blue +4.0 × 10−2 +8.2 × 10−2 +1.7 × 1042 +3.4 × 10−6 +6.0 × 10−4 +Red +1.7 × 10−1 +2.6 × 10−1 +4.5 × 1042 +1.2 × 10−5 +1.7 × 10−3 +Z CMa +Blue +1.6 × 10−2 +2.8 × 10−2 +4.9 × 1041 +7.2 × 10−7 +1.1 × 10−4 +Red +4.3 × 10−2 +1.1 × 10−1 +3.2 × 1042 +3.0 × 10−6 +7.2 × 10−4 +iPTF 15afq† +Blue +2.2 × 10−2 +3.6 × 10−2 +6.3 × 1041 +7.2 × 10−7 +1.1 × 10−4 +Red +2.6 × 10−2 +2.2 × 10−2 +2.7 × 1041 +1.7 × 10−7 +1.7 × 10−5 +GM Cha +Blue +3.8 × 10−4 +5.3 × 10−4 +7.6 × 1039 +1.6 × 10−7 +1.9 × 10−5 +Red +9.1 × 10−4 +2.1 × 10−3 +5.6 × 1040 +3.4 × 10−7 +7.5 × 10−5 +V346 Nor +Blue +1.2 × 10−2 +3.7 × 10−2 +1.3 × 1042 +2.8 × 10−6 +8.4 × 10−4 +Red +2.5 × 10−2 +8.5 × 10−2 +3.3 × 1042 +4.5 × 10−6 +1.4 × 10−3 +Note—The † labels the two FUors with tentative outflow detections. +We expanded on the work presented by K´osp´al et al. +(2017b), who analyzed the first half of the FUor sam- +ple, and we found that the FUors with the least massive +envelopes show the silicate feature in emission, while +those with more massive envelopes show it in absorption. +We found two exceptions to this trend: V899 Mon and +V960 Mon. For the latter, there could be two explana- +tions. As mentioned below, there are three young stellar +objects inside the beam of our observations, and thus, +we could be significantly overestimating the amount of +material in the line of sight to this FUor. Alternatively, +if we consider that the outflow is indeed driven by the +FUor then based on the Moment 0 maps, we found that +the direction of the outflow is aligned with the line of +sight, and, therefore, it could be that the outflow has al- +ready cleared the line of sight to the FUor, allowing the +detection of the silicate feature in emission while main- +taining a high envelope mass. It is harder to explain +the case of V899 Mon, as our observations indicate that +the direction of the outflow is perpendicular to the line +of sight. Under the assumption that the outflows are +perpendicular to the inclination of the disks, we tried to +verify the geometry of the systems using ALMA contin- +uum observations (K´osp´al et al. 2021). However, both +disks were barely resolved and thus the inclination of +uncertainties are large enough to allow the scenarios of +almost edge-on and almost face-on geometries. Observa- +tions with higher angular resolution and sensitivity are +needed to determine the geometry of these systems and +understand this discrepancy between envelope mass and +the silicate feature. +The transition between absorption and emission ap- +pears to occur between 0.1 and 0.2 M⊙. +Indeed, +V900 Mon, V582 Aur, and V883 Ori have compara- +ble envelope masses with only the latter FUor having +the silicate feature in absorption. Here, it is not clear if +the geometry of the system could explain this difference. +The inclination of V582 Aur is unknown because contin- +uum observations have not resolved the disk (´Abrah´am + +14 +Cruz-S´aenz de Miera et al. +40" +20" +0" +-20" +-40" +Dec +L1551 IRS 5 +Haro 5a IRS +Reipurth 50 +V2775 Ori +V899 Mon +V900 Mon +40" 20" +0" +-20" -40" +40" +20" +0" +-20" +-40" +RA +Dec +V960 Mon +40" 20" +0" +-20" -40" +RA +Z CMa +40" 20" +0" +-20" -40" +RA +IPTF15AFQ +40" 20" +0" +-20" -40" +RA +GM Cha +40" 20" +0" +-20" -40" +RA +V346 Nor +Figure 4. The red and blue contours show redshifted and blueshifted CO (J=3–2) emission integrated in the velocity ranges +indicated in Figure 3. The star symbols mark the stellar position as given in Table 1. The hatched circle in the bottom right +frame is the APEX beam size and the arrows indicate the orientation of the outflow. In the cases of V2775 Ori and V960 Mon, +the outflow appears to be expanding in the direction of the line of sight. For L1551 IRS 5, the blueshifted contours are 3, +13, 24, 35, 45, 56, 67 and 78σ for σ=0.38 K km s−1 while the redshifted contours are 3, 18, 33, 48, 63, 78, 93 and 109σ for +σ=0.33 K km s−1. For Haro 5a IRS, the blueshifted contours are 3, 7, 12, 16, 21, 25, 30 and 35σ for σ=0.61 K km s−1 while the +redshifted contours are 3, 6, 10, 14, 18, 22, 26 and 30σ for σ=0.48 K km s−1. For Reipurth 50, the blueshifted contours are 3, 5, 7, +9, 11, 13, 15 and 17σ for σ=0.48 K km s−1 while the redshifted contours are 3, 6, 10, 14, 17, 21, 25 and 29σ for σ=0.43 K km s−1. +For V2775 Ori, the blueshifted contours are 3, 12, 22, 31, 41, 50, 60 and 70σ for σ=0.73 K km s−1 while the redshifted contours +are 3, 7, 12, 17, 21, 26, 31 and 36σ for σ=0.68 K km s−1. For V899 Mon, the blueshifted contours are 3, 5, 7, 9, 11, 13, 15 and +17σ for σ=0.52 K km s−1 while the redshifted contours are 3, 4, 6, 8, 9, 11, 13 and 15σ for σ=0.43 K km s−1. For V900 Mon, the +blueshifted contours are 3, 4, 5, 6, 7, 8, 9 and 10σ for σ=0.62 K km s−1 while the redshifted contours are 3, 4, 5, 6, 7 and 8σ +for σ=0.47 K km s−1. For V960 Mon, the blueshifted contours are 3, 6, 9, 13, 16, 20, 23 and 27σ for σ=0.44 K km s−1 while the +redshifted contours are 3, 7, 11, 15, 19, 23, 27 and 32σ for σ=0.25 K km s−1. For Z CMa, the blueshifted contours are 3, 4, 6, +8, 9, 11, 13 and 15σ for σ=0.52 K km s−1 while the redshifted contours are 3, 8, 13, 19, 24, 30, 35 and 41σ for σ=0.39 K km s−1. +For iPTF 15afq, the blueshifted contours are 3, 5, 7, 9, 11, 13, 15 and 17σ for σ=0.44 K km s−1 while the redshifted contours +are 3, 6, 10, 13, 17, 20, 24 and 28σ for σ=0.32 K km s−1. For GM Cha, the blueshifted contours are 3, 5, 7, 9, 11, 13, 15 and +18σ for σ=0.43 K km s−1 while the redshifted contours are 3, 9, 16, 23, 29, 36, 43 and 50σ for σ=0.39 K km s−1. For V346 Nor, +the blueshifted contours are 3, 12, 22, 32, 41, 51, 61 and 71σ for σ=0.52 K km s−1 while the redshifted contours are 3, 11, 20, +29, 38, 47, 56 and 65σ for σ=0.56 K km s−1. +et al. 2018), and the latter two FUors have comparable +inclination angles (Cieza et al. 2018; K´osp´al et al. 2021). +We do not have 10 µm data for two sources: AR 6A +and iPTF 15afq. Based on its high envelope mass, we +could expect the silicate feature around AR 6A to be +in absorption. However, the peak of its CO emission is +off-center (Figure 1) so the direct line of sight to our +target could have less material and show the feature in +emission. In the case of iPTF 15afq, the peak of CO is +also slightly off-center however, its mass envelope falls +in the intermediate range of masses so we expect this to +depend on the geometry of the system. +This suggests that HBC 687 is the most evolved FUor +in our sample. The case for the least evolved FUor is less +clear as V723 Car is a massive young star and thus this +evolutionary trend might not apply to it, and Z CMa +is a binary with one of its stars being an intermediate- +mass star (Koresko et al. 1991). Therefore, we consider +Haro 5a IRS as youngest FUor in our sample as it is one +with the most massive envelope with the silicate feature +in absorption. +6.2. FUors with outflows +L1551 IRS 5 —This Class I protostar was among the +first detections of bipolar outflows from young stellar +objects (Snell et al. 1980). +Later observations recov- +ered the blueshifted and redshifted lobes of the bipolar +outflow in CO (2–1) (Moriarty-Schieven et al. 2006; Wu +et al. 2009) and CO (3–2) (Yıldız et al. 2015). Based on +our maps, the molecular outflows have the same geome- +try as seen in those previous works (e.g. Wu et al. 2009), +and the position angle of the outflow (∼45◦) is almost +perpendicular to the position angle of the circumstellar +disks in the system (∼160◦; Lim et al. 2016; Cruz-S´aenz +de Miera et al. 2019). Comparing our estimated outflow +properties to those calculated by Yıldız et al. (2015), we + +Molecular outflows in FUors with APEX +15 +40" +20" +0" +-20" +-40" +Dec +L1551 IRS 5 +Haro 5a IRS +Reipurth 50 +V2775 Ori +V899 Mon +40" 20" +0" -20"-40" +RA +V900 Mon +40" 20" +0" -20"-40" +40" +20" +0" +-20" +-40" +RA +Dec +V960 Mon +40" 20" +0" -20"-40" +RA +Z CMa +40" 20" +0" -20"-40" +RA +IPTF15AFQ +40" 20" +0" -20"-40" +RA +GM Cha +V346 Nor +Figure 5. Similar to Figure 4 but for 12CO (4–3). In the case of V900 Mon, the emission at this transition is not significant +enough to be seen in this map and the emission detected is in the outer parts of the field of view. In the iPTF 15afq map, the +emission of the outflow is weak compared to the surrounding gas. For L1551 IRS 5, the blueshifted contours are 4, 7, 11, 15, +19, 23, 27 and 31σ for σ=0.96 K km s−1 while the redshifted contours are 4, 8, 12, 17, 21, 26, 30 and 35σ for σ=0.84 K km s−1. +For Haro 5a IRS, the blueshifted contours are 4, 5, 7, 8, 10, 11, 13 and 15σ for σ=2.09 K km s−1 while the redshifted contours +are 4, 5, 6, 8, 9, 11, 12 and 14σ for σ=1.65 K km s−1. For Reipurth 50, the blueshifted contours are 5, 6, 7, 8 and 9σ for +σ=1.22 K km s−1 while the redshifted contours are 5, 7, 10, 13, 16, 19, 22 and 25σ for σ=1.13 K km s−1. For V2775 Ori, the +blueshifted contours are 5, 8, 11, 15, 18, 22, 25 and 29σ for σ=1.72 K km s−1 while the redshifted contours are 5, 6, 7, 8, 9, 10, 11 +and 12σ for σ=1.62 K km s−1. For V899 Mon, the blueshifted contours are 3, 4, 5, 6, 7, 8 and 9σ for σ=1.50 K km s−1 while the +redshifted contours are 3, 4, 5, 6, 7 and 8σ for σ=1.22 K km s−1. For V900 Mon, the blueshifted contours are 4, 5, 6, 7 and 8σ +for σ=2.16 K km s−1 while the redshifted contours are 4, 5 and 6σ for σ=1.64 K km s−1. For V960 Mon, the blueshifted contours +are 4, 5, 6 and 7σ for σ=1.38 K km s−1 while the redshifted contours are 4, 6, 9, 12, 14, 17, 20 and 23σ for σ=0.82 K km s−1. +For Z CMa, the blueshifted contours are 4, 7, 10, 13, 16, 19, 22 and 26σ for σ=1.26 K km s−1 while the redshifted contours are +4, 7, 10, 13, 16, 19, 22 and 25σ for σ=0.94 K km s−1. For iPTF 15afq, the blueshifted contours are 4, 5, 6, 8, 9, 11, 12 and 14σ +for σ=1.15 K km s−1 while the redshifted contours are 4, 5, 7, 9, 10, 12, 14 and 16σ for σ=0.88 K km s−1. For GM Cha, the +blueshifted contours are 3, 4, 5, 6 and 7σ for σ=1.21 K km s−1 while the redshifted contours are 4, 6, 9, 12, 14, 17, 20 and 23σ +for σ=1.09 K km s−1. For V346 Nor, the blueshifted contours are 4, 8, 13, 18, 23, 28, 33 and 38σ for σ=1.66 K km s−1 while the +redshifted contours are 4, 8, 12, 16, 20, 24, 28 and 33σ for σ=1.77 K km s−1. +HBC 687 +BBW 76 +FU Ori +IPTF15AFQ +V900 Mon +V582 Aur +V883 Ori +GM Cha +L1551 IRS 5 +V899 Mon +V346 Nor +V2775 Ori +V1647 Ori +OO Ser +Reipurth 50 +V960 Mon +Haro 5a IRS +Z CMa +Ar 6a +V723 Car +10 +2 +10 +1 +100 +Envelope mass [M +] +Emission +Absorption +Unknown +Figure 6. +Comparison between envelope masses and the +emission/absorption of the silicate feature. +find that our mass estimate is in agreement with their +result, while our force and luminosity are a factor of ∼6 +lower than theirs, even when taking into account the +inclination correction factor the authors applied. How- +ever, this difference can be due to the higher sensitivity +(their vmax is higher for both lobes) and the larger field +of view of their observations. +Haro 5a IRS —This is a Class I protostar is located +in the Orion star forming region and it was identified +as a FUor-like object by Reipurth et al. (2012). Pre- +vious CO observations of source releaved its outflow +(Takahashi et al. 2006, 2008) with the same geometry +as what we detected, including the slight overlap be- +tween the redshifted and blueshifted emission. K´osp´al +et al. (2017a) presented the J=4–3 and J=3–2 12CO +and J=3–2 13CO observations of this FUor, and found +narrow outflow in an almost East-West direction. Our +analysis, based on the same observations as them, recov- +ered the same morphology. Their estimates for outflow + +16 +Cruz-S´aenz de Miera et al. +Table 6. Outflow properties from 12CO (3–2) observations after optical depth correction. +Target +Side +Mof +Pof +Eof +Fof +Lof +[M⊙] +[M⊙ km s−1] +[erg] +[M⊙ yr−1 km s−1] +[L⊙] +L1551 IRS 5 +Blue +2.2 × 10−2 +5.2 × 10−2 +1.4 × 1042 +6.4 × 10−6 +1.5 × 10−3 +Red +3.5 × 10−2 +6.6 × 10−2 +1.4 × 1042 +7.6 × 10−6 +1.4 × 10−3 +Haro 5a IRS +Blue +9.5 × 10−2 +2.0 × 10−1 +4.2 × 1042 +8.2 × 10−6 +1.5 × 10−3 +Red +2.8 × 10−2 +8.1 × 10−2 +2.4 × 1042 +3.7 × 10−6 +9.1 × 10−4 +Reipurth 50 +Blue +2.9 × 10−2 +9.2 × 10−2 +2.9 × 1042 +4.9 × 10−6 +1.3 × 10−3 +Red +3.4 × 10−2 +1.6 × 10−1 +7.2 × 1042 +5.4 × 10−6 +2.0 × 10−3 +V2775 Ori +Blue +9.1 × 10−2 +2.7 × 10−1 +9.6 × 1042 +7.8 × 10−5 +2.3 × 10−2 +Red +9.3 × 10−2 +2.1 × 10−1 +5.3 × 1042 +4.8 × 10−5 +9.9 × 10−3 +V899 Mon +Blue +6.8 × 10−2 +8.9 × 10−2 +1.3 × 1042 +1.1 × 10−6 +1.3 × 10−4 +Red +8.5 × 10−2 +1.4 × 10−1 +2.2 × 1042 +3.6 × 10−6 +4.9 × 10−4 +V900 Mon† +Blue +2.5 × 10−2 +5.1 × 10−2 +1.0 × 1042 +1.3 × 10−6 +2.1 × 10−4 +Red +2.4 × 10−1 +1.8 × 10−1 +1.4 × 1042 +1.4 × 10−6 +9.2 × 10−5 +V960 Mon +Blue +1.9 × 10−1 +6.5 × 10−1 +2.4 × 1043 +6.2 × 10−5 +1.9 × 10−2 +Red +8.2 × 10−2 +3.1 × 10−1 +1.2 × 1043 +2.3 × 10−5 +7.1 × 10−3 +Z CMa +Blue +8.1 × 10−2 +1.7 × 10−1 +3.5 × 1042 +3.8 × 10−6 +6.6 × 10−4 +Red +1.2 × 10−1 +3.2 × 10−1 +9.2 × 1042 +7.8 × 10−6 +1.9 × 10−3 +iPTF 15afq† +Blue +2.2 × 10−2 +7.1 × 10−2 +2.3 × 1042 +2.2 × 10−6 +6.0 × 10−4 +Red +3.8 × 10−2 +1.8 × 10−1 +8.9 × 1042 +4.2 × 10−6 +1.7 × 10−3 +GM Cha +Blue +1.3 × 10−3 +2.0 × 10−3 +3.1 × 1040 +3.1 × 10−7 +3.9 × 10−5 +Red +1.2 × 10−3 +3.4 × 10−3 +1.0 × 1041 +5.4 × 10−7 +1.3 × 10−4 +V346 Nor +Blue +1.9 × 10−2 +6.9 × 10−2 +2.8 × 1042 +5.2 × 10−6 +1.8 × 10−3 +Red +4.6 × 10−2 +2.0 × 10−1 +9.9 × 1042 +9.8 × 10−6 +3.9 × 10−3 +Note—The † labels the two FUors with tentative outflow detections. +masses are higher than ours by less than a factor of 2, +which can be explained by the difference in distances +and excitation temperatures, and by the difference in +the velocity ranges used in the calculation of the out- +flow properties. Tobin et al. (2020) and K´osp´al et al. +(2021) presented continuum observations at millimeter +wavelengths with data from ALMA and VLA, and both +reported that this FUor is also a proto-binary star. +Reipurth 50 —A Class I protostar also referred to as +HBC 494. Ru´ız-Rodr´ıguez et al. (2017b) presented high +angular resolution observations with ALMA in which +they traced the emission from the outflow in 12CO (J=2– +1) and the envelope emission with the same transition of +13CO and C18O. The extension of the J=2–1 outflow ob- +tained with ALMA is smaller than the size of our beam. +This can be explained by the maximum recoverable scale +of their ALMA configuration (11′′), which is compara- +ble to the extended emission seen in their channel maps +(e.g. the 6 km s−1 channel in their Figure 3), thus its is +likely they have resolved out most of the extended emis- +sion of the outflow. Indeed, most of the emission they +recovered with ALMA originates from the dense cavity +walls of the bipolar outflow. Nevertheless, the position +angle obtained from the high-resolution interferometric +observations (∼145◦) is comparable to our estimation of +the position angle (150◦). The outflow mass from the +ALMA observations, calculated assuming an excitation +temperature of 50 K, is a factor of 60 higher the one +we determined using the J=3–2 transition. If we adjust +for the higher temperature used in our calculations (see +Section 6.6), the mass estimated from ALMA measure- +ments is still a factor of 50 higher. This difference hints +that most of the mass of the outflow of Reipurth 50 is +located in the narrow cavity walls, which are severely +diluted by our single-dish beam. +V2775 Ori —The first detection of a molecular outflow +on this object was done in the J=2–1 transitions of +12CO, 13CO and C18O with ALMA (Zurlo et al. 2017). +The authors found the system is almost face-on with an +inclination angle of ∼14°. Our observations recovered + +Molecular outflows in FUors with APEX +17 +a similar orientation of the outflow (see Appendix A). +In addition, we found significant extended emission at +both the systemic velocity and at redshifted velocities +(+3 km s−1, see also Figure 3). Zurlo et al. (2017) re- +ported different velocity ranges for 12CO and C18O (see +their Table 2). The velocities of the 12CO match those +of the redshifted excess emission (peaking at ∼6 km s−1, +see Figure 3), and the velocities of C18O match those +of the systemic emission we report in Table 1. The red- +shifted cloud emission appears at velocities where the +outflow is still detected. Therefore, for all the analy- +ses of the outflow in this FUor, we removed the cloud’s +contamination. Similar to the case of Reipurth 50, the +difference in beam sizes and sensitivities complicate the +comparison between our estimated physical properties +and those from Zurlo et al. (2017). However, we found +that the masses from the J=3–2 transition are higher +by a factor of ∼8 than those estimated from the ALMA +observations, which is even more surprising due to the +lower excitation temperature used by Zurlo et al. (2017). +We suggest that contrary on the case of Reipurth 50, the +extended emission, which is likely resolved out by their +interferometric observations, contains more of the mass +of the outflow of V2775 Ori than the narrow cavity walls. +V899 Mon —This source with a Flat or Class II SED +was originally reported as a FUor by Wils et al. (2009). +Follow-up observations indicated that the source was +dimming, which was interpreted as a decrease in ac- +cretion rate by Ninan et al. (2015). The authors also +recovered P Cygni profile for several forbidden lines, in- +dicating the presence of outflows. Our observations are +the first to recover an indication of a bipolar molecular +outflow in CO, which follows a Northeast-Southwest di- +rection. However, the outflow position angle disagrees +with the position angle of the disk (K´osp´al et al. 2021) +and of the jets detected at optical wavelengths (Park +et al. 2021), thus the analysis of outflows with higher +angular resolution is needed to resolve this discrepancy. +Based on its channel maps (Appendix A) there is also +significant extended emission at low velocities due to the +envelope. +V960 Mon —Based on its pre-outburst SED, this is a +Class II object (K´osp´al et al. 2015) and our observations +are the first to study the gas surrounding the system. +The 12CO line profiles show high-velocity wings (Fig- +ure 3), which we interpreted as an indication of a bipolar +molecular outflow. The integrated emission maps (Fig- +ure 4 and Figure 5) and the channel maps (Appendix A) +show the two outflow lobes overlapping, an indication of +the outflow having a direction along the line-of-sight. +High-angular resolution ALMA observations barely re- +solved the FUor disk, and indicate a disk inclination +between 16° and 60°, depending on the method used +(K´osp´al et al. 2021). Therefore, for the rest of the anal- +ysis, we assumed the lower inclination angle for the out- +flow. +K´osp´al et al. (2015) detected two sources close +to this FUor (one to the North and one to the South- +east), and K´osp´al et al. (2021) found a third one to the +East. As these sources are located within our beams, +our observations contain emission from these neighbor- +ing sources, and it is possible that a source other than +the FUor drives the outflow. Therefore, our results for +the outflow around this FUor must be taken with cau- +tion as an analysis of higher angular resolution observa- +tions is needed. +Z CMa —This source is a binary composed of the FUor +and a Herbig Be star with a separation of 0.′′1. Levreault +(1988) did not detect an outflow in the J=1–0 transi- +tion of 13CO, and in the J=2–1 and J=1–0 transitions +of 12CO. Evans et al. (1994) and Liljestr¨om & Olofsson +(1997) detected the bipolar outflow emanating from this +FUor in the J=3–2 and J=1–0 transitions of CO, respec- +tively. +Our observations recovered emission from the +outflow with a Northwest-Southeast orientation (similar +to that found in previous works), and we find the out- +flow is compact and has low velocities. Our estimations +of the outflow properties are in general agreement with +those of Evans et al. (1994). +It is unknown which of +the two binary components drives the outflow, however, +since both sources drive jets (Whelan et al. 2010), it is +possible that both sources drive outflows. +GM Cha —The outflows around this Class I/II object +had been previously reported using a single dish an- +tenna (e.g. Mottram et al. 2017) and ALMA (Hales et al. +2020). We recovered the East-West outflow orientation +found by these authors. +Comparing our results with +those of Mottram et al. (2017), we find the redshifted +lobe is more extended than the blueshifted side. Our es- +timation of mass for the blueshifted lobe is higher than +their estimations, which is explained by us integrating +lower velocity fluxes compared to them. The redshifted +mass and the other outflow properties are comparable +to those by Mottram et al. (2017). +V346 Nor —K´osp´al et al. (2017a) presented single dish +observations of the J=3–2 and J=4–3 transitions, and +K´osp´al et al. (2017b) presented ALMA Cycle 2 ob- +servations of the J=2–1 transition. Our analysis uses +the same observations as K´osp´al et al. (2017a) and the +properties of outflow have the same values within 10%. +The small differences are due to slight differences in the +methodology, such as different apertures and systemic +velocities. The orientation of the outflow is the same as + +18 +Cruz-S´aenz de Miera et al. +that obtained at high angular resolution (K´osp´al et al. +2017b). Based on the 12CO/13CO ratio used in the op- +tical depth correction, it appears even the rarer isotopo- +logue is optically thick at velocities close to the systemic. +6.3. FUors with tentative detections +Below we present the two FUors for which we can +only make a tentative detection of their outflows, and +thus we consider that these two sources require follow- +up observations. +V900 Mon —One of the most recently discovered FUors, +it is a Class I source bordering on Class II (Reipurth +et al. 2012). K´osp´al et al. (2017b) used the same data as +us and carried out a similar analysis as us, and did not +find outflow emission. +However, Takami et al. (2019) +presented high-angular resolution ALMA observations +of the J=2–1 transition of 12CO, 13CO and C18O, where +they identified a bipolar outflow where the redshifted +and blueshifted lobes are in the East and West direc- +tions, respectively. Following their results, we searched +for the velocity ranges that could be integrated in the +J=3–2 transition for which we could find emission that +following the one detected in the J=2–1 observations. +We found bipolar emission only in the J=3–2 transition +that follows a similar East-West alignment (see Figure 4) +using the velocity range indicated in Table 3, and thus +we considered this source to drive an outflow and es- +timated its properties. The J=4–3 transition does not +show significant emission (see Figure 5) which prompted +us to consider this as only a tentative detection. +iPTF 15afq —This Class I object is one of the latest dis- +covered FUors. It showed a ∼2.5 mag brightening in +2015 which lasted a few months (Miller et al. 2015), and +follow-up brightenings in 2018 and 2019 (Hillenbrand +2019). The 2019 outburst lasted until early 2021, and +was followed by another outburst which is ongoing as +of this writing4. Hillenbrand (2019) presented high res- +olution (R=37 000) spectra taken during outburst and +found that Hα and the Ca II triplet showed a P Cygni +profile, an indicator of high velocity winds. Our obser- +vations are the first sub-millimeter wavelength observa- +tions of this object. Its CO line profiles (Figure 3) show +high velocity line wings, in particular on the redshifted +side. Based on its J=3–2 integrated emission maps (Fig- +ure 4) and channels maps (Appendix A), there appears +to be an outflow whose blueshifted and redshifted lobes +are on the Southeast and Northwest directions, respec- +tively. The blueshifted component is broader and with +4 http://gsaweb.ast.cam.ac.uk/alerts/alert/Gaia19fct/ +lower velocities than its redshifted counterpart. How- +ever, we consider this FUor as only a tentative detection +because the emission is heavily dominated by the enve- +lope, and thus, it is hard to confirm that the morphology +seen in the J=3–2 transition as an outflow. +6.4. FUors without outflow detections +ALMA observations of the J=2–1 transition showed +outflow emission for two FUors: +V883 Ori (Ru´ız- +Rodr´ıguez et al. 2017a) and V1647 Ori (Principe et al. +2018). There could be multiple causes behind our lack +of detection: the combination of the higher sensitivity +and angular resolution in the ALMA observations, the +possible low temperatures in the system, and the low +velocities of the ALMA outflows. Indeed, in the case of +V883 Ori, White et al. (2019) found that the emission +of 13CO J=3–2 was a combination of the outflow at low +velocities and of a spherical-like envelope. In addition, +when considering interferometric observations, it is pos- +sible that they have resolved-out the contribution from +the envelope, which our single-dish observations did not, +therefore, the envelope emission at dominates in the low- +velocity channels of our observations. +A similar case +was found for FU Ori, for which previous observations +reported it did not drive an outflow (Levreault 1988) +but ALMA observations of J=2–1 hint towards an out- +flow, thus making its detection uncertain (P´erez et al. +2020). We detect emission in the Northeast-Southwest +direction, which is perpendicular to the position angles +of the resolved disks (P´erez et al. 2020); however, the +angular resolution of our observations prevents us from +determining if it is a bipolar outflow so we do not con- +sider this as a detection. +For the remaining FUors (AR 6A, BBW 76, V582 Aur, +V723 Car, and OO Ser) we did not detect outflows, and +we did not find previous publications that reported out- +flows. +6.5. Outflow parameters +We detect clear outflow emission in ∼55% of the FUors +in our sample (10 out of the 18), which is lower than the +92% found in Class 0 and Class I objects (e.g. Mot- +tram et al. 2017). Even including the two cases where +ALMA detected outflows when we did not (V883 Ori +and V1647 Ori) and the possible outflow in FU Ori, +we would only find outflows in ∼73% the FUors of our +sample. However, this is not surprising considering that +some of the FUors in our sample are classified as Flat +spectrum or Class II objects, and the outflows in these +evolved stages might be harder to detect due to the lower +densities of the enveloping material (Arce & Sargent +2006). Indeed, only two FUors of Class II had evidence +of outflows: V960 Mon and GM Cha. + +Molecular outflows in FUors with APEX +19 +After the optical depth correction, the outflow masses +increased by a median factor of 3, with values ranging +between 1 (V346 Nor) and 14 (V900 Mon). The values of +the kinematic properties (e.g. momentum, energy) after +the optical depth correction are also a factor of a few +higher than without the correction. Even after applying +this correction, we can still expect the outflow properties +presented in Table 6 to be underestimated as explained +in Section 5.4.3. +In Figure 7 we show a comparison of the outflow +masses determined for the FUors that had outflow detec- +tion in both transitions of 12CO. For this comparison, +we estimated the outflow mass uncertainties by multi- +plying the number of pixels used when calculating the +mass by the rms of each data cube, and then converted +these fluxes to masses using the same assumptions as +the outflows. We find that, within these uncertainties, +most outflows have comparable masses in both transi- +tions. Reipurth 50 is the only FUor in which the mass +estimate is higher in the J=4–3 transition than in J=3– +2 by a factor of ∼2. +The outflows of V2775 Ori and +V899 Mon are more massive in the lower transition by +factors of 4 and 5, respectively, thus, this could be an +indication of different excitation properties causing the +lower transition to be stronger. +However, these com- +parisons are limited by differences in the observations +(i.e. angular resolution and sensitivities), in the images +(i.e. pixel size and field of view), and by using the same +excitation for the two transitions in all the FUors. A +large-scale program to target multiple CO transitions +under comparable conditions would alleviate these lim- +itations and provide more insight on the masses of the +outflows. +6.6. Comparison with quiescent young stellar objects +We put into context our outflow properties by compar- +ing them with the values of similar studies based on qui- +escent sources. This comparison is not straightforward +due to the differences in the observational properties +(i.e. angular resolution and sensitivity), and methodol- +ogy (i.e. choosing velocities for integration, optical depth +correction and inclination correction), which have signif- +icant effects on the resulting values of the outflow prop- +erties. It is expected that FUor outbursts last for up to +a hundred years and the dynamical ages of the outflows +are in the other of thousands of years (see Table 3), thus, +the outflows we have detected around FUors are not re- +lated to the current outbursts. This means that we are +comparing the histories of the two samples, which could +provide hints towards the nature behind the outbursts. +We compared our sample with the values of the out- +flow properties published in the following studies: Dun- +10 +3 +10 +2 +10 +1 +MJ = 3 +2 [M +] +10 +3 +10 +2 +10 +1 +MJ = 4 +3 [M +] +Figure 7. Comparison of outflow masses determined from +the J=3–2 and J=4–3 transitions. The dashed line indicates +a ratio of 1. +ham et al. (2014), Yıldız et al. (2015) and Mottram et al. +(2017). The three studies cover a combination of Class 0 +and Class I objects and our calculations followed similar +methods to theirs. We included quiescent Class 0 ob- +jects even when the vast majority of FUors are Class I +objects (see Table 1) because we want to compare how +the FUor outbursts compare to the different stages of +the star-formation process. We compared the outflow +masses and forces as those were the only two properties +presented by all three studies from the literature. +Dunham et al. (2014) presented outflow properties +with and without optical depth correction, while Yıldız +et al. (2015) and Mottram et al. (2017) did not calculate +this correction, thus we used the optically thin values for +this comparison. +Dunham et al. (2014) assumed Tex = 50 K for their +calculations, while Yıldız et al. (2015), Mottram et al. +(2017) used the same temperature we did in our anal- +ysis, Tex = 75 K. To test the effect of using the lower +temperature, we calculated the outflow properties of the +FUors using Tex = 50 K, and found the values were a fac- +tor of ∼1.19 higher when using the higher temperature. +Thus, we multiplied the outflow properties of Dunham +et al. (2014) by this factor to minimize the differences +in methodology. The outflow forces estimated by Yıldız +et al. (2015) and Mottram et al. (2017) were corrected +due to the inclination of the systems based on Downes & +Cabrit (2007). These correction values were estimated +for Class 0 objects and are not recommended to correct +Class I objects; however, for the sake of a comparison + +20 +Cruz-S´aenz de Miera et al. +between our sample and the ones from the literature, we +applied this correction factor to the FUor outflows even +if they are at later stages (Class I or II). We estimated +the inclination of the outflows by assuming the inclina- +tion of the outflow is perpendicular to the inclination +of the disk. For most sources, we used the inclination +of the FUor disks obtained from high-angular resolution +observations with ALMA (Cieza et al. 2018; Hales et al. +2020; K´osp´al et al. 2021), while for Z CMa we used the +estimate by Antoniucci et al. (2016) estimated from an +analysis with data from optical interferometry, and in +the case of iPTF 15afq, we assumed an inclination of +45◦. However, the works studying the quiescent sample +used a coarse correction table, e.g. Table A.5 in Mottram +et al. (2017). Thus, we rounded our inclinations to the +closest values in the inclination correcion table, and the +angles assumed are listed in Table 3. We combined the +quiescent samples from the literature into one, divided +it by Class, and compared the two sub-samples with the +FUors. +In Figure 8 we plotted the outflow forces calculated +from the J=3–2 transition against the envelope masses +calculated from the 13CO emission (panel a), against +the outflow masses also calculated from the 12CO J=3– +2 transition (panel b), against the ratio between the +outflow mass and envelope mass (panel c), and against +the bolometric luminosities obtained from the literature +(panel d; Table 1). +Panels a and b of Figure 8 show that FUors outflows +follow the same trends as the quiescent sources from +the literature, i.e. higher envelope masses and higher +outflow masses indicate higher outflow forces. +In the +outflow mass to envelope mass ratio subplot, panel c +of Figure 8, the FUor sample is offset from the quies- +cent samples. This ratio has been used to discuss the +core-to-star formation efficiency in the quiescent sam- +ple (Mottram et al. 2017), thus it hints that FUors are +less efficient at driving mass from the envelope onto the +star, and this relationship will be discussed further be- +low. The values for this ratio are presented in Figure 7. +The correlation between outflow force and bolomet- +ric luminosity, panel d of Figure 8, has been well stud- +ied for quiescent sources (Cabrit & Bertout 1992; Bon- +temps et al. 1996; Yıldız et al. 2015; Mottram et al. +2017), and it would appear that FUors do not follow +this correlation. However, the FUor bolometric lumi- +nosities were estimated from photometry taken while in +outburst, and none have sufficient pre-outburst photo- +metric data to estimate their pre-outburst luminosities. +Even if we do not have sufficient information about the +individual FUors, when in quiescence, the protostars are +. +Table 7. Ratio between outflow masses and +envelope masses using the J=3–2 transition +of the two observe CO isotopologues, and the +core-t-star formation efficiency, ϵ +Target +Moutflow/Menvelope +ϵ +L1551 IRS 5 +0.068 +−2.91 +Haro 5a IRS +0.024 +0.50 +Reipurth 50 +0.018 +0.69 +V2775 Ori +0.178 +−19.23 +V899 Mon +0.159 +0.05 +V900 Mon† +0.140 +0.44 +V960 Mon +0.271 +−8.89 +Z CMa +0.055 +0.39 +iPTF 15afq† +0.496 +−4.73 +GM Cha +0.010 +0.23 +V346 Nor +0.159 +−2.77 +Note—The † labels the two FUors with ten- +tative outflow detections. +expected to be low-mass and low-luminosity objects (ex- +cept for V723 Car), and our measured outflow parame- +ters are consistent with this. Thus, our results suggest +that FUors, when in quiescence, produce molecular out- +flows with forces comparable to those from outflows in +quiescent stars. +In order to get a better estimate of how similar +FUor outflows are with their quiescent counterparts, +we present cumulative histograms comparing different +properties (Figure 9), and we carried out three comple- +mentary statistical tests to examine whether the sam- +ples of the quiescent young stars were drawn from +the same sample as the FUors. +The first was a +two-sided Kolmogorov-Smirnov (K-S) test that com- +pares the shapes of the distributions, the second was a +Mann–Whitney U-test (MWU), which is more sensitive +to the mean of the two samples rather than the shape +of both distributions, and the third was a k-sample +Anderson-Darling test (kAD), which is more sensitive +to the tails of the distributions. The three tests were +done using the SciPy functions kstest, mannwhitneyu, +and anderson ksamp, respectively. The results of the +statistical tests are presented in Table 8. +With a significance level of 5% we found that the dis- +tribution of FUor envelope masses is similar to that of +Class Is and different from the Class 0s (Figure 9, panel +a), the outflow masses of FUors are different to those of + +Molecular outflows in FUors with APEX +21 +10 +1 +100 +101 +Envelope mass [M +] +10 +8 +10 +7 +10 +6 +10 +5 +10 +4 +10 +3 +10 +2 +Outflow force [M + km s +1 yr +1] +a) +Class 0 +Class I +FUors +10 +4 +10 +3 +10 +2 +10 +1 +Outflow mass [M +] +b) +10 +3 +10 +2 +10 +1 +Outflow mass / Envelope mass +c) +100 +101 +102 +Bolometric luminosity [L +] +d) +Figure 8. Comparison of the outflows from quiescent sources in the literature with the FUor outflows. Outflow forces (J=3–2) +plotted against envelope masses (panel a), outflow masses (J=3–2, panel b), the ratio between outflow mass and envelope mass +(panel c), and bolometric luminosities (panel d). In case of the FUors, all bolometric luminosities are during outburst. The +two FUors with tentative outflow detections are marked with empty diamond symbols. The Lbol of iPTF 15afq is unknown and +thus it is not shown in panel d. +1.0 +0.5 +0.0 +0.5 +1.0 +log10 Envelope mass +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Fraction +a) +Class 0 +Class I +FUors +4 +3 +2 +1 +log10 Outflow mass +b) +3.0 +2.5 +2.0 +1.5 +1.0 +0.5 +log10 Outflow mass/Envelope mass +c) +8 +6 +4 +2 +log10 Outflow force +d) +Figure 9. Cumulative histograms of outflow properties, and the outflow mass to envelope mass ratio, for the Class 0 and +Class I objects from the literature (Dunham et al. 2014; Yıldız et al. 2015; Mottram et al. 2017), and the FUors from this work. +The bin widths for each histogram were selected using the Freedman–Diaconis rule. +Table 8. p-values of the three statistical tests done for the envelope masses, outflow +masses and forces, and the ratio between outflow and envelope masses. +Envelope masses +Outflow masses +O/E mass ratio +Outflow forces +Test +Class 0 +Class I +Class 0 +Class I +Class 0 +Class I +Class 0 +Class I +K-S +0.011 +0.605 +0.220 +1×10−3 +2×10−3 +2×10−4 +0.070 +0.247 +MWU +2×10−3 +0.667 +0.129 +0.002 +5×10−4 +2×10−4 +0.272 +0.726 +kAD +<0.001 +>0.250 +0.200 +0.002 +<0.001 +<0.001 +0.097 +0.208 +The p-values from anderson ksamp are capped between 0.001 and 0.250, thus these +values are upper or lower limits, respectively. + +22 +Cruz-S´aenz de Miera et al. +Class I objects and comparable with those of Class 0s +(Figure 9, panel b), the outflow mass to envelope mass +ratio is different in the FUors when compared to either +sample (Figure 9, panel c), and the outflow forces of +FUors are comparable with the two quiescent samples +(Figure 9, panel d). Our tests did not include the two +tentative detections. We ran the tests including these +two FUors and found that our statistical results would +not change. +The envelope mass result is not surprising because, +based on their SEDs, the FUors are also Class Is. Fol- +lowing the same logic, the result for the outflow masses is +surprising as the outflow masses are close to the Class 0s. +This could be interpreted as an indication that FUors +are in the very early stages of their Class I stage and +their outflows have not had sufficient time to dissipate. +However, the masses of outflows are a combination of +the material that has passed through the accretion disk +and is now being driven away from the star by the out- +flow, and the material in the envelope that has been +entrained by the outflow. +FUor outflows have higher +outflow masses but similar envelope masses compared +to the Class Is, pointing towards FUor outflows having +a higher percentage of material that was ejected from +the accretion disk, i.e. material that was not accreted +onto the star. This can be seen in the ratio between +the outflow mass and the envelope mass in Figure 8 and +Figure 9. +The separation between the quiescent sample and the +FUors might be biased due to the distance of the tar- +gets. The YSOs of the quiescent samples are all withing +500 pc from the Sun, while half of the FUor sample is +beyond this distance. Therefore, if we consider that in +most cases the extension of the outflows is larger than +that of the envelopes, our analysis might be biased to- +wards the FUors as it is likely that for we are measur- +ing the full extension of their outflows in comparison to +the quiescent sample. In Figure 10 we plotted the out- +flow/envelope mass ratio versus the distance for each +target. The distribution of points indicates that indeed +there might be a positive correlation between the mass +ratio and the distance. +However, the maps in Yıldız +et al. (2015) and Mottram et al. (2017) indicate that +40% of their outflows extend beyond the areas of the +sky covered by their respective observations. Therefore, +the mass ratios for those sources are lower limits, and +thus it raises the question whether this relationship is +real or not. An in-depth observational program cover- +ing outflows at a wide range of distances should shed +some light on this matter. +Mottram et al. (2017) used the outflow/envelope mass +ratio to examine the core-to-star efficiency in a group of +102 +103 +Distance [pc] +10 +3 +10 +2 +10 +1 +O/E mass ratio +Class 0 +Class I +FUors +Figure 10. Ratio between outflow mass and envelope mass +plotted against the distance to each target. The open dia- +monds are the two FUors with tentative outflow detections. +quiescent young stars. They assumed that during the +whole duration of the Class 0 and I phases, a star has +an outflow rate with small enough variations that it can +be approximated by a constant value, and calculate the +core-to-star formation efficiency, ϵ, as follows: +ϵ = 1 − Mof +Menv +τ0+I +τd +, +(4) +where τ0+I is the total duration of the Class 0 and I +phases, i.e. 0.5 Myr. The authors used their typical val- +ues of Mof/Menv = 10−2 and τd = 104 years, and found a +core-to-star formation efficiency of 0.5, which is in agree- +ment with the literature. We used our derived masses +and dynamical times to calculate ϵ for the FUors. The +values can be found in Figure 7. +As can be seen from our results, five of the FUors have +negative ϵ values. These can be explained by the under- +estimation of the dynamical age because either the out- +flow appears to be face-on (V2775 Ori and V960 Mon), +or it extends beyond the field of view of our observa- +tions (L1551 IRS 5, iPTF 15afq, and V346 Nor). Three +of the six FUors with positive values extend beyond our +field of view (Haro 5a IRS, Reipurth 50, and V899 Mon) +and thus their ϵ values are uncertain because it is un- +known how much of the mass and the extension of the +outflow is beyond our field of view. The three remaining +FUors (V900 Mon, Z CMa, and GM Cha) with positive +ϵ and with the full outflow inside the field of view, have +lower efficiencies than the quiescent sample. These val- +ues would suggest that a significant amount of material + +Molecular outflows in FUors with APEX +23 +that was fed from the envelope onto the disk was not ac- +creted onto the star but instead was driven outwards by +the outflow. However, two of these have strong caveats. +For V900 Mon, its outflow was only tentatively detected +and follow-up observations might reveal a different ge- +ometry than ours, which would lead to a different value +of ϵ. GM Cha is a Class II object, i.e. in a more evolved +stage than most FUors, and the equation used to cal- +culate the efficiencies was created for Class I objects +whose accretion rate is orders of magnitude higher than +for Class IIs (Fiorellino et al. 2022) and have higher en- +velope masses, which means that our estimated value is +not accurate. +As a final point, we address the similar distributions +of the outflow forces between the two quiescent samples +and the FUors. The outflow force is a property that is +commonly associated to the accretion history of a young +stellar object (Bontemps et al. 1996). This is because, +as can be seen in panel a of Figure 8, there is a positive +correlation between the outflow force and the envelope +mass, and the more evolved young stars have lower en- +velope masses and lower mass accretion rates. Here we +present some scenarios that can explain the lack of sep- +aration between the FUors and quiescent sample. +First, if we assume that the similar distributions are +because the accretion histories of the two samples are +the same then this can be interpreted as either the “qui- +escent” sample had outbursts that were undetected, or +the current outbursts in the FUors are the first ones in +their accretion histories. +If these are indeed the first +outbursts in each of the FUors, then their effects would +be undetected because the angular resolution of our ob- +servations is insufficient to resolve the inner parts of the +outflows where the effects of a <100 year old outburst +could be detected. Few FUor outbursts (∼20 Connelley +& Reipurth 2018) have been detected so the incidence +rate of these events is unknown, and as such it is difficult +to separate between these two scenarios. +Second, if we assume that the accretion histories are +different between quiescent and outbursting samples +then the likeness between distributions should be be- +cause of a physical property of the outflows. The out- +flow force is calculated using the outflow momentum +and the dynamical age of the outflow, and both of these +properties depend on the distribution of velocities in the +outflow. If the FUor outflows have masses comparable +to Class 0 outflows and similar velocity ranges as the +outflows from the literature, we would expect the FUor +outflow forces to be close to those of the Class 0 ob- +jects from the other samples. Therefore, the divergence +between Class 0 outflow forces and FUor outflow forces +indicates that the latter have lower velocities. This had +already been mentioned by Principe et al. (2018) when +comparing the V1647 Ori outflow to those of others out- +flows observed with ALMA. +However, as mentioned earlier, the outflow forces pre- +sented here are highly uncertain because they are cal- +culated as the ratio of two lower limits, the outflow mo- +mentum and the outflow dynamical age, and because of +the understudied effects of the inclination of the outflow +with respect to the plane of the sky. As such, there is a +need for a thorough program to study these molecular +outflows. +7. SUMMARY & CONCLUSIONS +We presented APEX observations of 20 FUors or +FUor-like objects from which we estimated the enve- +lope mass and searched for outflows. +Using a combi- +nation of line profiles and inspection of channel maps, +we detected outflows in 45% of our sample. These in- +clude the possible first detections of molecular outflows +in V899 Mon and V960 Mon, although these should +be observed with higher angular resolution to corrob- +orate them. We also found two tentative detections in +V900 Mon and iPTF 15afq, that require follow-up obser- +vations to confirm. In the case of V883 Ori, V1647 Ori, +and possibly FU Ori, we did not detect the outflows that +have been observed by ALMA. +Based on our 13CO measurements, envelopes with +masses higher/lower than 0.1–0.2 M⊙ show the silicate +feature at 10 µm in absorption/emission. +If the enve- +lope mass is close to this threshold level, the geometry +of the system determines whether the spectral feature is +in emission or absorption. The most significant outlier of +this trend is V960 Mon, which shows the 10 µm feature +in emission despite having an envelope of ∼0.6 M⊙. +The masses of outflows estimated from the 12CO 3– +2 and 4–3 transitions are in agreement, except for two +FUors: V900 Mon and V960 Mon. +We suggest that +these two sources could be colder than the rest of the +sample and, thus, the higher transition is dimmer. +V960 Mon is an outlier in both trends, thus we pro- +posed another possible explanation. +This FUor has +three companion YSOs in its proximity, with separa- +tions smaller than the sizes of our beams. Therefore, we +suggest that these additional sources move this object +away from the trend seen in the rest of the sample. +The kinematic outflow properties (momenta, energies, +forces and luminosities) are higher when estimated from +the J=3–2 transition than those from J=4–3. We at- +tribute this to the higher sensitivity of the lower transi- +tion, which causes a difference in the range of velocities +in which we detected outflow emission. + +24 +Cruz-S´aenz de Miera et al. +After applying an optical depth correction to the J=3– +2 transition using the 13CO emission, we found that the +mass of the outflows increased by a median factor of 3 +and up to an order of magnitude. The minimum im- +provement, seen in a few cases, showed that the outflow +mass increased only by a few percent. +We compared the outflows found in our FUor sam- +ple with three works from the literature and found that +outflows emanating from FUors are more massive than +those from quiescent Class I sources but with masses +comparable to outflows in Class 0 sources. We found +that FUors have a higher outflow/envelope mass ratio +than the quiescent sample, although this result could be +biased by the distance. We calculated the core-to-star +efficiencies of the FUors and although our results are +severely constrained by the geometry of the outflows, +it could indicate that a significant portion of the ma- +terial that was deposited into the accretion disk from +the envelope is not accreted onto the star but instead +is driven back to the envelope by the outflow. Finally, +we found that outflow forces from the FUor sample are +comparable to the two quiescent sources, which can be +interpreted as similar accretion histories or as very low +velocities in the FUor outflows. +This study focused on the outflow histories of the +FUors observable from the APEX site. +The dynami- +cal ages of the detected outflows indicate that they are +much older than any of the ongoing outbursts, which are +less than 100 years old. Indeed, any outflow emission di- +rectly related to the current outburst would be detected +at high velocities, close to the protostar and would have +small spatial scales that would be diluted by the beam of +our single-dish observations. Our comparison between +the outflow properties of FUors and of other quiescent +objects should be taken with caution due to the varying +quality of the individual observations, and the method- +ology used by each research group. A complete survey +of all known FUors in both hemispheres with similar ob- +servational setups and sensitivities, and a control sam- +ple of multiple quiescent YSOs at different evolutionary +stages, would greatly improve our analysis. +This project has received funding from the European +Research Council (ERC) under the European Union’s +Horizon 2020 research and innovation programme un- +der grant agreement No 716155 (SACCRED). T.Cs. +has received financial support from the French State in +the framework of the IdEx Universit´e de Bordeaux In- +vestments for the future Program. 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The minimum and maximum velocities in +the channels maps are those when the gas emission starts or finished being significant. The purple contours are used +for all the 13CO channel maps and for the 12CO maps when outflows were not detected. The blue and red contours +show the blueshifted and redshifted emission of outflows, and the green contours show the envelope emission. All +channel maps show the aperture used for the calculation of the envelope mass in the case of 13CO, and the outflow +properties in the case of both 12CO transition. The velocities shown in the plots were chosen so that the maximum and +minimum velocities are shown within 27 frames, which can cause some irregular velocity steps in the plot. However, +these differences are of one channel, and we do not expect to see significant changes in the distribution of CO between +two continuous channels. Fig. Set A1. +Channel maps +B. OPTICAL DEPTH CORRECTION +In Table B1 we present the parameters of the best-fitted parabola used to determine the optical depth correction +for the sources with outflows, and in Figure B1 we present the parabolic fit and the line profiles used in the fitting. +Fig. Set B1. +Optical depth correction +Table B1. +Parameters of best +fitted parabolas used for optical +depth correction. +Target +A +C +L1551 IRS 5 +2.388 +2.683 +Haro 5a IRS +1.327 +2.194 +Reipurth 50 +2.548 +0.643 +V2775 Ori +1.825 +4.320 +V899 Mon +1.138 +6.189 +V900 Mon +1.531 +3.946 +V960 Mon +1.409 +3.357 +Z CMa +0.924 +3.350 +iPTF 15afq +-6.751 +6.224 +GM Cha +-4.909 +19.505 +V346 Nor +-2.419 +8.246 + +28 +Cruz-S´aenz de Miera et al. +4.36 km / s +4.53 km / s +4.67 km / s +4.81 km / s +13CO (3 +2) +4.94 km / s +5.12 km / s +5.26 km / s +5.39 km / s +5.53 km / s +5.70 km / s +5.84 km / s +5.98 km / s +6.12 km / s +6.26 km / s +6.43 km / s +6.57 km / s +6.71 km / s +6.85 km / s +60" +30" +0" +-30" +60" +30" +0" +-30" + RA +Dec +7.02 km / s +7.16 km / s +7.30 km / s +7.43 km / s +7.61 km / s +7.75 km / s +7.88 km / s +8.02 km / s +8.16 km / s +-0.27 km / s +0.23 km / s +0.73 km / s +1.22 km / s +CO (3 +2) +1.72 km / s +2.21 km / s +2.74 km / s +3.24 km / s +3.74 km / s +4.23 km / s +4.73 km / s +5.26 km / s +5.75 km / s +6.25 km / s +6.75 km / s +7.24 km / s +7.77 km / s +8.27 km / s +60" +30" +0" +-30" +60" +30" +0" +-30" + RA +Dec +8.76 km / s +9.26 km / s +9.76 km / s +10.28 km / s +10.78 km / s +11.28 km / s +11.77 km / s +12.27 km / s +12.80 km / s +0.93 km / s +1.33 km / s +1.77 km / s +2.22 km / s +CO (4 +3) +2.66 km / s +3.06 km / s +3.51 km / s +3.95 km / s +4.40 km / s +4.80 km / s +5.24 km / s +5.69 km / s +6.14 km / s +6.58 km / s +6.98 km / s +7.43 km / s +7.87 km / s +8.32 km / s +60" +30" +0" +-30" +60" +30" +0" +-30" + RA +Dec +8.72 km / s +9.16 km / s +9.61 km / s +10.06 km / s +10.45 km / s +10.90 km / s +11.35 km / s +11.79 km / s +12.24 km / s +Figure A1. L1551 IRS 5 channel maps. +13CO (3–2) with contours at 3, 5, 7, 9, 11, 13, 15, 17, 19 and 21σ with σ = 0.32 K. +CO (3–2) with contours at 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36 and 39σ with σ = 0.37 K. CO (4–3) with contours at 3, 5, 7, +9, 11, 13, 15, 17, 19 and 21σ with σ = 0.65 K. The black circles indicate the 8000 au aperture used to extract the line profiles +used to analyze the outflow. The complete figure set (20 images) is available in the online journal. + +Molecular outflows in FUors with APEX +29 +6 +4 +2 +0 +2 +4 +6 +v [km s +1] +102 +103 +104 +105 +TMB [K] +6 +4 +2 +0 +2 +4 +6 +v [km s +1] +0 +10 +20 +30 +T12 / T13 +Figure B1. Optical depth correction for L1551 IRS 5. In the left panel we show the line profiles, where the green, purple and +black colors represent the 13CO, the observed 12CO and the corrected 12CO, respectively, and the vertical dashed lines indicate +the rage of velocities used in the parabola fit. The right panel shows the ratio of main beam temperatures (TMB), where the +light blue crosses are all the values of the ratio for each velocity channel, in pink dots with errorbars are the points used in the +parabolic fitting, and the black line is the resulting best-fitted parabola. The complete figure set (10 images) is available in the +online journal. + diff --git a/TdE1T4oBgHgl3EQfuQXJ/content/tmp_files/load_file.txt b/TdE1T4oBgHgl3EQfuQXJ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7826b53f4d1a80f98869e534a7e6207a923f3ec5 --- /dev/null +++ b/TdE1T4oBgHgl3EQfuQXJ/content/tmp_files/load_file.txt @@ -0,0 +1,2696 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf,len=2695 +page_content='Draft version January 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2023 Typeset using LATEX twocolumn style in AASTeX631 An APEX study of molecular outflows in FUor-type stars Fernando Cruz-S´aenz de Miera,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2 ´Agnes K´osp´al,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 4 P´eter ´Abrah´am,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 4 Timea Csengeri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 Orsolya F´eher,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 7 Rolf G¨usten,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='8 and Thomas Henning3 1Konkoly Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Research Centre for Astronomy and Earth Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' E¨otv¨os Lor´and Research Network (ELKH),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Konkoly-Thege Mikl´os ´ut 15–17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 1121 Budapest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Hungary 2CSFK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' MTA Centre of Excellence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Konkoly Thege Mikl´os ´ut 15–17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 1121,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Budapest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Hungary 3Max Planck Institute for Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' K¨onigstuhl 17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 69117 Heidelberg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Germany 4ELTE E¨otv¨os Lor´and University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Institute of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' P´azm´any P´eter s´et´any 1/A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 1117 Budapest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Hungary 5Laboratoire d’astrophysique de Bordeaux,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Bordeaux, CNRS, B18N, all´ee Geoffroy Saint-Hilaire, 33615 Pessac, France 6School of Physics and Astronomy, Cardiff University, Queen’s Buildings, The Parade, Cardiff CF24 3AA, UK 7IRAM, 300 Rue de la piscine, 38406 Saint-Martin-d’H`eres, France 8Max Planck Institute for Radioastronomy, Auf dem H¨ugel 69, 53121 Bonn, Germany (Received January 10, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Revised January 10, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Accepted January 10, 2023) Submitted to ApJ ABSTRACT FU Orionis-type objects (FUors) are low-mass pre-main-sequence objects which go through a short- lived phase (∼100 years) of increased mass accretion rate (from 10−8 to 10−4 M⊙ yr−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' These eruptive young stars are in the early stages of stellar evolution and, thus, still deeply embedded in a massive envelope that feeds material to the circumstellar disk that is then accreted onto the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Some FUors drive molecular outflows, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' low-velocity wide-angle magneto-hydrodynamical winds, that inject energy and momentum back to the surrounding envelopes, and help clear the material surrounding the young star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Here we present a 12CO (3–2), 13CO (3–2) and 12CO (4–3) survey of 20 FUor-type eruptive young stars observed with APEX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We use our 13CO (3–2) observations to measure the masses of the envelopes surrounding each FUor and find an agreement with the FUor evolutionary trend found from the 10 µm silicate feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We find outflows in 11 FUors, calculate their masses and other kinematic properties, and compare these with those of outflows found around quiescent young stellar objects gathered from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' This comparison indicates that outflows in FUors are more massive than outflows in quiescent sources, and that FUor outflows have a higher ratio outflow mass with respect to the envelope than the quiescent sample, indicating that the eruptive young stars have lower star-forming efficiencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Finally, we found that the outflow forces in FUors are similar to those of quiescent young stellar objects, indicating that their accretion histories are similar or that the FUor outflows have lower velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' INTRODUCTION Jets and molecular outflows are a ubiquitous phe- nomenon in the process of star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The for- mer are highly collimated gas streams at high veloci- ties (≥100 km s−1), and the latter have wider opening angles and velocities between 1 km s−1 and 50 km s−1 in the case of low-mass stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Jets are detected with optical, near-infrared, radio molecular lines, and radio Corresponding author: Fernando Cruz-S´aenz de Miera cruzsaenz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='fernando-at-csfk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='org continuum, while the slower outflows are typically de- tected with molecular line tracers (Frank et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Bally 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Both types of mass ejection events are driven by accre- tion, thus the physical properties of the outflows depend on the accretion history of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Indeed, evidence has shown that Class 0 objects (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' younger protostars with higher mass accretion rates) have elevated outflow mass loss rates and higher outflow forces compared to more evolved Class I or Class II objects (Mottram et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The mass accretion rates from protostellar disks to pro- tostars are expected to undergo episodic variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' De- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='03387v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='SR] 9 Jan 2023 2 Cruz-S´aenz de Miera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' tailed analysis of jet knots (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Ellerbroek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Garufi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2019) and molecular out- flow shells (Plunkett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Nony et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Vazzano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2021) show how the study of outflows can shed light on the accretion history of the protostars that drive them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' FU Orionis-type objects (FUors) are examples of the episodic nature of accretion (Hartmann & Kenyon 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Audard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Fischer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' These eruptive young stars are low-mass protostars charac- terized by a sudden increase in their mass accretion rate, going from typical values of ∼10−8 M⊙ yr−1 up to ∼10−4 M⊙ yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' These events are typically detected as a 3 − 5 magnitude brightening at optical and near- infrared wavelengths, and are expected to last up to a century, meaning that these events increase the final stellar mass by a significant amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' FUor-type events generally occur in Class I objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Accretion outbursts have been detected in earlier stages, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' the Class 0 HOPS 383 (Safron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2015), and in later stages, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' the Class II Gaia20eae (Cruz-S´aenz de Miera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2022), however, these are not considered FUors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' This differentiation is because to classify an object as a FUor, the near-infrared spectrum of the protostar must also present the spectral signatures found in the prototyp- ical FUors (Connelley & Reipurth 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' In the case a protostar shows these signatures and the photomet- ric outburst was not detected, the source is considered FUor-like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' And if a Class I protostar shows an outburst and none, or a minimal amount, of the spectral signa- tures, then it is considered as Peculiar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Outflows play an important role in the star forma- tion process as they remove angular momentum from the accretion disk, inject mass and energy into their sur- roundings, and clear material from the envelope (Arce & Sargent 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The circumstellar envelopes are the re- mains of the parent molecular cloud core that surround the protostar, and their properties (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' , mass and ex- tension) are deeply connected with how evolved a young star is, with younger objects having more massive and larger envelopes than their evolved counterparts (Andre & Montmerle 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Therefore, if the elevated accre- tion rates during the outbursts can inject more momen- tum into the envelopes via outflows, then these episodic events must play an important role in the evolution of their protostellar system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Indeed, it is expected that af- ter an eruption, the inner circumstellar disk becomes de- pleted and will be replenished by the surrounding enve- lope (Vorobyov & Basu 2006) until the system can erupt again (Bell & Lin 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Takami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Eventu- ally, the repetitive outbursts will clear out the envelope and the young system will move to its next evolutionary phase, from Class I to Class II (Green et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Quanz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Green et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Previous observations of CO rotational transitions have shown the presence of outflows in some known FUors: V1057 Cyg (Rodriguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 1990), V1735 Cyg (Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 1994), L1551 IRS 5 (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2009, and references therein), V883 Ori (Ru´ız-Rodr´ıguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2017a), Reipurth 50 (Ru´ız-Rodr´ıguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2017b), FU Ori (Hales et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2015), V1647 Ori (Principe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2018), V2775 Ori (Zurlo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2017), V346 Nor (K´osp´al et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2017b) and V900 Mon (Takami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' In other cases, optical and near-infrared spectroscopy have shown indication of high-velocity jets: Z CMa (Poetzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 1989), V899 Mon (Ninan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2015), iPTF 15afq (Hillenbrand 2019), and V346 Nor (K´osp´al et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Each of the aforementioned studies focused on a sin- gle FUor-type object or on a few of them, preventing a statistical analysis of their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' In this paper we present a systematic study of the envelopes surrounding FUors, and we search for outflows among our full sam- ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We then compare our results with outflows found in YSOs that are currently quiescent and for which it is unknown whether they experienced an outburst or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' As outflows found at thousands of astronomical units are an indication of the accretion history of a protostar, this comparison allows us to examine how comparable is the histories of the FUors with those of the quiescent sam- ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The triggering mechanisms behind the FUor-type outbursts is still not understood, however, it is possi- ble that an examination of the differences between the two samples might hint that FUors are protostar with intrinsic differences that caused the outburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Alterna- tively, it could show the samples are similar and, thus, we cannot rule out that quiescent sources experienced FUor-type outbursts in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The structure of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The observed sample is briefly introduced in Section 2, while in Sec- tion 3 we describe the observations and the data reduc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' In Section 4 we present the distribution of the gas in the environment surrounding the FUors, and the properties of the integrated line profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The main goal of this paper is to study the properties of the circum- stellar gas, this analysis is found in Section 5, including the characterization of the molecular outflows where de- tected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' In Section 6 we compare the outflows found in the FUor sample with non-outbursting sources and draw our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Finally, in Section 7 we summarize our work and present our main findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' SAMPLE Our sample is composed of 20 eruptive young stars, including most of the known FUors accessible from the Molecular outflows in FUors with APEX 3 APEX site (Audard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Connelley & Reipurth 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We note that not all the targets in our list are considered FUors as some objects are cataloged as FUor-like objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' This subclassification is used when the photometric outburst was not detected but their near-infrared spectrum shows features similar to those of the prototypical FUors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' BBW 76, Connelley & Reipurth 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The target list also includes erup- tive young stars with peculiar accretion histories (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' V1647 Ori) and a massive star with a powerful accre- tion outburst (V723 Car).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The non-FUor objects were included because of their sudden increases of their mass accretion rate, and thus the properties of their outflows could be affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The first part of the sample, com- posed of eight targets, was analyzed by K´osp´al et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2017a), where they found outflows in three objects: HBC 494, Haro 5a IRS and V346 Nor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Here we will analyze the full sample, including a re-processing of the target list presented in K´osp´al et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2017a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The sam- ple presented in this paper includes ∼50% of the cur- rently known FUors and FUor-like objects (Connelley & Reipurth 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The full target list is presented in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' OBSERVATIONS AND DATA REDUCTION We carried out two programs with the FLASH+ re- ceiver (Klein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2014) at the APEX telescope (G¨usten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2006) to measure the 12CO (3–2), 13CO (3–2), and 12CO (4–3) lines towards our targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Program 094.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='F-9508 was observed between 2014 August 23–28 and program 098.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='F-9505 between 2016 August 25 and 2016 September 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Both programs used the same tech- nical setup and reduction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The lower frequency channel was tuned to 344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 GHz in USB to cover the 13CO (3–2) at 330.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='588 GHz, and the 12CO (3–2) at 345.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='796 GHz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The higher frequency chan- nel was tuned to the 12CO (4–3) line at 461.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='041 GHz in USB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We used the XFFTS backends providing a nomi- nal 38 kHz spectral resolution for the J = 3–2 lines and 76 kHz for the J = 4–3 line, these resulted in spectral resolutions of ∼34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 m s−1, ∼32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='9 m s−1 and ∼49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='4 m s−1 for the 13CO (3–2), CO (3–2) and CO (4–3) lines, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='. For each target, 120′′×120′′ on-the-fly (OTF) maps were obtained at 6 ′′ s−1, using a relative reference off position 1000′′ away in right ascension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We removed a first order baseline from the spectra, and calibrated the data using a main beam efficiency of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='73 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='60 at 352 GHz and 464 GHz, respectively, and the values were converted to Jansky using 41 Jy K−1 and 48 Jy K−1 at 352 GHz and 464 GHz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We calculated the noise levels of each CO line by first se- lecting the first and the last 100 channels of each cube (individually confirmed to be free of line emission), cal- culated the noise levels for each FUor using these chan- nels, and then we calculated the median noise level of all FUors to obtain representative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The rms noise levels, at the native spectral resolution mentioned ear- lier, are 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='6 Jy for 13CO (3–2), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='7 Jy for 12CO (3–2), and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='4 Jy for 12CO (4–3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The telescope’s half-power beam-width is 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='′′2, and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='′′3 at the corresponding fre- quencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' As mentioned earlier, the first half of the sur- vey has already been published by K´osp´al et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2017a), and here we use their calibrated data for our analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' RESULTS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Distribution of gas We constructed velocity integrated emission maps (Moment 0) for 12CO (3–2) using all channels in our data cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The resulting maps are presented in Fig- ure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Some young eruptive stars are still deeply embed- ded, therefore, it is possible that the observed CO emis- sion originates from the remaining material in their sur- rounding envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' In order to verify that our CO detec- tions come from the FUors, we compared our Moment 0 maps with the dust continuum emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We searched for 250 µm continuum maps taken with Herschel/SPIRE maps in the Herschel Science Archive1 and found data at an angular resolution of 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='6′′ for 18 sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For the two remaining sources (V900 Mon and Z CMa), we searched the Canadian Astronomy Data Centre2 for archival 850 µm observations taken with the James Clerk Maxwell Telescope3 (JCMT) with an angular resolution of 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' In the cases of L1551 IRS 5, Haro 5a IRS, V883 Ori, Reipurth 50, V899 Mon, V960 Mon, Z CMa, V346 Nor, GM Cha, and HBC 687, the peaks of both the dust emission and the gas emission are located at the po- sition of the protostar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For five of our targets (V582 Aur, AR 6A, iPTF 15afq, V723 Car, and OO Ser) the brighter peaks of both gas and dust emission are offset from the position of the protostar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The continuum peaks to these five sources are 27′′ to the Southeast, 12′′ to the East, 28′′ to the Northwest, 22′′ to the West, and 6′′ to the Southwest, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' In the case of OO Ser, there is continuum emission toward the position of the FUor, 1 http://archives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='esac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='int/hsa/whsa/ 2 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='cadc-ccda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='hia-iha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='nrc-cnrc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='gc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='ca/en/ 3 The James Clerk Maxwell Telescope has historically been oper- ated by the Joint Astronomy Centre on behalf of the Science and Technology Facilities Council of the United Kingdom, the National Research Council of Canada and the Netherlands Or- ganisation for Scientific Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 4 Cruz-S´aenz de Miera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Objects observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Name Coordinates Classa FUor classification vLSRb Distancec Lbold [km s−1] [pc] [L⊙] L1551 IRS 5 04:31:34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='07 +18:08:04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='9 I FUor-like 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='46 147 25 V582 Aur 05:25:51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='97 +34:52:30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 F Bona fide FUor −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='85 1320 146 Haro 5a IRS 05:35:26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='75 −05:03:55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1 I FUor-like 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='90 391 50 V883 Ori 05:38:18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='09 −07:02:25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='9 F Bona fide FUor 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='10 388 400 Reipurth 50e 05:40:27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='45 −07:27:30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 I Peculiar 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='76 460 300 FU Ori 05:45:22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='36 +09:04:12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 I/II Bona fide FUor 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='96 402 420 V1647 Ori 05:46:13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='13 −00:06:04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='8 I/II Peculiar 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='06 388 39 V2775 Ori 05:42:48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='48 −08:16:34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='7 I Bona fide Fuor 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='08 428 25 V899 Mon 06:09:19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='24 −06:41:55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='8 F/II Peculiar 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='57 785 419 AR 6Af 06:40:59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='30 +09:35:52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3 I Peculiar 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='02 890 450 V900 Mon 06:57:22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='22 −08:23:17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='6 I Bona fide FUor 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='77 1130 106 V960 Mon 06:59:31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='58 −04:05:27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='7 II Bona fide FUor 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='81 2068 Z CMa 07:03:43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='15 −11:33:06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 I FUor-like 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='91 1150 500 iPTF 15afqg 07:09:21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='39 −10:29:34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='4 I/F Peculiar 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='04 920 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 BBW 76e 07:50:35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='59 −33:06:23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='9 I FUor-like 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='64 1040 287 V723 Car 10:43:23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='44 −59:33:55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3 I Peculiar −19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='58 2500 4000 GM Cha 11:09:28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='55 −76:33:28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1 I/II Peculiar 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='86 160 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 V346 Nor 16:32:32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='19 −44:55:30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='7 0/I Peculiar −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='08 700 135 OO Ser 18:29:49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='13 +01:16:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='6 I Peculiar 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='36 311 31 HBC 687h 19:29:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='87 +09:38:42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='9 II FUor-like 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='98 400 10 aHere we refer as Class to the classification based on the shape of its SED (Lada 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Andre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Greene et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' b See Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' c See Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' dObtained from the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Audard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Connelley & Reipurth 2018, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' eReipurth 50 and BBW 76 are labeled as HBC 494 and Bran 76 in K´osp´al et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2017a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' fAlso known as V912 Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' gAlso known as Gaia19fct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' hAlso known as Parsamian 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' however the brightest peak is the one previously men- tioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We find that for FU Ori, V1647 Ori, V2775 Ori, and V900 Mon, the dust emission is located at the po- sition of the protostar, however, the peak of the gas emission is offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Finally, in the case of BBW 76, the dust emission peaks at the position of the source, how- ever, the CO map shows that the emission is weak and extended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Systemic velocity As it is shown below, to estimate the kinematic prop- erties of the outflows, we need a reliable estimate of the systemic velocity for each target so that we can mea- sure the velocity of the outflow relative to the proto- star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The systemic velocities of our targets were esti- mated by fitting a Gaussian function to the line profile of 13CO, extracted using a circular aperture with ra- dius of 10 000 au, and using the center of the best-fitting Gaussian as the systemic velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Due to the proximity of L1551 IRS 5 and the field-of-view of our observations, we had to use a smaller aperture of 8000 au for this FUor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Molecular outflows in FUors with APEX 5 50" 0" 50" Dec L1551 IRS 5 V582 Aur Haro 5a IRS V883 Ori Reipurth 50 50" 0" 50" Dec FU Ori V1647 Ori V2775 Ori V899 Mon Ar 6a 50" 0" 50" Dec V900 Mon V960 Mon Z CMa IPTF15AFQ BBW 76 50" 0" 50" 50" 0" 50" RA Dec V723 Car 50" 0" 50" RA GM Cha 50" 0" 50" RA V346 Nor 50" 0" 50" RA OO Ser 50" 0" 50" RA HBC 687 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Integrated intensity (Moment 0) maps of our targets for the 12CO (3–2) line observed with APEX (orange contours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The Moment 0 maps were generated by integrating the full spectral cube in order to produce an unbiased map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The purple contours are the 250 µm continuum emission from Herschel for most of our targets, the two exceptions are V900 Mon and Z CMa where we show contours of the 850 µm continuum emission from the JCMT (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The CO and the dust contours are plotted with levels at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='9 of the peak intensity, and are meant to be representative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The star symbols indicate the nominal positions of the protostars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The bars at the bottom of each panel represent 10 000 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The line profiles and the best-fit Gaussians are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' A number of sources have complicated line profiles that could not be fitted by a single Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Some of these show asymmetric dips around the peak of the emission (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' V1647 Ori, GM Cha, and OO Ser), which can be due to the self-absorption of the envelope or due to the rotation of the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The former scenario is more likely based on the inspection of the channel maps of 13CO, thus, for these sources, we discarded the velocity range of the dip and fitted the Gaussian function using the remaining velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The line profiles of other ob- jects show asymmetric shapes out to the wings of the line profiles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' V582 Aur, Reipurth 50, and V346 Nor), an indication of multiple components (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' envelope, out- flows, Keplerian disk, or unrelated gas in the same line- of-sight) showing emission at 13CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For these objects, we fitted a combination of two or three Gaussian func- tions to the line profile, and used the best-fit mean of the Gaussian with the highest amplitude to estimate the systemic velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' To verify our estimated systemic velocities, we searched for the velocity around which the line profile is most symmetrical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' As expected, we found that the sources with asymmetrical line profiles have the largest differences, but still less than 1 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For the symmet- rical sources, the differences are less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' In addition, we examined the channel maps of each target to confirm our systemic velocity estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 6 Cruz-S´aenz de Miera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 5 10 0 1000 2000 3000 Flux [Jy] L1551 IRS 5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 0 10 20 V582 Aur 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 0 2000 4000 Haro 5a IRS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 0 250 500 750 V883 Ori 0 5 0 500 1000 Reipurth 50 10 15 0 200 400 Flux [Jy] FU Ori 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 0 500 1000 1500 V1647 Ori 0 5 0 500 V2775 Ori 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 0 500 V899 Mon 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 0 100 200 Ar 6a 10 15 0 100 200 Flux [Jy] V900 Mon 20 25 0 20 40 V960 Mon 10 15 0 1000 2000 Z CMa 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 0 500 1000 1500 IPTF15AFQ 15 20 0 10 BBW 76 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 vLSR [km s 1] 0 50 100 Flux [Jy] V723 Car 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 vLSR [km s 1] 0 500 1000 1500 GM Cha 5 0 vLSR [km s 1] 0 200 400 600 V346 Nor 5 10 vLSR [km s 1] 0 500 1000 1500 OO Ser 15 20 vLSR [km s 1] 0 25 50 HBC 687 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Line profiles of 13CO (black lines) extracted using a circular aperture with radius of 10 000 au to determine the systemic velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The blue lines indicate the best-fit Gaussian when using the full velocity range, and the green lines where the fit was done without including velocities close to the peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The gray horizontal line at 0 Jy indicates the range of velocities excluded from this second fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The red lines show the best-fit when using two or three Gaussians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The vertical dashed line indicates the systemic velocity of each FUor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' In the cases of V883 Ori, V2775 Ori and V723 Car, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' FUors with known emission from other sources in the same line of sight, we did not use additional Gaussians to fit the additional components because they can be easily separate from the single Gaussian fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' See Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' As a final step, we compared our estimates with those from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The differences between our estimates and those obtained from previous ob- servations are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='04 km s−1 for L1551 IRS 5 (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2009), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='02 km s−1 for V2775 Ori (Zurlo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2017), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='04 km s−1 for GM Cha (Mottram et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2017), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='20 km s−1 for V883 Ori (Ru´ız-Rodr´ıguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2017a), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='06 km s−1 for V1647 Ori (Principe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2018), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='35 km s−1 for V582 Aur (´Abrah´am et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2018), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='27 km s−1 for V900 Mon (Takami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2019), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='56 km s−1 for FU Ori North (P´erez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2020, who resolved the binary system with an angular resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='05” using ALMA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' In the cases of Haro 5a IRS, AR 6A, BBW 76, OO Ser, and HBC 687 the esti- mates by (K´osp´al et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2017a) are in agreement within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='30 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For V899 Mon, V960 Mon, Z CMa, iPTF 15afq, and V723 Car these are the first estimates of their systemic velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Two of our measurements deviate from those determined by interferometric obser- vations of C18O: V346 Nor and Reipurth 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' In the case of the former, K´osp´al et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2017b) found the line profile peaks at −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='55 km s−1, indicating a difference of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='47 km s−1 from our estimate, and in the case of the latter FUor, Ru´ız-Rodr´ıguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2017b) determined a systemic velocity of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='6 km s−1, a value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='77 km s−1 dif- ferent from ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' It is likely that the larger differences are due to the interferometric observations resolving out Molecular outflows in FUors with APEX 7 emission from the extended envelopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The final values for the systemic velocities are presented in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Line Profiles In order to examine the outflows using their line pro- files, we must select apertures that cover the gas emis- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We began by exploring the channel maps of the two 12CO transitions and checking which channels and which regions show emission above the 3σ contour level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The channels with wide extended emission that showed little variations from channel to channel were considered as envelope emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Then we inspected the blue- and red-shifted channel maps for emission similar to what is found in outflows, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' emission whose red-shifted chan- nels is in the opposite side from the blue-shifted channels with respect to the expected position of the star, and emission that is generally more extended in the chan- nels with velocities closer to the systemic velocity and more compact towards higher velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Finally, we cre- ated a polygon whose shape would cover this emission in both transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For the targets where the CO emission does not the morphology described above, the spectra were extracted using a 10 000 au aperture centered on the nominal position of the protostar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The only excep- tion is L1551 IRS 5, where we used a circular aperture with a radius of 8000 au, due to the proximity of this source (see below) and the size of our CO map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For each target, we used the same aperture in the three CO maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The aperture used for each target can be seen in their channel maps in Appendix A, and the spectral line profiles integrated over these apertures for all three observed CO lines are presented in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The line profiles and the channel maps show con- tamination caused by faint extended emission in four of the FUors: V582 Aur (at ∼−9 km s−1), V883 Ori (at ∼5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 km s−1), V2775 Ori (at ∼5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='9 km s−1), and V723 Car (∼−24 km s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The peaks in the V883 Ori profiles were reported by White et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2019), and the blueshifted broad feature in V582 Aur was discussed by ´Abrah´am et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' HBC 687, BBW 76, FU Ori, and V883 Ori show the narrowest lines in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The line profiles of V1647 Ori, V900 Mon, and Z CMa are slightly wider and do not show obvious indications of wings caused by high velocity outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The remainder of the sources exhibit much wider profiles with clear indication of line wings and possible outflows, mainly in the 12CO (4–3) and 12CO (3–2) transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The 12CO (4–3) line is the strongest line for most sources, except for AR 6A and V960 Mon where both transitions are equally strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Indeed, for most FUors, the ratio between line profiles, (J=4–3)/(J=3–2), is <1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 at the systemic velocity of each object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The two exceptions are FU Ori, where the J=3–2 transition almost reaches 0 due to strong self-absorption, and V582 Aur, where there is a ratio of ∼4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For the latter, this ratio suggests different excita- tion conditions, which can be explained by the intense radiation from two early B type stars within 30 pc of V582 Aur that are exciting the region surrounding the FUor (Kun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Some of our line profiles are different from those pre- sented by K´osp´al et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2017a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' These discrepancies are because of differences in the distance to the FUors and in the shape of the apertures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' An example of the for- mer is BBW 76, for which they used a distance 660 pc larger than ours (see below), thus their aperture covered fewer pixels, causing a difference in the integrated flux of a factor of ∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' A similar scenario applies to AR 6A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Concerning the different shapes of the apertures, K´osp´al et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2017a) used a 10,000 au circular aperture for all targets while we tailored the shape of our apertures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Haro 5a IRS, Reipurth 50 and V346 Nor are examples of this, where our apertures produced higher integrated fluxes by a factor of ∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Unsurprisingly, we find that the 13CO (3–2) transition produces the faintest line in all targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Its line profiles are single-peaked for most FUors with the maximum at velocities close to the systemic velocity (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' GM Cha, iPTF 15afq, and OO Ser are double-peaked with slightly less emission at the systemic velocity, a possible indication that 13CO (3–2) is optically thick at the line center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' ANALYSIS 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Distances The estimation of gas masses is dependent on the dis- tance to the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For FU Ori, V899 Mon, AR 6A, V900 Mon, V960 Mon, and BBW 76 we used photoge- ometric distances from the Gaia Early Data Release 3 (Bailer-Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' In the case of the more em- bedded objects (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' undetected by Gaia), L1551 IRS 5, V883 Ori, V1647 Ori, and V2775 Ori, we used the dis- tances estimated from the distance to their molecular clouds and the positions of the FUors within them (Con- nelley & Reipurth 2018, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For V582 Aur we used the distance estimated by Kun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2017) under the assumption the FUor is related to the Aur OB1 association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We followed Tapia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2015) and used the mean distance to the Great Carina Nebula (NGC 3372) for V723 Car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The distance to iPTF 15afq was estimated by Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2022) after comparing dif- ferent distance estimates based from kinematics, Gaia parallax and the distance to the CMa OB1 association to which this object belongs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For Reipurth 50, Z CMa, 8 Cruz-S´aenz de Miera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='7500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='Flux [Jy] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='L1551 IRS 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='V582 Aur ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5000 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='GM Cha ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='vLSR [km s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='V346 Nor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='OO Ser ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='vLSR [km s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='HBC 687 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' CO line profiles of our targets observed with APEX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The vertical dotted line is the systemic velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The vertical dashed lines are the range of velocities of the CO (3–2) outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The line profiles have been smoothed for presentation purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' GM Cha, V346 Nor, and OO Ser we used distances com- piled from the literature (Audard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2014, and ref- erences therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We compared our distances to those used by K´osp´al et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2017a) and found that four FUors have different distance estimates: Haro 5a IRS, AR 6A, V900 Mon and BBW 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The difference between our and their estimates are −79 pc, 90 pc, 30 pc and −660 pc, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' If we had used the same apertures and ve- locity integration ranges as K´osp´al et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2017a), these differences in distance would translate to a difference in masses of factors of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='69, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='24, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='06 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='37, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Envelope masses We used the 13CO (3–2) emission to calculate the masses of the envelopes surrounding the FUors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' To calculate the integrated fluxes, we used the line pro- files defined in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3, and integrated the channels that had emission above 3σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We assumed local thermo- dynamical equilibrium, used an excitation temperature of 20 K, a 13CO/12CO abundance ratio of 69 (Wilson 1999) and a 12CO/H2 abundance ratio of 10−4 (Bolatto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The velocity range used to calculate the line fluxes, the resulting line fluxes and the envelope mass es- timates are presented in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' To test the impact of our choice on gas temperature, we did the calculations using 10 K or 50 K, and found our estimated envelope masses would change by a factor of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='94 or ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='55, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' As we show later, some FUors have optically thick emission at velocities close to the systemic veloc- ity, therefore, the estimated masses are lower limits for these sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Molecular outflows in FUors with APEX 9 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Envelope masses of the FUors based on 13CO (3–2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' vmin and vmax indicate the velocity range used to integrate and calculate the line fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The emission or absorption of the 10 µm silicate feature is indicated, when known, and the reference used for each target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Name vmin vmax Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Flux Menv 10 µm feature Si reference [km s−1] [km s−1] [Jy km s−1] [M⊙] L1551 IRS 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='11 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='03 4920.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='193 Absorption Quanz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2007) V582 Aur −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='73 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='34 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='137 Emission K´osp´al et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2020a) Haro 5a IRS 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='92 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='21 3714.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='032 Absorption Postel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2019) V883 Ori 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='00 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='80 506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='138 Absorption Quanz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2007) Reipurth 50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='28 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='61 1567.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='603 Absorption Quanz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2007) FU Ori 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='20 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='73 323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='095 Emission Quanz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2007) V1647 Ori 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='95 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='79 1903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='521 Absorption Quanz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2007) V2775 Ori 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='72 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='53 1363.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='454 Absorption Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2016) V899 Mon 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='92 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='27 196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='220 Emission K´osp´al et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2020a) AR 6a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='24 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='40 899.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='39 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='295 Unknown V900 Mon 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='26 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='50 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='136 Emission K´osp´al et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2020a) V960 Mon 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='82 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='70 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='624 Emission K´osp´al et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2020a) Z CMa 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='96 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='73 499.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='94 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='201 Absorption Quanz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2007) iPTF 15afq 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='27 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='63 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='113 Unknown BBW 76 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='36 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='98 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='016 Emission Quanz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2007) V723 Car −21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='70 −16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='13 294.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='341 Absorption K´osp´al et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2020a) GM Cha 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='31 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='39 4099.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='191 Absorption Manoj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2011) V346 Nor −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='60 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='86 443.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='395 Absorption Quanz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2007) OO Ser 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='22 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='38 3071.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='540 Absorption Quanz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2007) HBC 687 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='71 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='40 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='010 Emission Quanz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2007) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Outflow detection Here we explain the process we followed to determine whether a FUor had an outflow detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We inspected all the sources in our sample, including the ones that K´osp´al et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2017a) considered as not having an out- flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' As mentioned earlier, high-velocity wings in the line profiles of 12CO are a common indicator of outflows, and these are present in some of our FUors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' However, this feature by itself is not enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Thus, we examined the 12CO channel maps for each target (found in Ap- pendix A) to verify the existence of the outflows via a visual inspection of the different distribution of the gas between the channels close to the systemic velocity and outwards to higher velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The envelope emission dominates the velocities closest to the systemic, which are approximately the same velocities covered by the 13CO emission (Table 2), so we focused on velocities beyond these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' If there emission is not detected at ve- locities higher the ones overran by the envelope then we consider the FUor as not having an outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' As out- flows originate from the protostars, it is expected that at lower velocities (with respect to the systemic) the out- flow is extended and its position is closer to the FUor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' So in the case there is emission in the maps beyond the envelope velocities, we checked the separation of this gas with respect the position of the FUor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Should the emission be close to the FUor we consider them to be an outflow, and in the case they are separated we do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' When compared to the outflow detections of K´osp´al et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2017a), our methodology resulted in almost the same detections and non-detections with the only dif- ference begin V900 Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' In this case, the line profiles do not show high velocity wings and thus resulted in a non-detection for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' After we detected an outflow, we determined the ve- locity ranges at which it was present in the blueshifted and redshifted sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' First, we found the velocity chan- nel on which the envelope is not dominant and consid- ered it as the “inner” velocity, vin, of the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Next, we located the velocity channel where a 3σ detection was not found and considered it as the “outer” velocity, vout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We then calculated the maximum velocity of each lobe of the outflow with respect to the systemic velocity 10 Cruz-S´aenz de Miera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' as vmax = vout − vsys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The list of FUors with outflows and these three velocities are presented in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' In Figure 3 we marked with vertical dotted lines the veloc- ity ranges where outflows are detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For two FUors, V900 Mon and iPTF 15afq, the 12CO (4–3) emission at velocities close to the systemic does not appear to be dominated by the envelope, thus, we used the systemic as vin for both lobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Finally, we used these velocities to produce blueshifted and redshifted integrated emission maps of the J=3–2 and J=4–3 transitions of 12CO, and their contour maps are presented in Figure 4 and Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We used these maps to estimate the position angle of the outflow, re- ported in Table 3, and to estimate the extension of each lobe (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Outflow properties One of the goals of this work is to compare the out- flows emanating from eruptive young stars to those from quiescent young stellar objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We carried out our cal- culations for the blueshifted and redshifted parts of the spectra separately and here we describe how we carried out these calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The results for the J=3–2 and J=4–3 transitions are presented in Table 4 and Table 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Mass, momenta, and energy The outflow masses were calculated assuming the wind emission is in local thermodynamical equilibrium with an excitation temperature of 75 K (van Kempen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Yıldız et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2015) and assuming a CO abundance of 10−4 with respect to H2 (Bolatto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We calculated the mass (Mv) for each velocity channel (v) for all pixels above 3σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Afterwards, we calculated the momentum and kinematic energy for each channel with Pv = Mv ×v and Ev = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5Mv ×v2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Finally, we integrated the three properties over the same velocity range to obtain the total values (Mof, Pof, Eof) for the blueshifted and redshifted lobes of the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Force and luminosity The outflow force and luminosity are calculated as Fof = Pof/τd and Lof = Eof/τd, respectively, where τd is the dynamical time of the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The dynamical time is defined as τd = Rlobe/vmax , where Rlobe is the projected extension of the outflow lobe and vmax is the maximum velocity of the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For the projected extension of the outflows, we used the integrated emission maps (Figure 4 and Figure 5) to measure the separation between the position of the star and the maximum length at which the outflow is above 3σ for each transition separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' However, the outflows around some of our targets extend beyond the field of view of our observations so our extension measurements are only a lower limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The lobes for which this is the case are indicated with an asterisk in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We esti- mated the maximum outflow velocity for each lobe in- dependently by calculating the difference between the systemic velocity and the minimum/maximum velocity where there is blueshifted/redshifted emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' In both Rlobe and vmax , the sensitivity and spatial resolution of the observations directly affect their measured values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Caveats Due to the nature of our observations and methodol- ogy, it is important to understand the limitations of our estimated values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' First, the large uncertainties in the estimations of the outflow masses due to the presence of envelopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' This surrounding material dominates emission at velocities close to the systemic velocity, thus we have calculated the outflow properties using only channels where the outflow is the predominant flux contributor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Therefore, we are knowingly underestimating the outflow masses by not integrating the emission at low-velocities to prevent this contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' However, it is likely that some of the envelope emission is still included into our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Indeed, in a couple of cases (V883 Ori and V1647 Ori) we are not able to separate their known outflows due to their emission being at velocities comparable to those of the surrounding cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Second, the sensitivity of the observations put strong constraints on the maximum velocities where outflows are detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For example, in the case of L1551 IRS 5, Yıldız et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2015) reported a sum of maximum ve- locities of 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 km s−1, while we obtained ∼13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1 km s−1 (see Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Their JCMT observations had typical noise values ≲0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1 K, while our APEX observations have a noise value of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='38 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Finally, the extension of the outflow can reach beyond the field of view of our observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' An extreme exam- ple is that L1551 IRS 5 whose outflow extends out to ∼20′ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Stojimirovi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2006) and our field of view is less than 1′ (see Figure 4 and Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Therefore, both the outflow masses and their dynam- ical ages should be considered as lower limits, while the outflow forces and luminosities must be considered as highly uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Optical depth correction Molecular outflows in FUors with APEX 11 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Position angles, velocities, extensions and dynamical times of outflows CO (3–2) CO (4–3) Target Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' PA Side vin vout |vmax | Rlobe τd vin vout |vmax | Rlobe τd [◦] [◦] [ km s ] [ km s ] [ km s ] [103 au] [103 yr] [ km s ] [ km s ] [ km s ] [103 au] [103 yr] L1551 IRS 5 70 65 Blue* 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='27 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='73 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='7 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='93 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='54 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='7 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 Red* 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='57 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='80 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='33 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='7 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='67 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='24 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='77 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='7 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='6 Haro 5a IRS 50 70 Blue* 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='20 5.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='36 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3 Note—The inclination angles here are the values used for the inclination correction in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5, see text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The position angles were estimated by hand using the CO (3–2) integrated emission maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The asterisk (*) indicates the lobes that extend beyond the field-of-view of our observations, thus their values of Rlobe and τd are lower limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The † labels the two FUors with tentative outflow detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We calculated the outflow parameters assuming the 12CO lines are optically thin, however, this isotopologue is typically optically thick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' One way to correct for this optical depth issue is by using the 13CO emission, under the assumption that that isotopologue is optically thin, and use it to correct the fluxes of 12CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' In this section we present our methodology to correct the emission of the J=3–2 transition of the 12CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' This correction was done following the procedure pre- sented by Dunham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We assumed that both CO isotopologues are in local thermodynamical equi- librium at the same excitation temperature, and with identical beam filling factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Under these conditions, the brightness temperature ratio between the two iso- topologues is given by Tmb,12 Tmb,13 = 1 − e−τ12 1 − e−τ13 , (1) where Tmb,12 and Tmb,13 are the brightness temperatures of 12CO and 13CO, respectively, and τ12 and τ13 are their respective opacities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Assuming that 13CO is optically thin, Equation 1 can be re-written as Tmb,12 Tmb,13 = [12CO] [13CO] 1 − e−τ12 τ12 , (2) where [12CO]/[13CO] is the abundance ratio, for which we use a value of 69 (Wilson 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We began the estimation of the correction factor (1 − exp (−τ12))/τ12 by calculating Tmb,12/Tmb,13 for each channel where both isotopologues were detected above 6σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' In some low-velocity channels for a few FUors, the 13CO appears to be optically thick, therefore, we dropped these points from the fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 12 Cruz-S´aenz de Miera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Outflow properties from 12CO (3–2) observations assuming they are optically thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Target Side Mof Pof Eof Fof Lof [M⊙] [M⊙ km s−1] [erg] [M⊙ yr−1 km s−1] [L⊙] L1551 IRS 5 Blue 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='9 × 10−2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='8 × 1041 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='4 × 10−6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='9 × 10−4 Red 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='8 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='7 × 10−2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='9 × 1041 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='9 × 10−6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='7 × 10−4 Haro 5a IRS Blue 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='6 × 10−2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='6 × 10−2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='7 × 1041 1.' metadata={'source': 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1041 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='4 × 10−7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='7 × 10−4 Red 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='7 × 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 × 10−1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='8 × 1042 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 × 10−6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} 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1040 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='8 × 10−7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3 × 10−5 Red 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 × 10−3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3 × 10−3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='8 × 1040 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3 × 10−7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3 × 10−4 V346 Nor Blue 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='7 × 10−2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='4 × 10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='7 × 1042 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='8 × 10−6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='7 × 10−3 Red 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='6 × 10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 × 10−1 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='8 × 1042 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='8 × 10−6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='9 × 10−3 Note—The † labels the two FUors with tentative outflow detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We then fitted a parabola Tmb,12 Tmb,13 = A + B (v − vsys) + C (v − vsys)2, (3) which will allow us to correct for the velocity channels where the 13CO emission was not detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We fixed B = 0 to keep the parabola symmetric with respect to the systemic velocity and prevent over-correcting one side of the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Finally, the correction factor se- lected for each channel was the lower value between the fitted parabola and the expected abundance ratio of 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The plots and the values of the fitted parabolas for each target are presented in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We note that in the case of iPTF 15afq, due to the complex emission of 13CO, we only used blueshifted points to fit the parabola (see Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' DISCUSSION Here we present our discussion about the envelope masses and their relationship with the FUor evolution- ary scheme presented by Quanz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Then we discuss the FUors for which we detected outflows, and we comment on the sources for which an outflow was not detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Finally, we make a statistical comparison between the properties of the outflows in our FUor sam- ple and those from other works in the literature focused on quiescent protostars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Envelope masses Quanz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2007) targeted 14 FUor-type objects and obtained mid-infrared spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' They found that the sil- icate feature at 10 µm could be present in either absorp- tion or emission, and suggested that when the feature is in absorption, it is an indication of higher content of mass in the envelope surrounding the FUor and, thus, an indication of the object being younger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' In Figure 6 we compare our estimations of envelope masses to the emission/absorption of the silicate feature based on the references for each object listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Molecular outflows in FUors with APEX 13 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Outflow properties estimated from the 12CO (4–3) observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Target Side Mof Pof Eof Fof Lof [M⊙] [M⊙ km s−1] [erg] [M⊙ yr−1 km s−1] [L⊙] L1551 IRS 5 Blue 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='8 × 10−3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='4 × 10−3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3 × 1041 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='4 × 10−7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='9 × 10−4 Red 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='9 × 10−3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='9 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='9 × 1041 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 × 10−7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='6 × 10−4 Haro 5a IRS Blue 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='8 × 10−2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3 × 10−2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='7 × 1041 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='6 × 10−6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='6 × 10−4 Red 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3 × 10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='9 × 10−2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='9 × 1041 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 × 10−6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='4 × 10−4 Reipurth 50 Blue 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3 × 10−3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3 × 1041 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='6 × 10−7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='8 × 10−5 Red 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='9 × 10−2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 × 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 × 1042 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='7 × 10−6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='7 × 10−4 V2775 Ori Blue 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='9 × 10−3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='4 × 10−2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='4 × 1041 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 × 10−6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='9 × 10−4 Red 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 × 10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3 × 10−2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='7 × 1041 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='7 × 10−6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 × 10−4 V899 Mon Blue 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 × 10−3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3 × 10−3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='7 × 1040 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 × 10−8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 × 10−6 Red 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='9 × 10−3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 × 10−3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3 × 1040 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1 × 10−7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='4 × 10−5 V900 Mon† Blue 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='8 × 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3 × 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1 × 1041 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='7 × 10−7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 × 10−5 Red 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3 × 10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='8 × 10−3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1 × 1039 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1 × 10−8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 × 10−7 V960 Mon Blue 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 × 10−2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 × 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='7 × 1042 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='4 × 10−6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 × 10−4 Red 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='7 × 10−1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='6 × 10−1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 × 1042 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 × 10−5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='7 × 10−3 Z CMa Blue 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='6 × 10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='8 × 10−2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='9 × 1041 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 × 10−7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1 × 10−4 Red 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3 × 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1 × 10−1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 × 1042 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 × 10−6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 × 10−4 iPTF 15afq† Blue 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 × 10−2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='6 × 10−2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3 × 1041 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 × 10−7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1 × 10−4 Red 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='6 × 10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 × 10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='7 × 1041 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='7 × 10−7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='7 × 10−5 GM Cha Blue 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='8 × 10−4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3 × 10−4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='6 × 1039 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='6 × 10−7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='9 × 10−5 Red 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1 × 10−4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1 × 10−3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='6 × 1040 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='4 × 10−7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 × 10−5 V346 Nor Blue 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 × 10−2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='7 × 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3 × 1042 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='8 × 10−6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='4 × 10−4 Red 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 × 10−2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 × 10−2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3 × 1042 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 × 10−6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='4 × 10−3 Note—The † labels the two FUors with tentative outflow detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We expanded on the work presented by K´osp´al et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2017b), who analyzed the first half of the FUor sam- ple, and we found that the FUors with the least massive envelopes show the silicate feature in emission, while those with more massive envelopes show it in absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We found two exceptions to this trend: V899 Mon and V960 Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For the latter, there could be two explana- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' As mentioned below, there are three young stellar objects inside the beam of our observations, and thus, we could be significantly overestimating the amount of material in the line of sight to this FUor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Alternatively, if we consider that the outflow is indeed driven by the FUor then based on the Moment 0 maps, we found that the direction of the outflow is aligned with the line of sight, and, therefore, it could be that the outflow has al- ready cleared the line of sight to the FUor, allowing the detection of the silicate feature in emission while main- taining a high envelope mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' It is harder to explain the case of V899 Mon, as our observations indicate that the direction of the outflow is perpendicular to the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Under the assumption that the outflows are perpendicular to the inclination of the disks, we tried to verify the geometry of the systems using ALMA contin- uum observations (K´osp´al et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' However, both disks were barely resolved and thus the inclination of uncertainties are large enough to allow the scenarios of almost edge-on and almost face-on geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Observa- tions with higher angular resolution and sensitivity are needed to determine the geometry of these systems and understand this discrepancy between envelope mass and the silicate feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The transition between absorption and emission ap- pears to occur between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Indeed, V900 Mon, V582 Aur, and V883 Ori have compara- ble envelope masses with only the latter FUor having the silicate feature in absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Here, it is not clear if the geometry of the system could explain this difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The inclination of V582 Aur is unknown because contin- uum observations have not resolved the disk (´Abrah´am 14 Cruz-S´aenz de Miera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 40" 20" 0" 20" 40" Dec L1551 IRS 5 Haro 5a IRS Reipurth 50 V2775 Ori V899 Mon V900 Mon 40" 20" 0" 20" -40" 40" 20" 0" 20" 40" RA Dec V960 Mon 40" 20" 0" 20" -40" RA Z CMa 40" 20" 0" 20" -40" RA IPTF15AFQ 40" 20" 0" 20" -40" RA GM Cha 40" 20" 0" 20" -40" RA V346 Nor Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The red and blue contours show redshifted and blueshifted CO (J=3–2) emission integrated in the velocity ranges indicated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The star symbols mark the stellar position as given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The hatched circle in the bottom right frame is the APEX beam size and the arrows indicate the orientation of the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' In the cases of V2775 Ori and V960 Mon, the outflow appears to be expanding in the direction of the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For L1551 IRS 5, the blueshifted contours are 3, 13, 24, 35, 45, 56, 67 and 78σ for σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='38 K km s−1 while the redshifted contours are 3, 18, 33, 48, 63, 78, 93 and 109σ for σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='33 K km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For Haro 5a IRS, the blueshifted contours are 3, 7, 12, 16, 21, 25, 30 and 35σ for σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='61 K km s−1 while the redshifted contours are 3, 6, 10, 14, 18, 22, 26 and 30σ for σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='48 K km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For Reipurth 50, the blueshifted contours are 3, 5, 7, 9, 11, 13, 15 and 17σ for σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='48 K km s−1 while the redshifted contours are 3, 6, 10, 14, 17, 21, 25 and 29σ for σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='43 K km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For V2775 Ori, the blueshifted contours are 3, 12, 22, 31, 41, 50, 60 and 70σ for σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='73 K km s−1 while the redshifted contours are 3, 7, 12, 17, 21, 26, 31 and 36σ for σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='68 K km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For V899 Mon, the blueshifted contours are 3, 5, 7, 9, 11, 13, 15 and 17σ for σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='52 K km s−1 while the redshifted contours are 3, 4, 6, 8, 9, 11, 13 and 15σ for σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='43 K km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For V900 Mon, the blueshifted contours are 3, 4, 5, 6, 7, 8, 9 and 10σ for σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='62 K km s−1 while the redshifted contours are 3, 4, 5, 6, 7 and 8σ for σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='47 K km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For V960 Mon, the blueshifted contours are 3, 6, 9, 13, 16, 20, 23 and 27σ for σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='44 K km s−1 while the redshifted contours are 3, 7, 11, 15, 19, 23, 27 and 32σ for σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='25 K km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For Z CMa, the blueshifted contours are 3, 4, 6, 8, 9, 11, 13 and 15σ for σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='52 K km s−1 while the redshifted contours are 3, 8, 13, 19, 24, 30, 35 and 41σ for σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='39 K km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For iPTF 15afq, the blueshifted contours are 3, 5, 7, 9, 11, 13, 15 and 17σ for σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='44 K km s−1 while the redshifted contours are 3, 6, 10, 13, 17, 20, 24 and 28σ for σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='32 K km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For GM Cha, the blueshifted contours are 3, 5, 7, 9, 11, 13, 15 and 18σ for σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='43 K km s−1 while the redshifted contours are 3, 9, 16, 23, 29, 36, 43 and 50σ for σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='39 K km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For V346 Nor, the blueshifted contours are 3, 12, 22, 32, 41, 51, 61 and 71σ for σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='52 K km s−1 while the redshifted contours are 3, 11, 20, 29, 38, 47, 56 and 65σ for σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='56 K km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2018), and the latter two FUors have comparable inclination angles (Cieza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' K´osp´al et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We do not have 10 µm data for two sources: AR 6A and iPTF 15afq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Based on its high envelope mass, we could expect the silicate feature around AR 6A to be in absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' However, the peak of its CO emission is off-center (Figure 1) so the direct line of sight to our target could have less material and show the feature in emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' In the case of iPTF 15afq, the peak of CO is also slightly off-center however, its mass envelope falls in the intermediate range of masses so we expect this to depend on the geometry of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' This suggests that HBC 687 is the most evolved FUor in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The case for the least evolved FUor is less clear as V723 Car is a massive young star and thus this evolutionary trend might not apply to it, and Z CMa is a binary with one of its stars being an intermediate- mass star (Koresko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Therefore, we consider Haro 5a IRS as youngest FUor in our sample as it is one with the most massive envelope with the silicate feature in absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' FUors with outflows L1551 IRS 5 —This Class I protostar was among the first detections of bipolar outflows from young stellar objects (Snell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Later observations recov- ered the blueshifted and redshifted lobes of the bipolar outflow in CO (2–1) (Moriarty-Schieven et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2009) and CO (3–2) (Yıldız et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Based on our maps, the molecular outflows have the same geome- try as seen in those previous works (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2009), and the position angle of the outflow (∼45◦) is almost perpendicular to the position angle of the circumstellar disks in the system (∼160◦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Lim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Cruz-S´aenz de Miera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Comparing our estimated outflow properties to those calculated by Yıldız et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2015), we Molecular outflows in FUors with APEX 15 40" 20" 0" 20" 40" Dec L1551 IRS 5 Haro 5a IRS Reipurth 50 V2775 Ori V899 Mon 40" 20" 0" -20"-40" RA V900 Mon 40" 20" 0" -20"-40" 40" 20" 0" 20" 40" RA Dec V960 Mon 40" 20" 0" -20"-40" RA Z CMa 40" 20" 0" -20"-40" RA IPTF15AFQ 40" 20" 0" -20"-40" RA GM Cha V346 Nor Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Similar to Figure 4 but for 12CO (4–3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' In the case of V900 Mon, the emission at this transition is not significant enough to be seen in this map and the emission detected is in the outer parts of the field of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' In the iPTF 15afq map, the emission of the outflow is weak compared to the surrounding gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For L1551 IRS 5, the blueshifted contours are 4, 7, 11, 15, 19, 23, 27 and 31σ for σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='96 K km s−1 while the redshifted contours are 4, 8, 12, 17, 21, 26, 30 and 35σ for σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='84 K km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For Haro 5a IRS, the blueshifted contours are 4, 5, 7, 8, 10, 11, 13 and 15σ for σ=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='09 K km s−1 while the redshifted contours are 4, 5, 6, 8, 9, 11, 12 and 14σ for σ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='65 K km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For Reipurth 50, the blueshifted contours are 5, 6, 7, 8 and 9σ for σ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='22 K km s−1 while the redshifted contours are 5, 7, 10, 13, 16, 19, 22 and 25σ for σ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='13 K km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For V2775 Ori, the blueshifted contours are 5, 8, 11, 15, 18, 22, 25 and 29σ for σ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='72 K km s−1 while the redshifted contours are 5, 6, 7, 8, 9, 10, 11 and 12σ for σ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='62 K km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For V899 Mon, the blueshifted contours are 3, 4, 5, 6, 7, 8 and 9σ for σ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='50 K km s−1 while the redshifted contours are 3, 4, 5, 6, 7 and 8σ for σ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='22 K km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For V900 Mon, the blueshifted contours are 4, 5, 6, 7 and 8σ for σ=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='16 K km s−1 while the redshifted contours are 4, 5 and 6σ for σ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='64 K km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For V960 Mon, the blueshifted contours are 4, 5, 6 and 7σ for σ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='38 K km s−1 while the redshifted contours are 4, 6, 9, 12, 14, 17, 20 and 23σ for σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='82 K km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For Z CMa, the blueshifted contours are 4, 7, 10, 13, 16, 19, 22 and 26σ for σ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='26 K km s−1 while the redshifted contours are 4, 7, 10, 13, 16, 19, 22 and 25σ for σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='94 K km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For iPTF 15afq, the blueshifted contours are 4, 5, 6, 8, 9, 11, 12 and 14σ for σ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='15 K km s−1 while the redshifted contours are 4, 5, 7, 9, 10, 12, 14 and 16σ for σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='88 K km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For GM Cha, the blueshifted contours are 3, 4, 5, 6 and 7σ for σ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='21 K km s−1 while the redshifted contours are 4, 6, 9, 12, 14, 17, 20 and 23σ for σ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='09 K km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For V346 Nor, the blueshifted contours are 4, 8, 13, 18, 23, 28, 33 and 38σ for σ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='66 K km s−1 while the redshifted contours are 4, 8, 12, 16, 20, 24, 28 and 33σ for σ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='77 K km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' HBC 687 BBW 76 FU Ori IPTF15AFQ V900 Mon V582 Aur V883 Ori GM Cha L1551 IRS 5 V899 Mon V346 Nor V2775 Ori V1647 Ori OO Ser Reipurth 50 V960 Mon Haro 5a IRS Z CMa Ar 6a V723 Car 10 2 10 1 100 Envelope mass [M ] Emission Absorption Unknown Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Comparison between envelope masses and the emission/absorption of the silicate feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' find that our mass estimate is in agreement with their result, while our force and luminosity are a factor of ∼6 lower than theirs, even when taking into account the inclination correction factor the authors applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' How- ever, this difference can be due to the higher sensitivity (their vmax is higher for both lobes) and the larger field of view of their observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Haro 5a IRS —This is a Class I protostar is located in the Orion star forming region and it was identified as a FUor-like object by Reipurth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Pre- vious CO observations of source releaved its outflow (Takahashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2006, 2008) with the same geometry as what we detected, including the slight overlap be- tween the redshifted and blueshifted emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' K´osp´al et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2017a) presented the J=4–3 and J=3–2 12CO and J=3–2 13CO observations of this FUor, and found narrow outflow in an almost East-West direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Our analysis, based on the same observations as them, recov- ered the same morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Their estimates for outflow 16 Cruz-S´aenz de Miera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Outflow properties from 12CO (3–2) observations after optical depth correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Target Side Mof Pof Eof Fof Lof [M⊙] [M⊙ km s−1] [erg] [M⊙ yr−1 km s−1] [L⊙] L1551 IRS 5 Blue 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 × 10−2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 × 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='4 × 1042 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='4 × 10−6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 × 10−3 Red 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 × 10−2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='6 × 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='4 × 1042 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='6 × 10−6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='4 × 10−3 Haro 5a IRS Blue 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 × 10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 × 10−1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 × 1042 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 × 10−6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 × 10−3 Red 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='8 × 10−2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1 × 10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='4 × 1042 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='7 × 10−6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1 × 10−4 Reipurth 50 Blue 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='9 × 10−2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 × 10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='9 × 1042 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='9 × 10−6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3 × 10−3 Red 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='4 × 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='6 × 10−1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 × 1042 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='4 × 10−6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 × 10−3 V2775 Ori Blue 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1 × 10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='7 × 10−1 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='6 × 1042 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='8 × 10−5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3 × 10−2 Red 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3 × 10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1 × 10−1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3 × 1042 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='8 × 10−5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='9 × 10−3 V899 Mon Blue 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='8 × 10−2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='9 × 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3 × 1042 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1 × 10−6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3 × 10−4 Red 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 × 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='4 × 10−1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 × 1042 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='6 × 10−6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='9 × 10−4 V900 Mon† Blue 2.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='4 × 1042 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='4 × 10−6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 × 10−5 V960 Mon Blue 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='9 × 10−1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 × 10−1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='4 × 1043 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 × 10−5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='9 × 10−2 Red 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 × 10−2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1 × 10−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 × 1043 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3 × 10−5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1 × 10−3 Z CMa Blue 8.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 × 1042 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='8 × 10−6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='9 × 10−3 iPTF 15afq† Blue 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 × 10−2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1 × 10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3 × 1042 2.' metadata={'source': 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Cha Blue 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3 × 10−3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 × 10−3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1 × 1040 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1 × 10−7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='9 × 10−5 Red 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 × 10−3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='4 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 × 1041 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='4 × 10−7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3 × 10−4 V346 Nor Blue 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='9 × 10−2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='9 × 10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='8 × 1042 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 × 10−6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='8 × 10−3 Red 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='6 × 10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 × 10−1 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='9 × 1042 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='8 × 10−6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='9 × 10−3 Note—The † labels the two FUors with tentative outflow detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' masses are higher than ours by less than a factor of 2, which can be explained by the difference in distances and excitation temperatures, and by the difference in the velocity ranges used in the calculation of the out- flow properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Tobin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2020) and K´osp´al et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2021) presented continuum observations at millimeter wavelengths with data from ALMA and VLA, and both reported that this FUor is also a proto-binary star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Reipurth 50 —A Class I protostar also referred to as HBC 494.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Ru´ız-Rodr´ıguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2017b) presented high angular resolution observations with ALMA in which they traced the emission from the outflow in 12CO (J=2– 1) and the envelope emission with the same transition of 13CO and C18O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The extension of the J=2–1 outflow ob- tained with ALMA is smaller than the size of our beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' This can be explained by the maximum recoverable scale of their ALMA configuration (11′′), which is compara- ble to the extended emission seen in their channel maps (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' the 6 km s−1 channel in their Figure 3), thus its is likely they have resolved out most of the extended emis- sion of the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Indeed, most of the emission they recovered with ALMA originates from the dense cavity walls of the bipolar outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Nevertheless, the position angle obtained from the high-resolution interferometric observations (∼145◦) is comparable to our estimation of the position angle (150◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The outflow mass from the ALMA observations, calculated assuming an excitation temperature of 50 K, is a factor of 60 higher the one we determined using the J=3–2 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' If we adjust for the higher temperature used in our calculations (see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='6), the mass estimated from ALMA measure- ments is still a factor of 50 higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' This difference hints that most of the mass of the outflow of Reipurth 50 is located in the narrow cavity walls, which are severely diluted by our single-dish beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' V2775 Ori —The first detection of a molecular outflow on this object was done in the J=2–1 transitions of 12CO, 13CO and C18O with ALMA (Zurlo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The authors found the system is almost face-on with an inclination angle of ∼14°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Our observations recovered Molecular outflows in FUors with APEX 17 a similar orientation of the outflow (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' In addition, we found significant extended emission at both the systemic velocity and at redshifted velocities (+3 km s−1, see also Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Zurlo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2017) re- ported different velocity ranges for 12CO and C18O (see their Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The velocities of the 12CO match those of the redshifted excess emission (peaking at ∼6 km s−1, see Figure 3), and the velocities of C18O match those of the systemic emission we report in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The red- shifted cloud emission appears at velocities where the outflow is still detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Therefore, for all the analy- ses of the outflow in this FUor, we removed the cloud’s contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Similar to the case of Reipurth 50, the difference in beam sizes and sensitivities complicate the comparison between our estimated physical properties and those from Zurlo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' However, we found that the masses from the J=3–2 transition are higher by a factor of ∼8 than those estimated from the ALMA observations, which is even more surprising due to the lower excitation temperature used by Zurlo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We suggest that contrary on the case of Reipurth 50, the extended emission, which is likely resolved out by their interferometric observations, contains more of the mass of the outflow of V2775 Ori than the narrow cavity walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' V899 Mon —This source with a Flat or Class II SED was originally reported as a FUor by Wils et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Follow-up observations indicated that the source was dimming, which was interpreted as a decrease in ac- cretion rate by Ninan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The authors also recovered P Cygni profile for several forbidden lines, in- dicating the presence of outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Our observations are the first to recover an indication of a bipolar molecular outflow in CO, which follows a Northeast-Southwest di- rection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' However, the outflow position angle disagrees with the position angle of the disk (K´osp´al et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2021) and of the jets detected at optical wavelengths (Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2021), thus the analysis of outflows with higher angular resolution is needed to resolve this discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Based on its channel maps (Appendix A) there is also significant extended emission at low velocities due to the envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' V960 Mon —Based on its pre-outburst SED, this is a Class II object (K´osp´al et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2015) and our observations are the first to study the gas surrounding the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The 12CO line profiles show high-velocity wings (Fig- ure 3), which we interpreted as an indication of a bipolar molecular outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The integrated emission maps (Fig- ure 4 and Figure 5) and the channel maps (Appendix A) show the two outflow lobes overlapping, an indication of the outflow having a direction along the line-of-sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' High-angular resolution ALMA observations barely re- solved the FUor disk, and indicate a disk inclination between 16° and 60°, depending on the method used (K´osp´al et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Therefore, for the rest of the anal- ysis, we assumed the lower inclination angle for the out- flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' K´osp´al et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2015) detected two sources close to this FUor (one to the North and one to the South- east), and K´osp´al et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2021) found a third one to the East.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' As these sources are located within our beams, our observations contain emission from these neighbor- ing sources, and it is possible that a source other than the FUor drives the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Therefore, our results for the outflow around this FUor must be taken with cau- tion as an analysis of higher angular resolution observa- tions is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Z CMa —This source is a binary composed of the FUor and a Herbig Be star with a separation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='′′1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Levreault (1988) did not detect an outflow in the J=1–0 transi- tion of 13CO, and in the J=2–1 and J=1–0 transitions of 12CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (1994) and Liljestr¨om & Olofsson (1997) detected the bipolar outflow emanating from this FUor in the J=3–2 and J=1–0 transitions of CO, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Our observations recovered emission from the outflow with a Northwest-Southeast orientation (similar to that found in previous works), and we find the out- flow is compact and has low velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Our estimations of the outflow properties are in general agreement with those of Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' It is unknown which of the two binary components drives the outflow, however, since both sources drive jets (Whelan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2010), it is possible that both sources drive outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' GM Cha —The outflows around this Class I/II object had been previously reported using a single dish an- tenna (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Mottram et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2017) and ALMA (Hales et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We recovered the East-West outflow orientation found by these authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Comparing our results with those of Mottram et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2017), we find the redshifted lobe is more extended than the blueshifted side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Our es- timation of mass for the blueshifted lobe is higher than their estimations, which is explained by us integrating lower velocity fluxes compared to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The redshifted mass and the other outflow properties are comparable to those by Mottram et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' V346 Nor —K´osp´al et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2017a) presented single dish observations of the J=3–2 and J=4–3 transitions, and K´osp´al et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2017b) presented ALMA Cycle 2 ob- servations of the J=2–1 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Our analysis uses the same observations as K´osp´al et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2017a) and the properties of outflow have the same values within 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The small differences are due to slight differences in the methodology, such as different apertures and systemic velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The orientation of the outflow is the same as 18 Cruz-S´aenz de Miera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' that obtained at high angular resolution (K´osp´al et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2017b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Based on the 12CO/13CO ratio used in the op- tical depth correction, it appears even the rarer isotopo- logue is optically thick at velocities close to the systemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' FUors with tentative detections Below we present the two FUors for which we can only make a tentative detection of their outflows, and thus we consider that these two sources require follow- up observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' V900 Mon —One of the most recently discovered FUors, it is a Class I source bordering on Class II (Reipurth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' K´osp´al et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2017b) used the same data as us and carried out a similar analysis as us, and did not find outflow emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' However, Takami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2019) presented high-angular resolution ALMA observations of the J=2–1 transition of 12CO, 13CO and C18O, where they identified a bipolar outflow where the redshifted and blueshifted lobes are in the East and West direc- tions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Following their results, we searched for the velocity ranges that could be integrated in the J=3–2 transition for which we could find emission that following the one detected in the J=2–1 observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We found bipolar emission only in the J=3–2 transition that follows a similar East-West alignment (see Figure 4) using the velocity range indicated in Table 3, and thus we considered this source to drive an outflow and es- timated its properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The J=4–3 transition does not show significant emission (see Figure 5) which prompted us to consider this as only a tentative detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' iPTF 15afq —This Class I object is one of the latest dis- covered FUors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' It showed a ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 mag brightening in 2015 which lasted a few months (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2015), and follow-up brightenings in 2018 and 2019 (Hillenbrand 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The 2019 outburst lasted until early 2021, and was followed by another outburst which is ongoing as of this writing4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Hillenbrand (2019) presented high res- olution (R=37 000) spectra taken during outburst and found that Hα and the Ca II triplet showed a P Cygni profile, an indicator of high velocity winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Our obser- vations are the first sub-millimeter wavelength observa- tions of this object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Its CO line profiles (Figure 3) show high velocity line wings, in particular on the redshifted side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Based on its J=3–2 integrated emission maps (Fig- ure 4) and channels maps (Appendix A), there appears to be an outflow whose blueshifted and redshifted lobes are on the Southeast and Northwest directions, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The blueshifted component is broader and with 4 http://gsaweb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='ast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='uk/alerts/alert/Gaia19fct/ lower velocities than its redshifted counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' How- ever, we consider this FUor as only a tentative detection because the emission is heavily dominated by the enve- lope, and thus, it is hard to confirm that the morphology seen in the J=3–2 transition as an outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' FUors without outflow detections ALMA observations of the J=2–1 transition showed outflow emission for two FUors: V883 Ori (Ru´ız- Rodr´ıguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2017a) and V1647 Ori (Principe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' There could be multiple causes behind our lack of detection: the combination of the higher sensitivity and angular resolution in the ALMA observations, the possible low temperatures in the system, and the low velocities of the ALMA outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Indeed, in the case of V883 Ori, White et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2019) found that the emission of 13CO J=3–2 was a combination of the outflow at low velocities and of a spherical-like envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' In addition, when considering interferometric observations, it is pos- sible that they have resolved-out the contribution from the envelope, which our single-dish observations did not, therefore, the envelope emission at dominates in the low- velocity channels of our observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' A similar case was found for FU Ori, for which previous observations reported it did not drive an outflow (Levreault 1988) but ALMA observations of J=2–1 hint towards an out- flow, thus making its detection uncertain (P´erez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We detect emission in the Northeast-Southwest direction, which is perpendicular to the position angles of the resolved disks (P´erez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' however, the angular resolution of our observations prevents us from determining if it is a bipolar outflow so we do not con- sider this as a detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For the remaining FUors (AR 6A, BBW 76, V582 Aur, V723 Car, and OO Ser) we did not detect outflows, and we did not find previous publications that reported out- flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Outflow parameters We detect clear outflow emission in ∼55% of the FUors in our sample (10 out of the 18), which is lower than the 92% found in Class 0 and Class I objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Mot- tram et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Even including the two cases where ALMA detected outflows when we did not (V883 Ori and V1647 Ori) and the possible outflow in FU Ori, we would only find outflows in ∼73% the FUors of our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' However, this is not surprising considering that some of the FUors in our sample are classified as Flat spectrum or Class II objects, and the outflows in these evolved stages might be harder to detect due to the lower densities of the enveloping material (Arce & Sargent 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Indeed, only two FUors of Class II had evidence of outflows: V960 Mon and GM Cha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Molecular outflows in FUors with APEX 19 After the optical depth correction, the outflow masses increased by a median factor of 3, with values ranging between 1 (V346 Nor) and 14 (V900 Mon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The values of the kinematic properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' momentum, energy) after the optical depth correction are also a factor of a few higher than without the correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Even after applying this correction, we can still expect the outflow properties presented in Table 6 to be underestimated as explained in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' In Figure 7 we show a comparison of the outflow masses determined for the FUors that had outflow detec- tion in both transitions of 12CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For this comparison, we estimated the outflow mass uncertainties by multi- plying the number of pixels used when calculating the mass by the rms of each data cube, and then converted these fluxes to masses using the same assumptions as the outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We find that, within these uncertainties, most outflows have comparable masses in both transi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Reipurth 50 is the only FUor in which the mass estimate is higher in the J=4–3 transition than in J=3– 2 by a factor of ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The outflows of V2775 Ori and V899 Mon are more massive in the lower transition by factors of 4 and 5, respectively, thus, this could be an indication of different excitation properties causing the lower transition to be stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' However, these com- parisons are limited by differences in the observations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' angular resolution and sensitivities), in the images (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' pixel size and field of view), and by using the same excitation for the two transitions in all the FUors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' A large-scale program to target multiple CO transitions under comparable conditions would alleviate these lim- itations and provide more insight on the masses of the outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Comparison with quiescent young stellar objects We put into context our outflow properties by compar- ing them with the values of similar studies based on qui- escent sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' This comparison is not straightforward due to the differences in the observational properties (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' angular resolution and sensitivity), and methodol- ogy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' choosing velocities for integration, optical depth correction and inclination correction), which have signif- icant effects on the resulting values of the outflow prop- erties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' It is expected that FUor outbursts last for up to a hundred years and the dynamical ages of the outflows are in the other of thousands of years (see Table 3), thus, the outflows we have detected around FUors are not re- lated to the current outbursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' This means that we are comparing the histories of the two samples, which could provide hints towards the nature behind the outbursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We compared our sample with the values of the out- flow properties published in the following studies: Dun- 10 3 10 2 10 1 MJ = 3 2 [M ] 10 3 10 2 10 1 MJ = 4 3 [M ] Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Comparison of outflow masses determined from the J=3–2 and J=4–3 transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The dashed line indicates a ratio of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' ham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2014), Yıldız et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2015) and Mottram et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The three studies cover a combination of Class 0 and Class I objects and our calculations followed similar methods to theirs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We included quiescent Class 0 ob- jects even when the vast majority of FUors are Class I objects (see Table 1) because we want to compare how the FUor outbursts compare to the different stages of the star-formation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We compared the outflow masses and forces as those were the only two properties presented by all three studies from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Dunham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2014) presented outflow properties with and without optical depth correction, while Yıldız et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2015) and Mottram et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2017) did not calculate this correction, thus we used the optically thin values for this comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Dunham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2014) assumed Tex = 50 K for their calculations, while Yıldız et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2015), Mottram et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2017) used the same temperature we did in our anal- ysis, Tex = 75 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' To test the effect of using the lower temperature, we calculated the outflow properties of the FUors using Tex = 50 K, and found the values were a fac- tor of ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='19 higher when using the higher temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Thus, we multiplied the outflow properties of Dunham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2014) by this factor to minimize the differences in methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The outflow forces estimated by Yıldız et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2015) and Mottram et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2017) were corrected due to the inclination of the systems based on Downes & Cabrit (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' These correction values were estimated for Class 0 objects and are not recommended to correct Class I objects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' however, for the sake of a comparison 20 Cruz-S´aenz de Miera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' between our sample and the ones from the literature, we applied this correction factor to the FUor outflows even if they are at later stages (Class I or II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We estimated the inclination of the outflows by assuming the inclina- tion of the outflow is perpendicular to the inclination of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For most sources, we used the inclination of the FUor disks obtained from high-angular resolution observations with ALMA (Cieza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Hales et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' K´osp´al et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2021), while for Z CMa we used the estimate by Antoniucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2016) estimated from an analysis with data from optical interferometry, and in the case of iPTF 15afq, we assumed an inclination of 45◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' However, the works studying the quiescent sample used a coarse correction table, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 in Mottram et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Thus, we rounded our inclinations to the closest values in the inclination correcion table, and the angles assumed are listed in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We combined the quiescent samples from the literature into one, divided it by Class, and compared the two sub-samples with the FUors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' In Figure 8 we plotted the outflow forces calculated from the J=3–2 transition against the envelope masses calculated from the 13CO emission (panel a), against the outflow masses also calculated from the 12CO J=3– 2 transition (panel b), against the ratio between the outflow mass and envelope mass (panel c), and against the bolometric luminosities obtained from the literature (panel d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Panels a and b of Figure 8 show that FUors outflows follow the same trends as the quiescent sources from the literature, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' higher envelope masses and higher outflow masses indicate higher outflow forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' In the outflow mass to envelope mass ratio subplot, panel c of Figure 8, the FUor sample is offset from the quies- cent samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' This ratio has been used to discuss the core-to-star formation efficiency in the quiescent sam- ple (Mottram et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2017), thus it hints that FUors are less efficient at driving mass from the envelope onto the star, and this relationship will be discussed further be- low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The values for this ratio are presented in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The correlation between outflow force and bolomet- ric luminosity, panel d of Figure 8, has been well stud- ied for quiescent sources (Cabrit & Bertout 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Bon- temps et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Yıldız et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Mottram et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2017), and it would appear that FUors do not follow this correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' However, the FUor bolometric lumi- nosities were estimated from photometry taken while in outburst, and none have sufficient pre-outburst photo- metric data to estimate their pre-outburst luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Even if we do not have sufficient information about the individual FUors, when in quiescence, the protostars are .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Ratio between outflow masses and envelope masses using the J=3–2 transition of the two observe CO isotopologues, and the core-t-star formation efficiency, ϵ Target Moutflow/Menvelope ϵ L1551 IRS 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='068 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='91 Haro 5a IRS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='50 Reipurth 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='69 V2775 Ori 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='178 −19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='23 V899 Mon 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='159 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='05 V900 Mon† 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='140 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='44 V960 Mon 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='271 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='89 Z CMa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='39 iPTF 15afq† 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='496 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='73 GM Cha 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='23 V346 Nor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='159 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='77 Note—The † labels the two FUors with ten- tative outflow detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' expected to be low-mass and low-luminosity objects (ex- cept for V723 Car), and our measured outflow parame- ters are consistent with this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Thus, our results suggest that FUors, when in quiescence, produce molecular out- flows with forces comparable to those from outflows in quiescent stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' In order to get a better estimate of how similar FUor outflows are with their quiescent counterparts, we present cumulative histograms comparing different properties (Figure 9), and we carried out three comple- mentary statistical tests to examine whether the sam- ples of the quiescent young stars were drawn from the same sample as the FUors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The first was a two-sided Kolmogorov-Smirnov (K-S) test that com- pares the shapes of the distributions, the second was a Mann–Whitney U-test (MWU), which is more sensitive to the mean of the two samples rather than the shape of both distributions, and the third was a k-sample Anderson-Darling test (kAD), which is more sensitive to the tails of the distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The three tests were done using the SciPy functions kstest, mannwhitneyu, and anderson ksamp, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The results of the statistical tests are presented in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' With a significance level of 5% we found that the dis- tribution of FUor envelope masses is similar to that of Class Is and different from the Class 0s (Figure 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' panel a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' the outflow masses of FUors are different to those of Molecular outflows in FUors with APEX 21 10 1 100 101 Envelope mass [M ] 10 8 10 7 10 6 10 5 10 4 10 3 10 2 Outflow force [M km s 1 yr 1] a) Class 0 Class I FUors 10 4 10 3 10 2 10 1 Outflow mass [M ] b) 10 3 10 2 10 1 Outflow mass / Envelope mass c) 100 101 102 Bolometric luminosity [L ] d) Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Comparison of the outflows from quiescent sources in the literature with the FUor outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Outflow forces (J=3–2) plotted against envelope masses (panel a), outflow masses (J=3–2, panel b), the ratio between outflow mass and envelope mass (panel c), and bolometric luminosities (panel d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' In case of the FUors, all bolometric luminosities are during outburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The two FUors with tentative outflow detections are marked with empty diamond symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The Lbol of iPTF 15afq is unknown and thus it is not shown in panel d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 log10 Envelope mass 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 Fraction a) Class 0 Class I FUors 4 3 2 1 log10 Outflow mass b) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 log10 Outflow mass/Envelope mass c) 8 6 4 2 log10 Outflow force d) Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Cumulative histograms of outflow properties, and the outflow mass to envelope mass ratio, for the Class 0 and Class I objects from the literature (Dunham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Yıldız et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Mottram et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2017), and the FUors from this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The bin widths for each histogram were selected using the Freedman–Diaconis rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' p-values of the three statistical tests done for the envelope masses, outflow masses and forces, and the ratio between outflow and envelope masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Envelope masses Outflow masses O/E mass ratio Outflow forces Test Class 0 Class I Class 0 Class I Class 0 Class I Class 0 Class I K-S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='605 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='220 1×10−3 2×10−3 2×10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='070 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='247 MWU 2×10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='667 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='129 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='002 5×10−4 2×10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='272 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='726 kAD <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='001 >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='002 <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='001 <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='097 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='208 The p-values from anderson ksamp are capped between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='001 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='250, thus these values are upper or lower limits, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 22 Cruz-S´aenz de Miera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Class I objects and comparable with those of Class 0s (Figure 9, panel b), the outflow mass to envelope mass ratio is different in the FUors when compared to either sample (Figure 9, panel c), and the outflow forces of FUors are comparable with the two quiescent samples (Figure 9, panel d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Our tests did not include the two tentative detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We ran the tests including these two FUors and found that our statistical results would not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The envelope mass result is not surprising because, based on their SEDs, the FUors are also Class Is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Fol- lowing the same logic, the result for the outflow masses is surprising as the outflow masses are close to the Class 0s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' This could be interpreted as an indication that FUors are in the very early stages of their Class I stage and their outflows have not had sufficient time to dissipate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' However, the masses of outflows are a combination of the material that has passed through the accretion disk and is now being driven away from the star by the out- flow, and the material in the envelope that has been entrained by the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' FUor outflows have higher outflow masses but similar envelope masses compared to the Class Is, pointing towards FUor outflows having a higher percentage of material that was ejected from the accretion disk, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' material that was not accreted onto the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' This can be seen in the ratio between the outflow mass and the envelope mass in Figure 8 and Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The separation between the quiescent sample and the FUors might be biased due to the distance of the tar- gets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The YSOs of the quiescent samples are all withing 500 pc from the Sun, while half of the FUor sample is beyond this distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Therefore, if we consider that in most cases the extension of the outflows is larger than that of the envelopes, our analysis might be biased to- wards the FUors as it is likely that for we are measur- ing the full extension of their outflows in comparison to the quiescent sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' In Figure 10 we plotted the out- flow/envelope mass ratio versus the distance for each target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The distribution of points indicates that indeed there might be a positive correlation between the mass ratio and the distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' However, the maps in Yıldız et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2015) and Mottram et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2017) indicate that 40% of their outflows extend beyond the areas of the sky covered by their respective observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Therefore, the mass ratios for those sources are lower limits, and thus it raises the question whether this relationship is real or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' An in-depth observational program cover- ing outflows at a wide range of distances should shed some light on this matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Mottram et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2017) used the outflow/envelope mass ratio to examine the core-to-star efficiency in a group of 102 103 Distance [pc] 10 3 10 2 10 1 O/E mass ratio Class 0 Class I FUors Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Ratio between outflow mass and envelope mass plotted against the distance to each target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The open dia- monds are the two FUors with tentative outflow detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' quiescent young stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' They assumed that during the whole duration of the Class 0 and I phases, a star has an outflow rate with small enough variations that it can be approximated by a constant value, and calculate the core-to-star formation efficiency, ϵ, as follows: ϵ = 1 − Mof Menv τ0+I τd , (4) where τ0+I is the total duration of the Class 0 and I phases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The authors used their typical val- ues of Mof/Menv = 10−2 and τd = 104 years, and found a core-to-star formation efficiency of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='5, which is in agree- ment with the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We used our derived masses and dynamical times to calculate ϵ for the FUors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The values can be found in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' As can be seen from our results, five of the FUors have negative ϵ values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' These can be explained by the under- estimation of the dynamical age because either the out- flow appears to be face-on (V2775 Ori and V960 Mon), or it extends beyond the field of view of our observa- tions (L1551 IRS 5, iPTF 15afq, and V346 Nor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Three of the six FUors with positive values extend beyond our field of view (Haro 5a IRS, Reipurth 50, and V899 Mon) and thus their ϵ values are uncertain because it is un- known how much of the mass and the extension of the outflow is beyond our field of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The three remaining FUors (V900 Mon, Z CMa, and GM Cha) with positive ϵ and with the full outflow inside the field of view, have lower efficiencies than the quiescent sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' These val- ues would suggest that a significant amount of material Molecular outflows in FUors with APEX 23 that was fed from the envelope onto the disk was not ac- creted onto the star but instead was driven outwards by the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' However, two of these have strong caveats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' For V900 Mon, its outflow was only tentatively detected and follow-up observations might reveal a different ge- ometry than ours, which would lead to a different value of ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' GM Cha is a Class II object, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' in a more evolved stage than most FUors, and the equation used to cal- culate the efficiencies was created for Class I objects whose accretion rate is orders of magnitude higher than for Class IIs (Fiorellino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2022) and have higher en- velope masses, which means that our estimated value is not accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' As a final point, we address the similar distributions of the outflow forces between the two quiescent samples and the FUors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The outflow force is a property that is commonly associated to the accretion history of a young stellar object (Bontemps et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' This is because, as can be seen in panel a of Figure 8, there is a positive correlation between the outflow force and the envelope mass, and the more evolved young stars have lower en- velope masses and lower mass accretion rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Here we present some scenarios that can explain the lack of sep- aration between the FUors and quiescent sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' First, if we assume that the similar distributions are because the accretion histories of the two samples are the same then this can be interpreted as either the “qui- escent” sample had outbursts that were undetected, or the current outbursts in the FUors are the first ones in their accretion histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' If these are indeed the first outbursts in each of the FUors, then their effects would be undetected because the angular resolution of our ob- servations is insufficient to resolve the inner parts of the outflows where the effects of a <100 year old outburst could be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Few FUor outbursts (∼20 Connelley & Reipurth 2018) have been detected so the incidence rate of these events is unknown, and as such it is difficult to separate between these two scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Second, if we assume that the accretion histories are different between quiescent and outbursting samples then the likeness between distributions should be be- cause of a physical property of the outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The out- flow force is calculated using the outflow momentum and the dynamical age of the outflow, and both of these properties depend on the distribution of velocities in the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' If the FUor outflows have masses comparable to Class 0 outflows and similar velocity ranges as the outflows from the literature, we would expect the FUor outflow forces to be close to those of the Class 0 ob- jects from the other samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Therefore, the divergence between Class 0 outflow forces and FUor outflow forces indicates that the latter have lower velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' This had already been mentioned by Principe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' (2018) when comparing the V1647 Ori outflow to those of others out- flows observed with ALMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' However, as mentioned earlier, the outflow forces pre- sented here are highly uncertain because they are cal- culated as the ratio of two lower limits, the outflow mo- mentum and the outflow dynamical age, and because of the understudied effects of the inclination of the outflow with respect to the plane of the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' As such, there is a need for a thorough program to study these molecular outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' SUMMARY & CONCLUSIONS We presented APEX observations of 20 FUors or FUor-like objects from which we estimated the enve- lope mass and searched for outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Using a combi- nation of line profiles and inspection of channel maps, we detected outflows in 45% of our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' These in- clude the possible first detections of molecular outflows in V899 Mon and V960 Mon, although these should be observed with higher angular resolution to corrob- orate them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We also found two tentative detections in V900 Mon and iPTF 15afq, that require follow-up obser- vations to confirm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' In the case of V883 Ori, V1647 Ori, and possibly FU Ori, we did not detect the outflows that have been observed by ALMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Based on our 13CO measurements, envelopes with masses higher/lower than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='2 M⊙ show the silicate feature at 10 µm in absorption/emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' If the enve- lope mass is close to this threshold level, the geometry of the system determines whether the spectral feature is in emission or absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The most significant outlier of this trend is V960 Mon, which shows the 10 µm feature in emission despite having an envelope of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='6 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The masses of outflows estimated from the 12CO 3– 2 and 4–3 transitions are in agreement, except for two FUors: V900 Mon and V960 Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We suggest that these two sources could be colder than the rest of the sample and, thus, the higher transition is dimmer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' V960 Mon is an outlier in both trends, thus we pro- posed another possible explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' This FUor has three companion YSOs in its proximity, with separa- tions smaller than the sizes of our beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Therefore, we suggest that these additional sources move this object away from the trend seen in the rest of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The kinematic outflow properties (momenta, energies, forces and luminosities) are higher when estimated from the J=3–2 transition than those from J=4–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We at- tribute this to the higher sensitivity of the lower transi- tion, which causes a difference in the range of velocities in which we detected outflow emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 24 Cruz-S´aenz de Miera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' After applying an optical depth correction to the J=3– 2 transition using the 13CO emission, we found that the mass of the outflows increased by a median factor of 3 and up to an order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The minimum im- provement, seen in a few cases, showed that the outflow mass increased only by a few percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We compared the outflows found in our FUor sam- ple with three works from the literature and found that outflows emanating from FUors are more massive than those from quiescent Class I sources but with masses comparable to outflows in Class 0 sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We found that FUors have a higher outflow/envelope mass ratio than the quiescent sample, although this result could be biased by the distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' We calculated the core-to-star efficiencies of the FUors and although our results are severely constrained by the geometry of the outflows, it could indicate that a significant portion of the ma- terial that was deposited into the accretion disk from the envelope is not accreted onto the star but instead is driven back to the envelope by the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Finally, we found that outflow forces from the FUor sample are comparable to the two quiescent sources, which can be interpreted as similar accretion histories or as very low velocities in the FUor outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' This study focused on the outflow histories of the FUors observable from the APEX site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The dynami- cal ages of the detected outflows indicate that they are much older than any of the ongoing outbursts, which are less than 100 years old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Indeed, any outflow emission di- rectly related to the current outburst would be detected at high velocities, close to the protostar and would have small spatial scales that would be diluted by the beam of our single-dish observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Our comparison between the outflow properties of FUors and of other quiescent objects should be taken with caution due to the varying quality of the individual observations, and the method- ology used by each research group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' A complete survey of all known FUors in both hemispheres with similar ob- servational setups and sensitivities, and a control sam- ple of multiple quiescent YSOs at different evolutionary stages, would greatly improve our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme un- der grant agreement No 716155 (SACCRED).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' has received financial support from the French State in the framework of the IdEx Universit´e de Bordeaux In- vestments for the future Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' This publication is based on data acquired with the Atacama Pathfinder Experiment (APEX) under programme IDs 094.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='F-9508 and 098.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='F-9505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' APEX is a collaboration between 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=', & Lim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2009, ApJ, 698, 184, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1088/0004-637X/698/1/184 Yıldız, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=', Kristensen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=', van Dishoeck, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2015, A&A, 576, A109, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1051/0004-6361/201424538 Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=', Arce, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=', Mardones, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2019, ApJ, 883, 1, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='3847/1538-4357/ab3850 Zurlo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=', Cieza, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=', Williams, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 2017, MNRAS, 465, 834, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='1093/mnras/stw2845 Molecular outflows in FUors with APEX 27 APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' CHANNEL MAPS Here we present the channel maps for the three observed transitions of L1551 IRS 5, and the complete figure set with the rest of the targets in the FUor sample is available in the online journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The minimum and maximum velocities in the channels maps are those when the gas emission starts or finished being significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The purple contours are used for all the 13CO channel maps and for the 12CO maps when outflows were not detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The blue and red contours show the blueshifted and redshifted emission of outflows, and the green contours show the envelope emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' All channel maps show the aperture used for the calculation of the envelope mass in the case of 13CO, and the outflow properties in the case of both 12CO transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The velocities shown in the plots were chosen so that the maximum and minimum velocities are shown within 27 frames, which can cause some irregular velocity steps in the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' However, these differences are of one channel, and we do not expect to see significant changes in the distribution of CO between two continuous channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Set A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Channel maps B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' OPTICAL DEPTH CORRECTION In Table B1 we present the parameters of the best-fitted parabola used to determine the optical depth correction for the sources with outflows, and in Figure B1 we present the parabolic fit and the line profiles used in the fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Set B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Optical depth correction Table B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Parameters of best fitted parabolas used for optical depth correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Target A C L1551 IRS 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='388 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='683 Haro 5a IRS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='327 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='194 Reipurth 50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='548 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='643 V2775 Ori 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='825 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='320 V899 Mon 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='138 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='189 V900 Mon 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='531 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='946 V960 Mon 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='409 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='357 Z CMa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='924 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='350 iPTF 15afq 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='751 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='224 GM Cha 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='909 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='505 V346 Nor 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='419 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='246 28 Cruz-S´aenz de Miera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='36 km / s 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='53 km / s 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='67 km / s 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='81 km / s 13CO (3 2) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='94 km / s 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='12 km / s 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='26 km / s 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='39 km / s 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='53 km / s 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='70 km / s 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='84 km / s 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='98 km / s 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='12 km / s 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='26 km / s 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='43 km / s 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='57 km / s 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='71 km / s 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='85 km / s 60" 30" 0" 30" 60" 30" 0" 30" RA Dec 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='02 km / s 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='16 km / s 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='30 km / s 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='43 km / s 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='61 km / s 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='75 km / s 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='88 km / s 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='02 km / s 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='16 km / s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='27 km / s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='23 km / s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='73 km / s 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='22 km / s CO (3 2) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='72 km / s 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='21 km / s 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='74 km / s 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='24 km / s 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='74 km / s 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='23 km / s 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='73 km / s 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='26 km / s 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='75 km / s 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='25 km / s 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='75 km / s 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='24 km / s 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='77 km / s 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='27 km / s 60" 30" 0" 30" 60" 30" 0" 30" RA Dec 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='76 km / s 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='26 km / s 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='76 km / s 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='28 km / s 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='78 km / s 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='28 km / s 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='77 km / s 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='27 km / s 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='80 km / s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='93 km / s 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='33 km / s 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='77 km / s 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='22 km / s CO (4 3) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='66 km / s 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='06 km / s 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='51 km / s 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='95 km / s 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='40 km / s 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='80 km / s 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='24 km / s 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='69 km / s 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='14 km / s 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='58 km / s 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='98 km / s 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='43 km / s 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='87 km / s 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='32 km / s 60" 30" 0" 30" 60" 30" 0" 30" RA Dec 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='72 km / s 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='16 km / s 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='61 km / s 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='06 km / s 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='45 km / s 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='90 km / s 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='35 km / s 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='79 km / s 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='24 km / s Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' L1551 IRS 5 channel maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' 13CO (3–2) with contours at 3, 5, 7, 9, 11, 13, 15, 17, 19 and 21σ with σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='32 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' CO (3–2) with contours at 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36 and 39σ with σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='37 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' CO (4–3) with contours at 3, 5, 7, 9, 11, 13, 15, 17, 19 and 21σ with σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content='65 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The black circles indicate the 8000 au aperture used to extract the line profiles used to analyze the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The complete figure set (20 images) is available in the online journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Molecular outflows in FUors with APEX 29 6 4 2 0 2 4 6 v [km s 1] 102 103 104 105 TMB [K] 6 4 2 0 2 4 6 v [km s 1] 0 10 20 30 T12 / T13 Figure B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' Optical depth correction for L1551 IRS 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' In the left panel we show the line profiles, where the green, purple and black colors represent the 13CO, the observed 12CO and the corrected 12CO, respectively, and the vertical dashed lines indicate the rage of velocities used in the parabola fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The right panel shows the ratio of main beam temperatures (TMB), where the light blue crosses are all the values of the ratio for each velocity channel, in pink dots with errorbars are the points used in the parabolic fitting, and the black line is the resulting best-fitted parabola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} +page_content=' The complete figure set (10 images) is available in the online journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE1T4oBgHgl3EQfuQXJ/content/2301.03387v1.pdf'} diff --git a/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf b/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..69125b336da02338e04a70e705942e6495f22b03 --- /dev/null +++ b/TtE0T4oBgHgl3EQfVAA3/content/2301.02257v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d4c71bd0766feedee9c040b915abde919e8109668f945e028130e7bcfc60d682 +size 12233226 diff --git a/TtFLT4oBgHgl3EQfQS-o/content/tmp_files/2301.12032v1.pdf.txt b/TtFLT4oBgHgl3EQfQS-o/content/tmp_files/2301.12032v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2324aee3d7446cc100ae534d57f5a471d56d9228 --- /dev/null +++ b/TtFLT4oBgHgl3EQfQS-o/content/tmp_files/2301.12032v1.pdf.txt @@ -0,0 +1,2365 @@ +BinaryVQA: A Versatile Test Set to Evaluate the Out-of-Distribution +Generalization of VQA Models +Ali Borji +Quintic AI +aliborji@gmail.com +Abstract +We introduce a new test set for visual question answering +(VQA) called BinaryVQA to push the limits of VQA models. +Our dataset includes 7,800 questions across 1,024 images +and covers a wide variety of objects, topics, and concepts. +For easy model evaluation, we only consider binary ques- +tions. Questions and answers are formulated and verified +carefully and manually. Around 63% of the questions have +positive answers. The median number of questions per im- +age and question length are 7 and 5, respectively. The state +of the art OFA model achieves 75% accuracy on BinaryVQA +dataset, which is significantly lower than its performance +on the VQA v2 test-dev dataset (94.7%). We also analyze +the model behavior along several dimensions including: a) +performance over different categories such as text, counting +and gaze direction, b) model interpretability, c) the effect +of question length on accuracy, d) bias of models towards +positive answers and introduction of a new score called the +“ShuffleAcc”, and e) sensitivity to spelling and grammar +errors. Our investigation demonstrates the difficulty of our +dataset and shows that it can challenge VQA models for +next few years. Data and code are publicly available at: +DATA and CODE. +1. Introduction +Visual question answering [5, 10] is a multidisciplinary +task at the intersection of computer vision, NLP, knowledge +representation, reasoning, common sense knowledge, etce- +tra. The goal is to answer a text-based question given an +input still image or a video. +Recent VQA models are able to answer binary questions +above 95% accuracy, which is astonishing considering that +in principle, any questions can be asked on an image. At +the same time, though, this alarms that perhaps we are not +using the test sets that have the right level of difficulty. Us- +ing the same test set over the years has the risk of over- +fitting, as researchers often tune their models towards the +statistics of the test sets (even when the annotations are held +hidden). To mitigate this issue, it is crucial to have sev- +Figure 1. Samples from our dataset. Our dataset covers a wide va- +riety of concepts including counting, crowd, emotions, drawings, +paintings, camouflage, clothing, time, weather, body parts, age, +text, gaze direction, etc. It also includes questions that address +spatial understanding of models (e.g. the blue rectangle in the last +image of the 3rd row). See Appendix A for more examples. +eral versatile independent test sets to evaluate models and +to track the progress. While several test sets are available +for problems such as image classification (e.g. [6, 14, 28]) +and object detection (e.g. [20,22,24]), the VQA field lacks +enough difficult test sets. +Our study is an effort in this +direction. This discussion naturally relates to the out-of- +distribution studies showing that models are biased towards +the test sets that are similar to the sets over which they have +been trained on. Likewise, they underperform over test sets +that are even slightly different [28, 32, 35]. In this regard, +here we also testing the out-of-distribution performance of +the VQA models. +Our test set contains 1,024 images crawled from +publicly-available and free-to-distribute sources. We used +Google and Bing search engines with different search +phrases to collect the images. We made sure that no im- +arXiv:2301.12032v1 [cs.CV] 28 Jan 2023 + +Dataset +# Images +# Questions +Question Type(s) +DAQUAR [25] +1449 +12468 +Object identification +COCO-QA [29] +123287 +115000 +Questions automatically generated from COCO captions +VQA [5] +204721 +614163 +Combining vision, language and common-sense +Visual Madlibs [41] +10738 +360001 +Fill in the blanks +Visual7W [43] +47300 +2201154 +7Ws, locating objects +CLEVR [17] +100000 +853554 +Synthetic question generation using relations +Tally-QA [1] +165000 +306907 +Counting objects on varying complexities +KVQA [31] +24602 +183007 +Questions based on Knowledge Graphs +VizWiz [13] +31000 +31000 +Questions by visually impaired users +TextVQA [33] +28408 +45336 +Questions demanding reasoning about text +Table 1. Overview of VQA datasets described in this paper. +age contains sensitive material, has poor resolution, or vi- +olates the copyright law1. The gathered data encompass +a wide variety of visual concepts over both RGB images, +paintings, drawings, cartoons, and clip arts (Fig. 1). We +have made sure that all the questions are unambiguous and +answers are correct. Our test set contains more questions +per image (∼7) than the VQA v2 test set (∼3). We only +consider the binary questions, since essentially any question +can be converted to a “yes/no” question. This simplifies the +model evaluation and eliminates the complicated process +of matching sentences of predicted answers with actual an- +swers. Notice that this argument does not necessarily mean +that we only need models that give binary answers. +Although our test set is smaller than the VQA test set, it +comes with the benefit of better control over the complexity +of the questions and quality of the answers. Controlling the +difficulty level of the questions generated by the Amazon +Mechanical Turk (AMT) workers is challenging, as work- +ers may choose to ask simple and short questions to save +time. Unlike the questions in the VQA dataset [5] that are +supposed to fool a toddler, alien, or a smart robot, some Bi- +naryVQA questions can even challenge adults. To answer +the majority of the questions, one has to carefully analyze +the images. Further, small versatile and carefully curated +test sets like ours can alleviate the legal issues concerning +consents, licensing, privacy and security which are harder +to control in datasets containing millions of images. +In curating the BinaryVQA, we have made three choices. +First, this test set is intentionally not paired with a train- +ing set. This is to encourage generalization and to prohibit +models to take advantage of correlations between testing +and training sets. These correlations are easily accessible +to models but are not detectable by humans [9]. Second, +our dataset comes with a license that disallows researchers +to update the parameters of any model for any reason on +it. This is again to avoid over-fitting. Third, to mitigate the +danger of leaking our data to other training sets, we mark +every image by a one pixel green border that must be re- +moved on the fly before testing. +In addition to the test set, we also introduce new dimen- +sions along which VQA models can be tested, in particu- +lar sensitivity of the models to small perturbations in the +1We choose images that were public domain, did not have copyright, +or were released by the government. +questions. We find that, unlike humans, current models are +highly sensitive to minor grammar mistakes. Further, we +study the bias of models towards generating positive an- +swers, whether models indeed require the image to answer +the questions, and whether they choose the right image re- +gions to do so. In a nutshell, our results show that state of +the art VQA models struggle on our dataset. This suggests +that, in conjunction with other datasets, our dataset can be +used to push the VQA models to become better. +2. VQA Datasets +Several VQA datasets have been introduced [18,26,40]. +In these datasets, images are either taken from an existing +vision dataset (e.g. MSCOCO; [24]) or are artificially cre- +ated (e.g. Abstract Scenes; [5], computer graphics; [4,17]). +Further, questions are generated either automatically [4,17, +18, 25, 29, 41], from crowd workers [5, 8, 11, 18, 21, 43], or +from in-house participants [18, 38]. Unlike these datasets, +questions in our dataset are carefully constructed by experts +such that to answer them a detailed inspection of the image +is necessary. Some prominent VQA datasets are listed in +Table 1. Relevant ones to our work are described next. +COCO-QA [29] includes 123,287 images from the +MSCOCO (72,783 for training and 38,948 for testing) and +each image has one question/answer pair. Questions are au- +tomatically generated from the image descriptions and are +categorized into four types based on the type of expected +answer: object, number, color, and location. A downside of +the COCO-QA dataset is that 9,072 (23.29%) of test ques- +tions also appear in the training questions. +VQA [5, 11] is one of the most widely used datasets +(https://visualqa.org/). It comprises two parts, +one using natural images called VQA-real (sourced from +MSCOCO), and a second one with cartoon images called +VQA-abstract. The latest more comprehensive version of +this dataset, VQA v2.0 consists of 1.1 million (image, ques- +tion) pairs with 13 million associated answers. +Visual Genome [21] is aimed to enhance the progress +on cognitive tasks, especially spatial relationship reasoning. +It contains over 108K images, with about 35 objects, 26 +attributes, and 21 pairwise relationships between objects. +Visual7W [43] includes seven types of WH questions +(what, where, when, who, why, which and how) to examine +capability of a model in visual understanding. Questions + +are asked in the multiple-choice format. There are four can- +didates for each question, and only one candidate is the cor- +rect answer. +Visual Madlibs [41] consists of 360,001 targeted de- +scriptions spanned across 12 different types of templates +and their corresponding images. +VizWiz [13] is constructed from interactions of visually +impaired users with a mobile application. +It consists of +31,000 visual questions together with 10 crowdsourced an- +swers per question. Images often have poor quality due to +poor lighting, focus, and framing of the content of interest. +Further, questions are on average more conversational and +are sometimes incomplete. +TextVQA [33] contains 45,336 questions on 28,408 im- +ages that require reasoning about text to be answered. Im- +ages are taken from the Open Images v3 dataset [20]. +TextVQA is available at https://textvqa.org. +In addition to above, some non-photo-realistic datasets +such as CLEVR [17], NLVR [34], and FigureQA [19] have +also been introduced to study visual reasoning independent +of language. Some datasets such as Fact-Based VQA [37] +explicitly require external knowledge to answer questions. +GQA [16] is a popular dataset, which also involves phrases +to address the relations. +Our work relates to research that addresses the functional +diagnostics of pre-trained language models (e.g. [27, 30]). +It also relates to works that examine adversarial robust- +ness and out-of-distribution generalization of VQA models +(e.g. [7,23]). For example, [23] shows that non-expert an- +notators can easily attack the best VQA models. +We construct an adversarial dataset to challenge the best +VQA models. Although there are few such datasets for free- +form VQA (e.g. VQA-CP [3]), here we show that even that +answering yes/no questions is not yet solved. +3. BinaryVQA Dataset +Our dataset contains 7,800 questions across 1,024 im- +ages. Majority of the questions start with “Is” and “Are” +as shown in the sunburst plot in Fig. 2. The most com- +mon terms in the questions are person, wearing, +people, and image (right panel in Fig. 2). +We do +not include WH questions and all questions have “yes” or +“no” answers. We ensured that each image is valid through +human review. We formulated the questions and then pre- +sented them along with their answers to three AMT workers +for verification. Please see Appendix D for details. Out of +all questions, only 41 QA pairs received the incorrect ma- +jority vote, which were fixed subsequently. +Statistics of the BinaryVQA dataset are shown in Fig. 3. +Out of the 7,800 questions, 4,897 have positive answers and +the remaining 2,903 have negative answers, resulting in a +ratio of about 62.7% (positive/all images). The median pos- +itive to all questions ratio per image is 0.625. 38 images +Q type +List of words +sky +sky +spatial +rectangle +vegetation +tree, plant, flower +gaze direction looking +real/drawing +painting, drawing +[in/out]doors +indoors, outdoors +daytime +daytime, nighttime +emotions +happy, sad, angry, upset +time +clock, time, watch, hour, minute, seconds +gender +man, woman, female, male, boy, girl +text +text, number, English, Roman, word, written +age +age, old, young, child, kid, baby, adult, teenager +weather +weather, snowy, sunny, cloudy, rainy, stormy, foggy +color +color, white, red, blue, yellow, black, purple, green, silver, blond +actions +fighting, walking, sitting, standing, running, climbing, lying, +dancing, partying +direction +right, left, top, bottom, above, below, side, leftmost, rightmost, next +counting +more than, less than, two, three, ten, fifteen, twenty, +two hundred, exactly, only +body parts +face, head, hand, leg, foot, feet, eye, torso, ear, +belly, belly button, finger, hair, shoulder, neck, mouth, nose, body +clothing +shoe, jean, jeans, dress, tie, shirt, short, long sleeve, sock, hat, cap, +earring, watch, piercing, necklace, scarf, eyeglasses, +belt, cloths, wearing +animals +animal, cat, dog, elephant, tiger, horse, owl, chicken, hen, +rooster, wolf, fox, octopus, sheep, eagle, lion, giraffe, monkey, +cow, scorpion, turtle, fly, mosquito, dinosaur, panda, pigeon, spider +fruits +fruit, apple, banana, acorn, tomato, potato, pomegranate, +pear, peach, orange, grape, melon, watermelon, cherry, strawberry, +corn, pumpkin, pineapple, lemon,pepper, avocado, cabbage, +lettuce, coconut, cucumber, eggplant, broccoli +Table 2. List of words per question type in the BinaryVQA dataset. +(3.7%) have all of their questions answered “yes”, while no +image has all of its questions answered “no”. The median +number of questions per image is 7 which means that half of +the images have more than 7 questions. The median num- +ber of positive questions (questions with answer “yes”) is +4 and the median number of negative questions is 3. The +mean number of questions per image in BinaryVQA is 7.62 +which is higher than 5.4 for VQA v2. BinaryVQA questions +range from 3 to 20 words. The mean and median question +length are 5.64 and 5 words, respectively. VQA v2 ques- +tions range from 4 to 10 words (average 5). The average +image resolution is 840.3 × 650.4 (w × h) with the average +aspect ratio of 1.32. +Sample images are shown in Fig. 1. BinaryVQA images +and questions cover a wide variety of topics and concepts +including drawings, paintings, uncommon views of objects, +hybrid animals, out of context objects and odd scenes (ele- +phant in the room, car in the swimming pool, black sheep +among white sheep), weather conditions, time, interactions +among people, actions (fighting, running, walking, danc- +ing), emotions (sadness, happiness, surprise, anger), counts +and quantity, gender, age, race, gaze direction, object mate- +rials, objects in the mirror, body parts (e.g. whether mouth +or eyes are open, whether teeth are visible), animals, fruits, +clothing (T-shirt, long sleeve, pants), shadow, color, crowd, +clouds, tattoos, camouflage, illusions, non-existing objects, +and logical reasoning. +In formulating the questions, we tried to remove any am- +biguity (e.g. in giving addresses relative to the image, ob- +jects, people in the scene, or image viewer; left side of the +rightmost person; left of the image). When only some peo- +ple in the image (e.g. standing ones) are doing an action, we +did not ask “Are these people doing X”. Instead, we asked +“Are the standing people in this image doing X”. + +Figure 2. Left: Distribution of questions in our dataset by their first three words. The ordering of the words starts towards the center and +radiates outwards. The arc length is proportional to the number of questions containing the word. Right: Venn-style word clouds of words +in the questions. The most frequent word is ‘person’ indicating that questions are often about people in the images. +Figure 3. BinaryVQA dataset statistics. Left: Distribution of the number of questions and its breakdown on positive and negative answers. +Half of the images have more than 7 questions. Middle: Ratio of positive to all questions. On average images contain more positive +questions than negative ones. Right: Distribution of question length. Half of the questions have length greater than five. +Some questions test whether models can tell the type of +the image (e.g. “Is this a drawing?” and “Is this a paint- +ing?”) and whether they can answer questions over different +types of images (e.g. drawings, paintings, cartoons, clip art, +black and white images). Some questions ask about the text, +for example “Is there text?”, “Is the word X written some- +where in this image?”, “Is the text written in English?”, “Is +the number 53813 written somewhere in the image?”. Ex- +ternal knowledge and common sense are needed to answer +some questions (e.g. ”Is this a map of Japan?, “Is this per- +son a celebrity?”). In order to further test the spatial under- +standing of the models, we placed a blue rectangle around +some objects in the image and targeted the questions only +on those regions (See Fig. 1). An example question is “Is +the spatula inside the blue rectangle blue?”. To test the con- +sistency of models and see whether they truly understand +the image, for some images we include questions that con- +tradict each other (e.g. “Is the boy standing?” vs “Is the boy +sitting?”). Some other sample questions are “Is the whole +body of the person visible?”, “Is she holding a wine in her +left hand?”, “Are some birds printed on her skirt?”, “Is her +right hand in her right pocket?”, “Is the person on the left +taller?”, “Is anyone looking at the camera?”, Is this per- +son an adult?”, “Is the sky clear?”, “Are his feet touching +the ground?”, “Are there more X objects than Y objects?”, +“Is object X to the left of object Y?”, “Is the person in the +image female?”, and “Is the person opening the door with +his right hand?”. We clustered the questions based on the + +there +people +the +an +te +plaking +ny +these +DO +Does +all +wearing +they +person's +people +Binary +the +Are +VQA +two +/s +person +two +the +there +more +woman +some +anyone +this +man +any +wearing +personwearing +black +nighttime outdoors +camera +open +carrying +people +clock +only +house +sitting +9 background +white man +daytime +6op +hat +building +standing +textfour more +water +shirt +say other +face + scene +like red +bottle +eatingfacing +wall +sky +running object sign +tree +five hair hands sideroad +wine +blue +person +cars +sseb +each +center real someone +inside +all +Woman touching +bird +fish +drawing +table +written +kid visible +two +ground +handeyes +shoes made + mouth +some +Watch +car +person'slooking +women +anyone +painting +holding +trees +imageleft +anything +playing +same +walking +eyeglasses +any +animal +paso +right0.25 +negative (med:3.0, mean: 2.83) +positive (med:4.0, mean: 4.78) +0.20 +all (med:7.0, mean: 7.62) +0.15 +0.10 +0.05 +0.00 +0 +5 +10 +15 +20 +25 +numberofquestions#positives /all answers,median:0.625.mean:0.63 +200 +150 +100 +50 +0 +0.2 +0.4 +0.6 +0.8 +1.0 +ratio# words in question, median: 5.0, mean: 5.64 +2000 +1500 +Count +1000 +500 +0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +questionlengthterms that appeared in them, as shown in Table 2. For ex- +ample, questions with words gender, man, woman, +female, male, boy, girl address the gender. No- +tice that a question may fall into more than one category. +These categories will be used later to analyze the models. +We did not incorporate any bias towards gender, age, or +race during data collection, and tried to be as inclusive as +possible in gathering images and formulating questions. We +include and balance questions that address different ages +and genders. The age groups are (baby, 26), (kid, +42), (children, 26), (Teenager, 5), (Young, +16), and (old, 12). The gender groups are (woman, +350), (women, 38), (man, 448), and (men, 79). +We did not include any question that ask about race. These +issues are more important to address over large training sets. +This is because sometimes models trained on such datasets +are directly deployed in the real-world. +The BinaryVQA dataset is substantially different from +the VQA v2 validation set (the real images) measured in +terms of the Fr´echet Inception Distance (FID) [15]. The +FID is equal to 50.9 indicating a large distribution shift, and +hence high diversity (using 7K images). To put this number +in perspective, the FID between VQA v2’s validation and +its test set is approximately 23.8. Notice that the lower the +FID, the more similar the two distributions. +4. Analyses and Results +To see how well the state of the art VQA models perform +on our dataset2, we choose the OFA model [39] which is +currently the leading scorer on the VQA v2 test-std dataset3. +It achieves 94.66% accuracy on “yes/no” questions. We also +include a simple baseline model [5,42] to see whether tran- +sitioning from simple to complicated models in VQA has +indeed been meaningful4. To put the results in perspective, +we also ran the Pythia model5. In this section, we focus +on explaining the results using the OFA model. Summary +results for both models are shown in Table 3. +The distribution of model scores on the BinaryVQA +dataset is shown in the left panel of Fig. 4. The average ac- +curacy of the OFA model is 75% which is much higher than +the 62% accuracy of the baseline model. The OFA model, +however, does significantly worse on our dataset than the +VQA v2 dataset (around 20% absolute performance drop). +We attribute this to the more complex nature of the ques- +tions and images in our dataset. Sample predictions of both +models are shown in Fig. 5. +The OFA is able to correctly answer all questions for +160 images (15.6%) whereas the baseline is right for only +50 images (4.8%). The OFA model fails all questions over +2We used a 12 GB NVIDIA Tesla K80 GPU to do the experiments. +3https : / / paperswithcode . com / sota / visual - question - +answering-on-vqa-v2-test-std +4https://github.com/iamaaditya/VQA_Demo.git +5https://github.com/Eurus-Holmes/Pythia-VQA +314 images (30.7%) while the baseline answers all ques- +tions wrong over 673 images (65.7%). +Performance of the models over question types is shown +in the right panel of Fig. 4. The OFA model does better than +the baseline in the majority of the question types. It per- +forms below the baseline model over counting (57.2%), text +(59.7%), and spatial (63%) categories. It does, however, +perform very well on weather (100%), daytime/nighttime +(95.5%) and indoors/outdoors (96%) categories. Surpris- +ingly, the OFA model does relatively well in answering +questions pertaining to gaze direction (68.7%) without us- +ing any ad-hoc module to process faces, eyes, and gaze an- +gles. The same argument holds over the real/drawing cate- +gory (80.6%). We find that models have indeed improved +drastically over the years, but there is still a large gap to +close. Further, our dataset is significantly harder than the +VQA v2 dataset (in “yes/no” questions) making it a great +auxiliary test set to the existing ones. +We found that models perform about the same over +real images, paintings, or drawings. +OFA model scores +∼ 74.12% over the paintings or drawings (568 questions +across 69 drawings/paintings) which is slightly lower than +its 75.47% accuracy on real images (7,232 questions over +955 images). The corresponding numbers for the baseline +model are 60.03% and 63.47%. The OFA model is correct +in answering the counting questions 57.2% of the time. This +model is accurate 69% of the time over the number category +on the VQA v2 dataset. Some difficult questions for the +OFA model are shown in Fig. 6 over different categories. +4.1. Model interpretability +VQA models are very efficient in answering the ques- +tions, but how much do they really understand the images? +Are their answers grounded on image content, or are merely +due to some correlations? Several attempts have been made +to address this (e.g. [2, 12]) and limiting the image area to +a spatial location as is done here (i.e. images containing the +blue rectangles) is one way to do so. In this section, we +propose a new way to interpret the models by masking the +image content and study its effect. To this end, we run the +OpenCV face detector [36] and mask the faces in images. +We then evaluate the OFA model on these images and plot +the performance per category as shown in the left panel of +Fig. 7. Notice that here we limit our analysis to those im- +ages for which at least one face is detected (309 out of 1024 +images). Some question categories that highly depend on +face information such as “gaze direction”, “age”, “gender”, +and “emotions” are severely degraded, which suggests that +models indeed use the right information. Degradation or +enhancement over some categories such as “text” or “ani- +mals” may be partially attributed to the false detections of +the face detector. This, however, needs further investiga- +tion. Note that our masking approach can also be extended +to more common objects. + +Figure 4. Left: Distribution of per image accuracy for models. OFA model is correct ∼ 75% of the time. Middle: Number of questions per +question type. Right: Accuracy per question type for models. OFA model does better than the baseline on most of the question types. +Figure 5. Sample images along with the question, ground truth answer (GT), prediction of the OFA model (M1) and prediction of the +baseline model (M2). See appendix for more examples. + +Is therean umbrella Are there two people Is thelicenseplate +Is this piece of a n +Are there any trees +Are there two people +in the scene? +wearing caps in the +on the left side of +ewspaper? +in the scene? +in the image? +GT:YesM1:YesM2:Yes +image? +the table? +GT:Yes +Ml:Yes +M2:No +GT: Yes M1: Yes +M2:Yes +GT:Yes M1:Yes M2:No +GT: Yes M1: Yes +M2:Yes +GT: No M1:No +M2:Yes +Is there a car in th +Are there six slices +Is there a person in +Is there a male and +e image? +Does anyone have eyels there a manikin i +of bread? +the image? +a female person in t +GT: Yes +Ml:Yes +M2:Yes +glasses? +n the image? +GT: Yes M1: Yes +M2:Yes +GT:No M1:No M2:Yes +he image? +GT:YesM1:Yes +M2:No +GT:Yes M1:No M2:Yes +GT: Yes M1: Yes M2: Yes +Are there two umbrelDoes the text on pil +Are there five peopl +Are there only two b +Are there eleven peo Does the letter say +las in the scene? +low say"mustard"? +e in the image? +irds in the image? +ple in the image? +"U"? +GT:YesMi:YesM2:Yes +GT:Yes M1:Yes M2:No +GT: Yes M1: No M2: No +GT: No M1:No M2:No +GT:YesM1:Yes M2:Yes +GT:Yes M1:Yes M2:No +Is there reflection +Is the pillow white? +Are the three people +Is the left bird try +Are more than two pels the letter litten +in the water? +in front wearing ey +ing to grab a piece +ople wearing suits? +GT:Yes M1:Yes M2:Yes +GT:No +M1:NoM2:Yes +eglasses? +of pretzel? +GT:NoM1:Yes M2:Yes +GT:Yes M1:No M2:Yes +GT: Yes M1:Yes M2:Yes +GT:YesM1:YesM2:No +Is this face of ape +Is there a person in +Is there a black ted +Is there a toaster i +Does the person haveIs there a flying va +rson? +the image? +dy bear attached +to +n +the image? +a beard? +n in the image? +GT:NoM1:YesM2:No +GT:NoM1:No M2:Yes +the car door? +GT: Yes +M1:Yes +M2:Yes +GT:YesM1:Yes +M2:Yes +GT:Yes M1:No M2:No +GT:YesM1:YesM2:Yes +Does this bell peppe +Is there an object h +Is there a mirror in +Is the person riding +Is the van on the gr +r look like a face? +anging from the ceil +Is there a mirror in +the image? +his bike? +ound? +GT:Yes M1:Yes +M2:Yes +ing? +the image? +GT: Yes M1: Yes +M2:Yes +GT:NoM1:No +M2:No +GT: No +M1:YesM2:No +GT:Yes +M1:YesM2:Yes +GT:Yes M1:Yes +M2:Yesaccperimage +Baseline:median:0.625,mean:0.62 +OFA:median:0.75.mean:0.75 +300 +OFA +Baseline +200 +Count +100 +0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0#questionsperquestiontype +600 +400 +200 +0# accuracy over questiontypes +counting +text +spatial +body parts +direction +time +gaze direction +fruits +animals +clothing +OFA +age +actions +Baseline +gender +emotions +real/drawing +vegetation +color +sky +indoors/outdoors +daytime/nighttime +weather +0.0 +0.2 +0.4 +0.6 +0.8 +1.0Figure 6. Failure cases of the OFA model over different categories +of the BinaryVQA dataset. +4.2. Impact of question length on accuracy +Questions in VQA datasets have different levels of com- +plexity. +Intuitively, a longer question may be harder to +answer than a short one, since it involves unpacking and +understanding the dependencies among words in the sen- +tences and their corresponding objects in the image. The +right panel of Fig. 7 shows the model accuracy as a func- +tion of question length. Due to rarity, questions longer than +10 words are discarded (only 150 occurrences). As it can be +noticed, accuracy decays as the question length grows. The +mean accuracy of the OFA model over questions less than +8 words is 72.3%. Its accuracy over questions longer than 8 +words (and less than 10) is 51.6%. The corresponding num- +bers for the baseline model in order are 62.3% and 52.8%. +This result corroborates previous findings over the VQA +dataset and shows that models underperform over longer +questions. Since our dataset contains longer questions than +the VQA dataset, it can better test this aspect of models. +4.3. Analysis of “yes” bias in models +VQA datasets usually contain more questions with “yes” +answers than questions with “no” answers. This is partially +due to the tendency of annotators to query the existing con- +tent in images. Consequently, a smart chance model that +often produces positive answers may win over a sophisti- +cated model. One approach to combat this issue, as is done +over the VQA v2 dataset, is to balance the distribution of +positive and negative questions. Here, we introduce a new +score called “ShuffleAcc” to automatically address this. A +subset of 2n questions consisting of n positive and n nega- +tive questions are randomly selected (here n = 2000). The +average model accuracy over m such subsets is then com- +puted (here m = 50). A model that consistently generates +a “yes” (or “no”) answer will achieve 50% accuracy. The +same argument holds for a model that randomly chooses +“yes” 50% of the time. The ShuffleAcc scores of OFA and +baselines models in order are 75% and 62.4% which are +about the same as their performance using the traditional +accuracy score. This entails that these models do not suffer +from inherent biases towards positive answers. +4.4. Sensitivity to spelling and grammar errors +Studies on understanding and evaluating VQA models +have been primarily focused on the visual component of +VQA. Less attention, however, has been paid to diagnosing +errors in the NLP component, in particular the sensitivity of +models to perturbations on asked questions. This is particu- +larly important to study since we know humans are still able +to correctly answer questions even in presence of significant +spelling and grammar mistakes, so long the meaning of the +question remains the same. Here, we study three simple +perturbations that are unlikely to change the answer. +Within-word character swap. Here, we first randomly +select a word (with length > 3) in the question. +Next, +we randomly choose two characters in this word and swap +them. For example, the question “Is there a person in the +image?” will turn into “Is there a peosrn in the image?”. +We then evaluate the OFA model by varying the number of +words, from 1 to 3, for which we swap two characters. OFA +accuracy drops to 61.4% with swap in one word, 53.5% +with swaps in two words, and 49.1% with swaps in three +words. These results clearly show that spelling errors dras- +tically hinder the models. Humans often do not notice these +changes during reading. +To test whether this result generalizes to other datasets, +we repeated these experiments over the VQA-v2 test set. +The accuracy of the OFA model drops to 91.7%. This num- +ber drops to 84.7% with swap in one word, 77.3% with +swaps in two words, and 65.5% with swaps in three words. +Similar observations are made for the baseline model. + +Arethetwopeopleinfrontfighting?GT:Yes,Pred:No +ActionsArethesepeoplerunning?GT:Yes,Pred:NoIs thisakid?GT:Yes,Pred:No +AgeIsthisanoldperson?GT:No,Pred:YesIstheonion on person's left hand?GT:Yes,Pred:No +BodyPartsIsthereahandintheimage?GT:Yes,Pred:NoIs this person wearing a suit?GT:No,Pred:Yes +ClothingIsthispersonwearingearrings?GT:Yes,Pred:NaIsthisared fish?GT:No,Pred:Yes +ColorIstheswan'sheadred?GT:Yes,Pred:NoAremorethantwopeoplewearing suits?GT:No,Pred:Yes +CountingAre thereten bananasintheimage?GT:No, Pred:YesFigure 7. Left: Performance of the OFA with and without faces masked. Sample images with faces masked are also shown. Right: +Performance of the OFA model as a function of question length. +Omission of the articles. Here, all the articles (“the”, +“a”, “an”) are removed from the question. For instance, the +question “Is the person on the right holding a camera?” will +be converted to “Is person on right holding camera?”. The +performance of the OFA model drops to 73.8% indicating +that this model, similar to humans, is robust to the omission +of the articles. +Negating the question. Questions in the BinaryVQA +dataset are formulated positively without using the word +“not”. +Logically, if the question is negated the answer +should also be negated6 For example, if the answer to the +question “Is there a firefighter on the crane?” is “yes”, then +the answer to the question “Is there not a firefighter on the +crane?” should be “no”. For this analysis, we focus only +on “Is there” type questions. Out of 1,841 such questions, +the OFA model maintained its decision in 738 cases when +the question was negated. This amounts to about 40% of +the cases, which is far above 0%. Ideally, the model should +always reverse its decision. +4.5. Ablation analyses +Following our interpretability analysis above, here we +conduct two analyses which can be considered as sanity +checks or baselines for models. Models can be right for +wrong reasons, and vice versa. In the first analysis, we ask +all the questions over a black image or a white noise image. +The OFA model performs well below chance, about 36.4% +and 36.89% over these images, respectively. This indicates +that this model indeed requires the image to produce the +right answer. +The second analysis investigates whether a model +can consistently produce the “no” answer to questions +for which we know the answer is surely “no”. +We +asked 15 questions in the form of “Is there a/an X in +the image?” +where X represents one of the following +6Of course there are exceptions in the conversational language, +e.g. Isn’t there a person in the room? Answer: No! (assuming there are +no people in the room). +Model +Avg Acc. ShuffledAcc Char Swap +Article +Question∗ +Acc on +(one word) Omission Negation (%) +VQA v2+ +Baseline +62.5 +62.4 +51.5 +59.3 +35 +80.5 +OFA +75 +75 +61.4 +73.8 +40 +94.66 +Pythia +72.1 +72.2 +58.8 +69.4 +46 +86.7† +Table 3. Summary of model performance on BinaryVQA dataset. +∗ = Percentage of questions for which the model retained its answer after negation. ++ = Human perf. is about 95.48 from https://visualqa.org/roe.html +† = Pythia v0.1 the winning entry in 2018 VQA benchmark https://visualqa. +org/roe_2018.html +objects +‘white orange’, ‘dragon’, ‘blue +horse’, ‘backgammon board’, ‘parrot’, +‘boxer dog’, ‘ostrich’, ‘dinosaur +egg’, ‘galaxy’, ‘mermaid’, ‘telescope’, +‘unicorn’, ‘centipede’, ‘yellow cow’, +‘yeti’ over all the 1024 images. +The mean accuracy +of OFA model across all 15 × 1024 questions is 93.1% +using original images. The breakdown per each of these +questions is shown in Appendix C. Interestingly, when we +asked these questions on white noise images, the accuracy +jumped to 100%. These results again demonstrate that the +OFA model indeed highly relies on the image content. +5. Discussion and Conclusion +Understanding complex questions in VQA is a big chal- +lenge, so is the understanding of complex scenes. +Our +dataset is better suited to address the latter, whereas other +datasets can address the former. It can be used to test mod- +els that already perform above 95% on binary questions of +VQA-v2 dataset. Our dataset contains a lot of questions +which are really challenging and need close examination +of the image to be answered. +Such questions ask about +non-standard objects, surreal imagery, and/or other oddities +(e.g. an eagle with a banana for a beak, water spout wear- +ing sneakers, an odd clothespin-like object on one side and +spoon on the other, a face with multiple pairs of eyes). +We share a zip file containing images, questions, meta- +data, and detailed documentation. 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Sample images along with the question, ground truth answer (GT), prediction of the OFA model (M1) and prediction of the +baseline model (M2). + +Is there an umbrella Are there two people Is the license plate +Is this piece of a n +Are there any trees +Are there two people +in the scene? +wearing caps in the +on the left side of +ewspaper? +in the scene? +in the image? +GT: Yes M1: Yes M2: Yes +image? +the table? +GT: Yes M1: Yes M2: No +GT: Yes M1: Yes M2: Yes +GT: Yes M1: Yes M2: No +Is there a car in th +GT:YesM1:YesM2:Yes +GT:NoM1:NoM2:Yes +Are there six slices +Is there a person in Is there a male and +e image? +Does anyone have eyels there a manikin i +of bread? +the image? +a female person in t +GT: Yes M1: Yes M2: Yes +glasses? +n the image? +GT: Yes M1: Yes M2: Yes +GT: No M1: No M2: Yes +he image? +GT: Yes M1: Yes M2: No +GT:YesM1:NoM2:Yes +GT: Yes M1: Yes M2: Yes +Are there two umbrelDoes the text on pil +Are there five peopl +Are there only two b Are there eleven peo Does the letter say +las in the scene? +low say "mustard"? +e in the image? +irds in the image? +ple in the image? +"U"? +GT: Yes M1: Yes M2: Yes +GT: Yes M1: Yes M2: No +GT: Yes M1: No M2: No +GT: No M1: No M2: No +GT: Yes M1: Yes M2: Yes +GT: Yes M1: Yes M2: No +Is there reflection +Is the pillow white? +Are the three people Is the left bird try +Are more than two pels the letter litten +in the water? +in front wearing ey +ing to grab a piece +ople wearing suits? +GT: Yes M1: Yes M2: Yes +GT: No M1: No M2: Yes +eglasses? +of pretzel? +GT: No M1: Yes M2: Yes +GT: Yes M1: No M2: Yes +GT: Yes M1: Yes M2: Yes +GT: Yes M1: Yes M2: No +Is this face of a pe +Is there a person in +Is there a black ted +Is there a toaster i +Does the person have Is there a flying va +rson? +the image? +dy bear attached to +n the image? +a beard? +n in the image? +GT: No M1: Yes M2: No +GT: No M1: No M2: Yes +the car door? +GT: Yes M1: Yes M2: Yes +GT: Yes M1: Yes M2: Yes +GT: Yes M1: No M2: No +Does this bell peppe + Is there an object h +GT:Yes M1:YesM2:Yes +Is there a mirror in +Is the person riding +Is the van on the gr +r look like a face? +anging from the ceil +l Is there a mirror in +the image? +his bike? +ipuno +GT: Yes M1: Yes M2: Yes +ing? +the image? +GT: Yes M1: Yes M2: Yes +GT: No M1: No M2: No +GT: No M1: Yes M2: No +GT: Yes M1: Yes M2: Yes +GT: Yes M1: Yes M2: Yes +Is this a giraffe? +Does this object loo +Is this a picture of +Is there a car in th +Is the person touchi Are these kids playi +GT: Yes M1: Yes M2: No +k like a computer mo a tattoo? +e image? +ng the ground? +ng basketball? +use? +GT: Yes M1: Yes M2: No +GT: Yes M1: Yes M2: Yes +GT: No M1: No M2: No +GT: Yes M1: Yes M2: No +Are the eyes of gira +ffe visible? +Is the car fully vis +Is the person touchi Are there only three +GT: Yes M1: No M2: No +Does this appear to +ns visible? +ible? +ng the hoop? +kids in the image? +be a happy computer GT: No M1: No M2: Yes +GT: No M1: No M2: No +GT: Yes M1: Yes M2: Yes +GT: Yes M1: Yes M2: No +mouse? +GT: Yes M1: No M2: No +Are there three peop Are these people shi A +Are these people cro Does this picture ap. +Is this person looki +Is this a drawing of +le in the image? +rtless? +ssing the street? +pear to taken from i ng at the camera? +face? +GT: No M1: Yes M2: No +GT: Yes M1: Yes M2: Yes +GT: Yes M1: Yes M2: Yes +nside a car? +GT: Yes M1: Yes M2: No +GT: Yes M1: Yes M2: No +Is the laptop in the +Are these people wea Do two people carryi +GT:YesM1:Yes M2:Yes +Is this picture take +Is this face drawing +image turned on? +ring T shirts? +ng umbrellas and theIs there a toy insid +n in a kitchen? +bald? +GT: Yes M1: Yes M2: No +GT: No M1: No M2: No +other two people ho e the car? +GT: Yes M1: Yes M2: Yes +GT: Yes M1: Yes M2: No +lding papers? +GT: Yes M1: Yes M2: YesFigure 9. Sample images along with the question, ground truth answer (GT), prediction of the OFA model (M1) and prediction of the +baseline model (M2). + +Are there animals in Is there a person in +Is there a person in +Is this a castle? +Is this a huge sewin +Does this look like +this image? +the image? +the image? +GT:Yes Ml:YesM2:Yes +g machine? +a computer mouse? +GT: Yes M1: No M2: Yes +GT: Yes M1: Yes M2: Yes +GT: Yes M1: Yes M2: Yes +GT:Yes M1:Yes M2:Yes +GT: Yes M1: No M2: Yes +Are there a person c +Are these animals re Is there a water bot +Is the person fully +arved on the tree? +Does this object see +Does this look like +al? +tle in the image? +visible? +GT: Yes M1: No M2: Yes +m like a factory or +a hand? +GT: No M1: No M2: No +GT: Yes M1: Yes M2: Yes +GT: No M1: Yes M2: No +a house? +GT:Yes M1:Yes M2:No +GT: Yes M1: Factory M2: We +11 +12 +10 +5 +Is this a giant bear +Is this a bottle? +Are all numbers fromAre these slippers? +Is this closed up of +Is there a dog in th +GT:YesMi:NoM2:No +1 to 15 present in +GT:YesM1:NoM2:No +a pencil? +e image? +GT:Yes M1:Yes M2: No +this image? +GT: Yes M1: No M2: Yes +Is the bottle wrappe +GT: Yes M1: Yes M2: Yes +GT:No M1:Yes M2:No +Are these slippers m +Is this a real bear? +d? +ade from bread? +Is the tip of the pe +Are the colored dots +GT: Yes M1: No M2: No +Is there a"o"in th +GT:Yes M1:Yes M2: No +ncil touching the su +in the image? +GT: No M1: No M2: No +is image? +rface? +GT: Yes M1: Yes M2: No +GT:YesM1:YesM2:Yes +GT: No M1: No M2: Yes +Has there been an in Are these bagels? +Are these birds? +Can this image be co Is there a sign with +Is there a huge roos +cident in this image +GT:YesMi:No M2:No +GT:YesM1:YesM2:No +nsidered a piece of +"Originals"written +ter in this image? +art? +on it? +GT: Yes M1: Yes M2: No +GT:Yes M1:Yes M2:Yes +Are there 8 bagels s +M1:Yes M2:Yes +GT: Yes M1: No M2: Yes +hown here? +pen? +Is this a red rooste +Is anyone being inju +GT: No M1: No M2: No +GT: Yes M1: Yes M2: No +Are there two pears +Is the person on the r? +red? +on the floor? +right with the bike +GT: No M1: No M2: No +GT: No M1: Yes M2: No +GT: Yes M1: No M2: Yes +wearing a helmet? +GT: YesMi:Yes M2: Yes +Is this a kiwi? +Is this a picture of +Is this giant fish? +Is this a handycam? +Is there a single ca +Are these earrings? +GT: Yes M1: Yes M2: No +a banana attached t GT: Yes Mi: Yes M2: No +GT:Yes M1:NoM2:No +r in the image? +GT: Yes M1: Yes M2: Yes +Is this a regular ki +o a snake? +GT: No M1: No M2: Yes +GT:Yes M1: Yes M2:Yes +Is this a shoe? +Is this handycam mac +Do these earrings lo +wi? +GT: No M1: No M2: Yes +e of plastic? +Are the cars covered ok like butterfly? +GT: No M1: No M2: No +Is there more than a +GT: No M1: No M2: No +by snow? +GT: Yes M1: Yes M2: No +banana in the image +GT: Yes M1: Yes M2:Yes +GT: No M1: No M2: Yes +Is this a rope coil +Is this house upside +Is this a sandwich? +Is there a chimpanze Is this a hammer? +Is this a hammer? +that looks like coco +down? +GT:YesM1:YesM2:Yes +e in this image? +GT: YesMi:Yes M2:Yes +GT: No M1: Yes M2: Yes +nut? +GT: Yes M1: Yes M2: Yes +GT: Yes M1: No M2: Yes +GT:Yes M1:Yes M2:Yes +Is this sandwich mad +Is this a funny obje +Is this a pipe wrenc +Is this daytime? +e of logo? +Is the chimpanzee lo ct? +h? +Is this a real cocon +GT:Yes M1:Yes M2:Yes +GT: Yes M1: No M2: No +oking at the person? GT:Yes Mi:No M2:Yes +GT: Yes M1: Yes M2: No +ut? +GT: No M1: Yes M2: No +GT: No M1: No M2: NoFigure 10. Sample images along with the question, ground truth answer (GT), prediction of the OFA model (M1) and prediction of the +baseline model (M2). + +Does this pigeon loo +Is this an animal? +Is this a binder cli +Is this a small golf +Is this a wooden obj +Is this a giant rat? +k like a teapot? +p? +ball? +ect? +GT:Yes M1:Yes M2: Yes +GT: Yes M1: Yes M2: No +GT: No M1: No M2: No +GT:Yes M1:Yes M2:Yes +GT: Yes M1: No M2: No +Is this a sheep? +Is there a nest in t +GT: Yes Mi:NoM2:No +Is this a dragonfly? +Is this a golf ball? +Is there a face on t +Is this inside a roo +he image? +he wood? +m? +GT: Yes M1:Yes M2: Yes +GT:Yes M1:Yes M2: No +GT:Yes M1:Yes M2:Yes +GT: Yes M1: Yes M2: Yes +GT:Yes M1:No M2:No +O +Is this a watering c +Is this a duck? +Is there a person in +Is this mousing bein Does this cloud look +Is this person sitti +an? +the image? +g forced to take a b +like a seahorse? +ng on the road? +GT: Yes M1:Yes M2: Yes +Is there a hand in t +GT: Yes M1: Yes M2: Yes +ath? +GT: Yes M1: No M2: No +GT: Yes M1: Yes M2: No +GT:Yes M1:Yes M2:Yes +Is this watering can +he image? +Is this person weari +Is this daytime? +Is there a TV on the +made of metal? +GT: Yes M1: No M2: Yes +ng a tie? +Isthemouth scratch GT:Yes Mi:Yes M2:Yes +image? +GT: No M1:Yes M2: Yes +GT:Yes M1: Yes M2: Yes +ing the wall? +GT: Yes M1: Yes M2: Yes +GT: Yes M1: Yes M2: No +Is this a horse? +Does this object loo +Is there a car in th +Does this rat seem t Is this a snake comi Are these mens paint +GT:Yes M1: No M2: No +k like a mosquito? +e image? +o be working out? +ng out of a tube? +ing tennis balls? +Does this horse look +GT: Yes M1:Yes M2:Yes +GT: No M1: No M2: Yes +GT: Yes M1: No M2: No +GT: Yes M1: No M2: No +GT: No M1: No M2: No +like a rock? +Does this object loo +Is this a huge clip? +Is the rat looking t +Is the snake yellow? +Is there a mirror in +GT: Yes M1: Yes M2: Yes +k like a dragonfly? +owards the camera? +the image? +GT: No M1: Yes M2: No +GT: Yes M1: Yes M2: Yes +GT:Yes M1:Yes M2: No +GT: No M1: No M2: No +GT: No M1: No M2: Yes +Is this objectmade +Does this object loo +Is this a car? +Is this a strange st +Are there three isla +Is this a spanner? +of only two logos? +k like a chimpanzee +GT:YesM1:YesM2:No +rawberry? +nds in the image? +GT: Yes M1: Yes M2: Yes +GT: No M1: No M2: No +or a monkey? +GT: Yes M1: Yes M2: No +GT: Yes M1: Yes M2: Yes +GT: Yes Mi: Minecraft M2: +Is there any number +Is this a depiction +Is there a yellow lo +in the image? +Is the strawberry in Do all these islands +of a fox? +go piece in the imag Is this animal stand +GT:No M1: No M2:No +person's hand? +look like fish? +GT: Yes M1: Yes M2: No +e? +ing up? +GT: Yes M1: No M2: Yes +GT: No M1: No M2: No +GT:Yes M1:Yes M2:Yes +Is this a depiction +Does this object loo +Is this a piece of c +Is there an animal i +Is this a shoe that +Is there a butterfly +of a dog? +k like a turtle? +ake? +n this image? +also looks like a ho +in the image? +GT: No M1: No M2: No +GT: Yes M1: Yes M2: No +GT: Yes M1: Yes M2: No +GT: Yes M1: No M2:Yes +rse? +GT: Yes M1: Yes M2: Yes +GT: Yes M1: Yes M2:Yes +Is this a depiction +Is the turtle made o Is there a candle be Is there a dog in th +Does this butterfly +of an eagle? +f an orange? +sides the cake? +is image? +Is the shoe made of +look like an elephan +GT: Yes M1:Yes M2: No +GT:Yes M1: Yes M2: No +GT: No M1: No M2: Yes +GT:Yes M1: No M2: Yes +metal? +GT: Yes M1: Yes M2: Yes +GT: Yes M1: No M2: YesFigure 11. Sample images along with the question, ground truth answer (GT), prediction of the OFA model (M1) and prediction of the +baseline model (M2). + +Is this a picture of +Is this a fish? +Is there fire in thi +Is there a fork in t +Are these zebras? +Is the person holdin +a wall? +s image? +his image? +g a garlic? +GT: Yes M1: Yes M2: Yes +GT:YesMl:YesM2:Yes +GT: No M1: No M2: Yes +GT: No M1: No M2: No +Is this a hen? +Does this image also +Is there a cigar in +GT: No M1: No M2: No +Is this image taken +Is this drawing of a +show a lion face? +Is the person holdin +the image? +daytime? +spoon? +GT: Yes M1: No M2: No +g an onion? +GT: Yes M1: No M2: Yes +GT: No M1: No M2: Yes +GT: Yes M1: No M2: No +GT: Yes M1: Yes M2: No +Does this object loo +Is this an eagle? +Does this object loo +Is this a fish? +Is this a chair? +Is this a strange fi +k like an animal? +GT:YesMi:NoM2:No +k like an ax? +GT:YesM1:NoM2:No +GT: YesMi:Yes M2:No +sh? +GT: Yes M1: Yes M2: Yes +GT: Yes M1: Yes M2: No +GT: Yes M1: Yes M2: Yes +Is the eagle's neck +Does this fish look +Does this object als +Is this object made +yellow? +Is this a megaphone? like a person? +o look like a coffee +Is this fish in the +of plastic? +GT: Yes M1: Yes M2: No +GT: Yes M1: No M2: No +saucer? +water? +GT: No M1: No M2: No +GT:Yes M1:Yes M2: No +GT: Yes M1: Yes M2: No +GT: No M1: No M2: No +Is there a book in t +Does this cloud look +Does this object loo +Is this a person? +Is there a rabbit? +Is this person weari +his image? +an animal? +k like a hourglass? +GT:Yes Ml:Yes M2: Yes +GT: Yes Mi:NoM2:No +ng shoes? +GT: Yes M1:Yes M2: Yes +GT:Yes M1: No M2: No +GT: Yes M1: Yes M2: Yes +GT: Yes M1: No M2: No +Is this a real panda +Is the rabbit runnin +Is there an object l +Does this cloud look +Are there pills insi +g? +Is she running? +ooking like a fox in +a dog? +de the inside the ho GT: No Mi: No M2: No +GT: No M1: No M2: No +GT: Yes M1: No M2: No +this image? +GT:Yes M1: No M2: No +urglass? +GT: Yes MI: Yes M2: Yes +GT: Yes M1: No M2: No +8008 +Is this an airplane? +Does this object loo +Does this image look Is this a hen? +Is there any person +Is this a tree? +k like a cherry? +like a fight scene? +GT:Yes M1:YesM2:No +in the image? +GT:Yes M1:Yes M2: No +GT: Yes M1:Yes M2: No +GT: Yes M1: No M2: No +GT: No M1: No M2: No +GT:Yes Mi:No M2:No +Is this hen made of +Is there a butterfly +Are there more than Is there only one di +egg shells? +Is this a model of c +on the tree? +two coffee cups? +ce in the image? +Is there a pickle in +GT:Yes M1:Yes M2: No +olosseum? +GT: Yes M1: No M2: Yes +GT: No M1: No M2: No +GT: No M1: No M2: No +the image? +GT: Yes M1:Yes M2:No +GT: Yes M1: Yes M2: Yes +Is this butterfly? +Is there a dog in th +Is this a dog? +Is this a sandwich? +Is the a picture of +Is this a deer? +GT: Yes M1: No M2: No +e image? +the cloudy sky? +Is this a toy? +GT: Yes M1: No M2: Yes +GT: Yes M1: Yes M2: No +Is this picture of a +Is this sandwich mad +Is the whole body of +GT: Yes M1:Yes M2: Yes +Is there a mountain? real dog? +e of bread and paper Does the cloud in th +the deer visible? +GT: Yes M1: Yes M2: No +S +e center look a dog? GT:No Mi:No M2:No +GT: Yes M1: Yes M2: Yes +GT: Yes M1: Yes M2: No +GT: No M1: No M2: NoFigure 12. Sample images along with the question, ground truth answer (GT), prediction of the OFA model (M1) and prediction of the +baseline model (M2). + +Are these people fig +Is there a crab? +Is there a person in +Is there a text in t +Is this a gym? +Is this a camera? +hting? +GT:Yes Mi:Yes M2:Yes +the image? +he image saying "LIL GT: Yes MI: No M2: No +GT: Yes M1: Yes M2: Yes +GT: No M1: No M2: No +GT: Yes M1: No M2: Yes +Y"? +Is there water? +GT:Yes M1:No M2:Yes +Is there a person on Is this a real camer +Are all of these peo +GT:YesMi:YesM2:Yes +Is there an ice crea +the treadmill? +a? +ple standing? +m? +Is there an adult in +GT:Yes M1:No M2:Yes +GT: No M1: No M2: No +GT: No M1: No M2: No +GT:Yes M1: No M2:No +this image? +GT: No M1: No M2: No +Are there some animels this a deer? +Is there a mirror onI +Is this a monkey? +Is there a chair in +Does this object loo +Is in this scene? +GT: Yes M1: Yes M2: No +the wall? +the image? +k like a boat? +GT: Yes M1: Yes M2: Yes +GT: Yes M1: Yes M2: Yes +GT: Yes M1:Yes M2:Yes +GT: Yes M1: No M2: Yes +Is this nighttime? +Is this a dog? +Isthere onlyone anGT:No M1:No M2:No +Is there a reflectio +GT: No M1: No M2: Yes +Is this a bird? +Is this a melon crus +imal? +n on the mirror? +GT: No M1: No M2: No +t? +GT: No M1: No M2: No +GT: Yes M1: Yes M2: Yes +GT:Yes M1:Yes M2: No +Is this a dog? +Is there an elephant Is this a house? +Is there a huge golf +Is this a stool? +Is this a pair of sc +GT: No M1: No M2: No +in the image? +GT:YesM1:YesM2:Yes +ball in the image? +issors? +GT: Yes M1: Yes M2: No +Does this house look +GT: Yes M1: Yes M2: Yes +GT: Yes M1: Yes M2: Yes +Is this a rabbit? +Is this a person? +GT: Yes M1: No M2: No +Is there a pillow on +like a fish? +Is this daytime? +GT: No M1: Yes M2: Yes +Are two people kissi +the chair? +GT: Yes M1: Yes M2: Yes +GT:Yes M1:Yes M2: Yes +ng in this image? +GT:Yes M1:Yes M2:Yes +GT: Yes M1: No M2: No +Is this a paint brus +Is this a hand? +Is this a banana? +Is there a dog in th +Is there a text in t +Is this a chair? +h? +GT:YesM1:YesM2:No +e image? +his image? +GT: Yes M1: Yes M2: Yes +GT: Yes M1: Yes M2: No +GT: Yes M1: No M2: Yes +GT: Yes M1: Yes M2: Yes +Is this a giraffe? +Is this a real banan +Does this chair look +Is there a hole in t +GT:YesMi:YesM2:Yes +a? +Is the there a blank +Is there a road in t +like a dog? +he paint brush? +GT: No M1: Yes M2: No +et in the image? +he image? +GT: No M1: Yes M2: Yes +GT: Yes M1:Yes M2: Yes +GT:Yes M1:Yes M2:Yes +GT: Yes M1: Yes M2:Yes +Is this a chair? +Does this object loo +Is there a fish in t +Is there a fish in t +Is there a fish in t +Is there a fish in t +GT:Yes M1:Yes M2: No +k like a dress? +his image? +his image? +his image? +his image? +Can someone sit on t +GT: Yes M1: Yes M2: No +GT: Yes M1: No M2: No +GT: Yes M1: No M2: No +GT: Yes M1: Yes M2: No +GT: Yes Mi: Yes M2: No +his object comfortab Is this object hangi +Is this a white fish +Is this a white fish +Is this a red fish? +Is there only one fi +ly? +ng? +GT: No M1: Yes M2: No +sh in this image? +GT: Yes M1: No M2: Yes +GT: Yes M1: Yes M2: Yes +GT: No M1: NoM2: No +GT: No M1: No M2: No +GT: No M1: No M2: NoFigure 13. Sample images along with the question, ground truth answer (GT), prediction of the OFA model (M1) and prediction of the +baseline model (M2). + +Is there a fish in t +Are there two forks? Is this a globe? +Does this object loo +Is the fish made of +Does the letter look +his image? +GT:YesM1:YesM2:No +k like a pair of sho +stone? +like"w"? +GT:Yes M1:Yes M2: No +GT: Yes M1: Yes M2: No +Is this globe made o +es? +GT:Yes M1:Yes M2:Yes +GT: Yes M1: Yes M2: Yes +GT:Yes M1:Yes M2:No +Is this a big fish? +Are the shadows of f f an orange? +Is there a kid on th +Is the French fries +GT: Yes M1: Yes M2: No +orks visible? +GT: Yes M1: Yes M2: Yes +Is this object yello +e left side of the f +in the image? +GT: Yes M1: Yes M2: Yes +w? +ish? +GT: Yes M1: Yes M2: Yes +GT: No M1: No M2: Yes +Is there a frog in t +Is there milk? +Are there some coins Is this an apple? +Are these people wea Is this a toy? +his image? +GT:Yes Mi:Yes M2:Yes +in this image? +GT: Yes M1: Yes M2: No +ring the same shoes? GT: Yes M1: Yes M2: Yes +GT: Yes M1: No M2: Yes +Is there some milk s +GT: Yes M1: Yes M2: Yes +Is this apple uncut? +GT:Yes M1:Yes M2:No +Is this a teddy bear +Does the frog look1 +pilt? +Are there six elepha +ike leaves? +GT: Yes M1: Yes M2: No +nts in this image? +GT: No M1: Yes M2: No +Are these shoes red? GT: Yes Mi: Yes M2: Yes +GT:Yes M1:No M2: No +GT: Yes M1: No M2: No +GT:YesMi:YesM2:Yes +Is this a shadow? +Is this a wine opene +Is this a giant bird +Are these slippers? +Is this an eagle? +Is there a car in th +GT: Yes M1: Yes M2: Yes +r? +GT: Yes M1:Yes M2:No +e pool? +GT: Yes M1: No M2: Yes +GT: Yes M1: Yes M2: No +GT: Yes M1: Yes M2: No +Is this shadow of a +Do these slippers ha Is the eagle's beak +person? +Does this object loo +Is this a white bird +ve a butterfly patte +made of a banana? +Is the car fully und +GT: Yes M1:Yes M2: No +k like a person? +rn? +GT: Yes M1: Yes M2: Yes +erwater? +GT:Yes M1: No M2: No +GT: Yes M1: No M2: No +GT:Yes M1:Yes M2: No +GT: Yes M1: No M2: No +Is this a hand? +Are these earrings? +Are these bananas? +Does this object loo +Is there a lemon in +Are these eyeglasses +GT: Yes M1: Yes M2: Yes +GT:YesM1:YesM2:No +k like a bread? +the image? +GT: Yes M1: Yes M2: No +GT: Yes M1: Yes M2: Yes +GT: Yes M1: Yes M2: No +Is this hand holding +Are these earrings m Does these bananas l +a person? +ade of lego? +ook like ducks? +Is the bread sliced? +Is the lemon sliced? +Is there a shadow in +GT: Yes M1: Yes M2: No +GT: Yes M1: Yes M2: Yes +GT: Yes M1: Yes M2: No +the image? +GT: Yes M1: No M2: No +GT: No M1: No M2: No +GT: Yes M1: Yes M2: Yes +Is there a bird in t +Is this an strange e +Does this object loo +Is this a snake? +Is this a rooster? +Is there an apple an +he image? +yeglass? +k like an octopus? +GT:YesM1:NoM2:No +GT:YesMi:NoM2:No +d a pear in this ima +GT: Yes M1: No M2: Yes +GT:Yes M1: No M2: No +GT: Yes M1: Yes M2: No +Is the snake's head +Is there a black leg +ge? +GT:Yes M1: No M2:Yes +Is there a fruit in +Does this eyeglass h Is the octopus holdi +i on the right side of + o piece in the image +the image? +ave two lenses? +ng a cup? +the image? +Are these fruits mad +GT: Yes M1:Yes M2:Yes +GT: No M1: Yes M2: Yes +GT: Yes M1: No M2: Yes +GT: Yes M1: No M2: No +GT: No M1: No M2: Yes +e of cloth? +GT: Yes M1: Yes M2: NoB. Samples images from the BinaryVQA +dataset + +Figure 14. Sample clock images with Roman numerals (left) and English numerals (right). +Figure 15. Sample clock images with (left) and without eye glasses (right). + +2CN +rosesFigure 16. Additional images from the BinaryVQA dataset. + +art +materialperception +sentiment +pop out +illusions +transformations +face illusions +counting +fake vs. real +drawings +perceptual grouping +I scene congruency +humor +MANAC! +rC. Breakdown over the words in ablation study +The accuracy of the OFA model over questions asking +about existence of non-existing objects in the image. +Object +Accuracy over the original image +Accuracy over the white noise image +white orange +0.702 +1 +dragon +0.916 +1 +blue horse +0.958 +1 +backgammon board +0.953 +1 +parrot +0.983 +1 +boxer dog +0.965 +1 +ostrich +0.990 +1 +dinosaur egg +0.985 +1 +galaxy +0.863 +1 +mermaid +0.956 +1 +telescope +0.900 +1 +unicorn +0.983 +1 +centipede +0.981 +1 +yellow cow +0.933 +1 +yeti +0.891 +1 +Table 4. performance of the OFA model over questions of the type +“Is there a/an X in the image? Replace X with the object name in +the first column. + +D. Data collection +We adopt the following high-level process to collect the +images and (question,answer) pairs. +First, we generated +some phrases and then searched Flickr or Google search to +find matching images. We limited the search results to only +those images that had the creative commons licences. Some +sample search queries include: “A couple of kids watch- +ing TV in a room while sitting on the floor?”, “A woman +looking at the camera while eating a burger?”, “A couple +of people in a meeting room?”, “Two people fighting”, “A +cat in the clouds”, “A sheep made of lego”, “A man with +blond hair”, etc. We then formulated some questions on +these images along with answers. The (question,answer) +pairs were presented to three AMT workers for further ver- +ification. Few questions for which AMT workers did not +agree were then corrected. +Our AMT interface for collecting the verification of our +answers to the questions. Workers were paid 25 cents per +question. The experiment took 30 hours per participant. +Figure 17. Our AMT interface for collecting the verification of our +answers to the questions. +We have 17 images (from 0700.jpeg to 0716.jpeg) that +have blue rectangles. 25 questions were asked on these rect- +angles. These questions either asked about an object or a +person inside the rectangle (e.g. Is there a spatula inside the +blue rectangle?) or something about the rectangle itself (Is +the blue rectangle on the bottom right corner of the image?). + +Instructions:Youarepresentedwithan image,a questionaskedaboutit,and ananswerto +the question.Pleaseverifythe answertothe question and choose"Correct"if the answeris +correct and vice versa. +Question:Aremorethantwopeoplewearingsuits? +Answer:No +Correct +Incorrect +prev +nextE. Dataset License +BinaryVQA dataset is free to use only for research and +academic purposes (not commercial). +It is licensed un- +der Creative Commons Attribution 4.0 with three additional +clauses: +1. BinaryVQA may never be used to tune the parameters +of any model. +2. The images containing people should not to be posted +anywhere unless the people in the images are appro- +priately de-identified. Even in this case, written agree- +ment from dataset creators is required. This is to check +whether all the clauses are properly followed. +To stop or limit the misuse of our BinaryVQA by bad ac- +tors, we have made a dataset request form7. We review the +requests that we receive and allow access for a legitimate +use. The dataset we share contains images and questions is +a zip file. The package also contains the detailed documen- +tation with all relevant metadata specified to users. +7https://bit.ly/3bDY0MS + +F. Experimental details and evaluation setup +We have used the validation set of the balanced real scens +from the VQAv2 dataset from https://visualqa. +org/download.html. We are only using the binary +questions. Images are resized and normalized. A question- +mark is added to the questions if it is missing. BOS and +EOS tokens are also added to the question. Model parame- +ters for each of the tested models are listed below. +Parameter settings for VQA baseline: +• VGG 16 model +• 4096 D feature vector for the image representation +• Image size 224 x 224 +• Each word in the question is a Glove vector 300D +OFA model: +• Checkpoint: ofa large 384.pt +• Images are resized to and normalized +• A questionmark is added if missing +• BOS and EOS tokens are added +Pythia model: +• TARGET IMAGE SIZE = [448, 448] +• CHANNEL MEAN = [0.485, 0.456, 0.406] +• CHANNEL STD = [0.229, 0.224, 0.225] + diff --git a/TtFLT4oBgHgl3EQfQS-o/content/tmp_files/load_file.txt b/TtFLT4oBgHgl3EQfQS-o/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d32ad8ced56a1f8e26a36e5be1049fedafcad285 --- /dev/null +++ b/TtFLT4oBgHgl3EQfQS-o/content/tmp_files/load_file.txt @@ -0,0 +1,1442 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf,len=1441 +page_content='BinaryVQA: A Versatile Test Set to Evaluate the Out-of-Distribution Generalization of VQA Models Ali Borji Quintic AI aliborji@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='com Abstract We introduce a new test set for visual question answering (VQA) called BinaryVQA to push the limits of VQA models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Our dataset includes 7,800 questions across 1,024 images and covers a wide variety of objects, topics, and concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' For easy model evaluation, we only consider binary ques- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Questions and answers are formulated and verified carefully and manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Around 63% of the questions have positive answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The median number of questions per im- age and question length are 7 and 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The state of the art OFA model achieves 75% accuracy on BinaryVQA dataset, which is significantly lower than its performance on the VQA v2 test-dev dataset (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='7%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' We also analyze the model behavior along several dimensions including: a) performance over different categories such as text, counting and gaze direction, b) model interpretability, c) the effect of question length on accuracy, d) bias of models towards positive answers and introduction of a new score called the “ShuffleAcc”, and e) sensitivity to spelling and grammar errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Our investigation demonstrates the difficulty of our dataset and shows that it can challenge VQA models for next few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Data and code are publicly available at: DATA and CODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Introduction Visual question answering [5, 10] is a multidisciplinary task at the intersection of computer vision, NLP, knowledge representation, reasoning, common sense knowledge, etce- tra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The goal is to answer a text-based question given an input still image or a video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Recent VQA models are able to answer binary questions above 95% accuracy, which is astonishing considering that in principle, any questions can be asked on an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' At the same time, though, this alarms that perhaps we are not using the test sets that have the right level of difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Us- ing the same test set over the years has the risk of over- fitting, as researchers often tune their models towards the statistics of the test sets (even when the annotations are held hidden).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' To mitigate this issue, it is crucial to have sev- Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Samples from our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Our dataset covers a wide va- riety of concepts including counting, crowd, emotions, drawings, paintings, camouflage, clothing, time, weather, body parts, age, text, gaze direction, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' It also includes questions that address spatial understanding of models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' the blue rectangle in the last image of the 3rd row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' See Appendix A for more examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' eral versatile independent test sets to evaluate models and to track the progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' While several test sets are available for problems such as image classification (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' [6, 14, 28]) and object detection (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' [20,22,24]), the VQA field lacks enough difficult test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Our study is an effort in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' This discussion naturally relates to the out-of- distribution studies showing that models are biased towards the test sets that are similar to the sets over which they have been trained on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Likewise, they underperform over test sets that are even slightly different [28, 32, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' In this regard, here we also testing the out-of-distribution performance of the VQA models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Our test set contains 1,024 images crawled from publicly-available and free-to-distribute sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' We used Google and Bing search engines with different search phrases to collect the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' We made sure that no im- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='12032v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='CV] 28 Jan 2023 Dataset # Images # Questions Question Type(s) DAQUAR [25] 1449 12468 Object identification COCO-QA [29] 123287 115000 Questions automatically generated from COCO captions VQA [5] 204721 614163 Combining vision,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' language and common-sense Visual Madlibs [41] 10738 360001 Fill in the blanks Visual7W [43] 47300 2201154 7Ws,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' locating objects CLEVR [17] 100000 853554 Synthetic question generation using relations Tally-QA [1] 165000 306907 Counting objects on varying complexities KVQA [31] 24602 183007 Questions based on Knowledge Graphs VizWiz [13] 31000 31000 Questions by visually impaired users TextVQA [33] 28408 45336 Questions demanding reasoning about text Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Overview of VQA datasets described in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' age contains sensitive material, has poor resolution, or vi- olates the copyright law1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The gathered data encompass a wide variety of visual concepts over both RGB images, paintings, drawings, cartoons, and clip arts (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' We have made sure that all the questions are unambiguous and answers are correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Our test set contains more questions per image (∼7) than the VQA v2 test set (∼3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' We only consider the binary questions, since essentially any question can be converted to a “yes/no” question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' This simplifies the model evaluation and eliminates the complicated process of matching sentences of predicted answers with actual an- swers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Notice that this argument does not necessarily mean that we only need models that give binary answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Although our test set is smaller than the VQA test set, it comes with the benefit of better control over the complexity of the questions and quality of the answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Controlling the difficulty level of the questions generated by the Amazon Mechanical Turk (AMT) workers is challenging, as work- ers may choose to ask simple and short questions to save time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Unlike the questions in the VQA dataset [5] that are supposed to fool a toddler, alien, or a smart robot, some Bi- naryVQA questions can even challenge adults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' To answer the majority of the questions, one has to carefully analyze the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Further, small versatile and carefully curated test sets like ours can alleviate the legal issues concerning consents, licensing, privacy and security which are harder to control in datasets containing millions of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' In curating the BinaryVQA, we have made three choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' First, this test set is intentionally not paired with a train- ing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' This is to encourage generalization and to prohibit models to take advantage of correlations between testing and training sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' These correlations are easily accessible to models but are not detectable by humans [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Second, our dataset comes with a license that disallows researchers to update the parameters of any model for any reason on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' This is again to avoid over-fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Third, to mitigate the danger of leaking our data to other training sets, we mark every image by a one pixel green border that must be re- moved on the fly before testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' In addition to the test set, we also introduce new dimen- sions along which VQA models can be tested, in particu- lar sensitivity of the models to small perturbations in the 1We choose images that were public domain, did not have copyright, or were released by the government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' We find that, unlike humans, current models are highly sensitive to minor grammar mistakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Further, we study the bias of models towards generating positive an- swers, whether models indeed require the image to answer the questions, and whether they choose the right image re- gions to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' In a nutshell, our results show that state of the art VQA models struggle on our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' This suggests that, in conjunction with other datasets, our dataset can be used to push the VQA models to become better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' VQA Datasets Several VQA datasets have been introduced [18,26,40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' In these datasets, images are either taken from an existing vision dataset (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' MSCOCO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' [24]) or are artificially cre- ated (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Abstract Scenes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' [5], computer graphics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' [4,17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Further, questions are generated either automatically [4,17, 18, 25, 29, 41], from crowd workers [5, 8, 11, 18, 21, 43], or from in-house participants [18, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Unlike these datasets, questions in our dataset are carefully constructed by experts such that to answer them a detailed inspection of the image is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Some prominent VQA datasets are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Relevant ones to our work are described next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' COCO-QA [29] includes 123,287 images from the MSCOCO (72,783 for training and 38,948 for testing) and each image has one question/answer pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Questions are au- tomatically generated from the image descriptions and are categorized into four types based on the type of expected answer: object, number, color, and location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' A downside of the COCO-QA dataset is that 9,072 (23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='29%) of test ques- tions also appear in the training questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' VQA [5, 11] is one of the most widely used datasets (https://visualqa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='org/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' It comprises two parts, one using natural images called VQA-real (sourced from MSCOCO), and a second one with cartoon images called VQA-abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The latest more comprehensive version of this dataset, VQA v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='0 consists of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='1 million (image, ques- tion) pairs with 13 million associated answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Visual Genome [21] is aimed to enhance the progress on cognitive tasks, especially spatial relationship reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' It contains over 108K images, with about 35 objects, 26 attributes, and 21 pairwise relationships between objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Visual7W [43] includes seven types of WH questions (what, where, when, who, why, which and how) to examine capability of a model in visual understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Questions are asked in the multiple-choice format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' There are four can- didates for each question, and only one candidate is the cor- rect answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Visual Madlibs [41] consists of 360,001 targeted de- scriptions spanned across 12 different types of templates and their corresponding images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' VizWiz [13] is constructed from interactions of visually impaired users with a mobile application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' It consists of 31,000 visual questions together with 10 crowdsourced an- swers per question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Images often have poor quality due to poor lighting, focus, and framing of the content of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Further, questions are on average more conversational and are sometimes incomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' TextVQA [33] contains 45,336 questions on 28,408 im- ages that require reasoning about text to be answered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Im- ages are taken from the Open Images v3 dataset [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' TextVQA is available at https://textvqa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' In addition to above, some non-photo-realistic datasets such as CLEVR [17], NLVR [34], and FigureQA [19] have also been introduced to study visual reasoning independent of language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Some datasets such as Fact-Based VQA [37] explicitly require external knowledge to answer questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GQA [16] is a popular dataset, which also involves phrases to address the relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Our work relates to research that addresses the functional diagnostics of pre-trained language models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' [27, 30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' It also relates to works that examine adversarial robust- ness and out-of-distribution generalization of VQA models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' [7,23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' For example, [23] shows that non-expert an- notators can easily attack the best VQA models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' We construct an adversarial dataset to challenge the best VQA models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Although there are few such datasets for free- form VQA (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' VQA-CP [3]), here we show that even that answering yes/no questions is not yet solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' BinaryVQA Dataset Our dataset contains 7,800 questions across 1,024 im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Majority of the questions start with “Is” and “Are” as shown in the sunburst plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The most com- mon terms in the questions are person, wearing, people, and image (right panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' We do not include WH questions and all questions have “yes” or “no” answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' We ensured that each image is valid through human review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' We formulated the questions and then pre- sented them along with their answers to three AMT workers for verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Please see Appendix D for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Out of all questions, only 41 QA pairs received the incorrect ma- jority vote, which were fixed subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Statistics of the BinaryVQA dataset are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Out of the 7,800 questions, 4,897 have positive answers and the remaining 2,903 have negative answers, resulting in a ratio of about 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='7% (positive/all images).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The median pos- itive to all questions ratio per image is 0.' 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orange,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' grape,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' melon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' watermelon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' cherry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' strawberry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' corn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' pumpkin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' pineapple,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' lemon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='pepper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' avocado,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' cabbage,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' lettuce,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' coconut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' cucumber,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' eggplant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' broccoli Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' List of words per question type in the BinaryVQA dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='7%) have all of their questions answered “yes”, while no image has all of its questions answered “no”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The median number of questions per image is 7 which means that half of the images have more than 7 questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The median num- ber of positive questions (questions with answer “yes”) is 4 and the median number of negative questions is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The mean number of questions per image in BinaryVQA is 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='62 which is higher than 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='4 for VQA v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' BinaryVQA questions range from 3 to 20 words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The mean and median question length are 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='64 and 5 words, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' VQA v2 ques- tions range from 4 to 10 words (average 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The average image resolution is 840.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='3 × 650.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='4 (w × h) with the average aspect ratio of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Sample images are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' BinaryVQA images and questions cover a wide variety of topics and concepts including drawings,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' paintings,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' uncommon views of objects,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' hybrid animals,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' out of context objects and odd scenes (ele- phant in the room,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' car in the swimming pool,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' black sheep among white sheep),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' weather conditions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' interactions among people,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' actions (fighting,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' running,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' walking,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' danc- ing),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' emotions (sadness,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' happiness,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' surprise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' anger),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' counts and quantity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' gender,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' age,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' race,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' gaze direction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' object mate- rials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' objects in the mirror,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' body parts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' whether mouth or eyes are open, whether teeth are visible), animals, fruits, clothing (T-shirt, long sleeve, pants), shadow, color, crowd, clouds, tattoos, camouflage, illusions, non-existing objects, and logical reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' In formulating the questions, we tried to remove any am- biguity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' in giving addresses relative to the image, ob- jects, people in the scene, or image viewer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' left side of the rightmost person;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' left of the image).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' When only some peo- ple in the image (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' standing ones) are doing an action, we did not ask “Are these people doing X”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Instead, we asked “Are the standing people in this image doing X”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Left: Distribution of questions in our dataset by their first three words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The ordering of the words starts towards the center and radiates outwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The arc length is proportional to the number of questions containing the word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Right: Venn-style word clouds of words in the questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The most frequent word is ‘person’ indicating that questions are often about people in the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' BinaryVQA dataset statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Left: Distribution of the number of questions and its breakdown on positive and negative answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Half of the images have more than 7 questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Middle: Ratio of positive to all questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' On average images contain more positive questions than negative ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Right: Distribution of question length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Half of the questions have length greater than five.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Some questions test whether models can tell the type of the image (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' “Is this a drawing?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' and “Is this a paint- ing?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=') and whether they can answer questions over different types of images (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' drawings, paintings, cartoons, clip art, black and white images).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Some questions ask about the text, for example “Is there text?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=', “Is the word X written some- where in this image?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=', “Is the text written in English?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=', “Is the number 53813 written somewhere in the image?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Ex- ternal knowledge and common sense are needed to answer some questions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' ”Is this a map of Japan?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=', “Is this per- son a celebrity?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' In order to further test the spatial under- standing of the models, we placed a blue rectangle around some objects in the image and targeted the questions only on those regions (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' An example question is “Is the spatula inside the blue rectangle blue?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' To test the con- sistency of models and see whether they truly understand the image, for some images we include questions that con- tradict each other (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' “Is the boy standing?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' vs “Is the boy sitting?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Some other sample questions are “Is the whole body of the person visible?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=', “Is she holding a wine in her left hand?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=', “Are some birds printed on her skirt?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=', “Is her right hand in her right pocket?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=', “Is the person on the left taller?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=', “Is anyone looking at the camera?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=', Is this per- son an adult?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=', “Is the sky clear?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=', “Are his feet touching the ground?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=', “Are there more X objects than Y objects?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=', “Is object X to the left of object Y?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=', “Is the person in the image female?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=', and “Is the person opening the door with his right hand?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' We clustered the questions based on the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='there ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='people ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='an ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='te ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='plaking ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='ny ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='these ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='DO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='Does ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='all ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='wearing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='they ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content="person's " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='people ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='Binary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='Are ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='VQA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='two ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='/s ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='0 ratio# words in question, median: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='0, mean: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='64 2000 1500 Count 1000 500 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='0 questionlengthterms that appeared in them, as shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' For ex- ample, questions with words gender, man, woman, female, male, boy, girl address the gender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' No- tice that a question may fall into more than one category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' These categories will be used later to analyze the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' We did not incorporate any bias towards gender, age, or race during data collection, and tried to be as inclusive as possible in gathering images and formulating questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' We include and balance questions that address different ages and genders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The age groups are (baby, 26), (kid, 42), (children, 26), (Teenager, 5), (Young, 16), and (old, 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The gender groups are (woman, 350), (women, 38), (man, 448), and (men, 79).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' We did not include any question that ask about race.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' These issues are more important to address over large training sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' This is because sometimes models trained on such datasets are directly deployed in the real-world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The BinaryVQA dataset is substantially different from the VQA v2 validation set (the real images) measured in terms of the Fr´echet Inception Distance (FID) [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The FID is equal to 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='9 indicating a large distribution shift, and hence high diversity (using 7K images).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' To put this number in perspective, the FID between VQA v2’s validation and its test set is approximately 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Notice that the lower the FID, the more similar the two distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Analyses and Results To see how well the state of the art VQA models perform on our dataset2, we choose the OFA model [39] which is currently the leading scorer on the VQA v2 test-std dataset3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' It achieves 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='66% accuracy on “yes/no” questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' We also include a simple baseline model [5,42] to see whether tran- sitioning from simple to complicated models in VQA has indeed been meaningful4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' To put the results in perspective, we also ran the Pythia model5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' In this section, we focus on explaining the results using the OFA model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Summary results for both models are shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The distribution of model scores on the BinaryVQA dataset is shown in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The average ac- curacy of the OFA model is 75% which is much higher than the 62% accuracy of the baseline model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The OFA model, however, does significantly worse on our dataset than the VQA v2 dataset (around 20% absolute performance drop).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' We attribute this to the more complex nature of the ques- tions and images in our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Sample predictions of both models are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The OFA is able to correctly answer all questions for 160 images (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='6%) whereas the baseline is right for only 50 images (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='8%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The OFA model fails all questions over 2We used a 12 GB NVIDIA Tesla K80 GPU to do the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' 3https : / / paperswithcode .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' com / sota / visual - question - answering-on-vqa-v2-test-std 4https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='com/iamaaditya/VQA_Demo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='git 5https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='com/Eurus-Holmes/Pythia-VQA 314 images (30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='7%) while the baseline answers all ques- tions wrong over 673 images (65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='7%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Performance of the models over question types is shown in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The OFA model does better than the baseline in the majority of the question types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' It per- forms below the baseline model over counting (57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='2%), text (59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='7%), and spatial (63%) categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' It does, however, perform very well on weather (100%), daytime/nighttime (95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='5%) and indoors/outdoors (96%) categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Surpris- ingly, the OFA model does relatively well in answering questions pertaining to gaze direction (68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='7%) without us- ing any ad-hoc module to process faces, eyes, and gaze an- gles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The same argument holds over the real/drawing cate- gory (80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='6%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' We find that models have indeed improved drastically over the years, but there is still a large gap to close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Further, our dataset is significantly harder than the VQA v2 dataset (in “yes/no” questions) making it a great auxiliary test set to the existing ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' We found that models perform about the same over real images, paintings, or drawings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' OFA model scores ∼ 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='12% over the paintings or drawings (568 questions across 69 drawings/paintings) which is slightly lower than its 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='47% accuracy on real images (7,232 questions over 955 images).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The corresponding numbers for the baseline model are 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='03% and 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='47%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The OFA model is correct in answering the counting questions 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='2% of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' This model is accurate 69% of the time over the number category on the VQA v2 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Some difficult questions for the OFA model are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' 6 over different categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Model interpretability VQA models are very efficient in answering the ques- tions, but how much do they really understand the images?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Are their answers grounded on image content, or are merely due to some correlations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Several attempts have been made to address this (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' [2, 12]) and limiting the image area to a spatial location as is done here (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' images containing the blue rectangles) is one way to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' In this section, we propose a new way to interpret the models by masking the image content and study its effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' To this end, we run the OpenCV face detector [36] and mask the faces in images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' We then evaluate the OFA model on these images and plot the performance per category as shown in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Notice that here we limit our analysis to those im- ages for which at least one face is detected (309 out of 1024 images).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Some question categories that highly depend on face information such as “gaze direction”, “age”, “gender”, and “emotions” are severely degraded, which suggests that models indeed use the right information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Degradation or enhancement over some categories such as “text” or “ani- mals” may be partially attributed to the false detections of the face detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' This, however, needs further investiga- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Note that our masking approach can also be extended to more common objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Left: Distribution of per image accuracy for models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' OFA model is correct ∼ 75% of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Middle: Number of questions per question type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Right: Accuracy per question type for models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' OFA model does better than the baseline on most of the question types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Sample images along with the question, ground truth answer (GT), prediction of the OFA model (M1) and prediction of the baseline model (M2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' See appendix for more examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is therean umbrella Are there two people Is thelicenseplate Is this piece of a n Are there any trees Are there two people in the scene?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' wearing caps in the on the left side of ewspaper?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' in the scene?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' in the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT:YesM1:YesM2:Yes image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' the table?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT:Yes Ml:Yes M2:No GT: Yes M1: Yes M2:Yes GT:Yes M1:Yes M2:No GT: Yes M1: Yes M2:Yes GT: No M1:No M2:Yes Is there a car in th Are there six slices Is there a person in Is there a male and e image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Does anyone have eyels there a manikin i of bread?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' a female person in t GT: Yes Ml:Yes M2:Yes glasses?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' n the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2:Yes GT:No M1:No M2:Yes he image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT:YesM1:Yes M2:No GT:Yes M1:No M2:Yes GT: Yes M1: Yes M2: Yes Are there two umbrelDoes the text on pil Are there five peopl Are there only two b Are there eleven peo Does the letter say las in the scene?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' low say"mustard"?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' e in the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' irds in the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' ple in the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' "U"?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT:YesMi:YesM2:Yes GT:Yes M1:Yes M2:No GT: Yes M1: No M2: No GT: No M1:No M2:No GT:YesM1:Yes M2:Yes GT:Yes M1:Yes M2:No Is there reflection Is the pillow white?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Are the three people Is the left bird try Are more than two pels the letter litten in the water?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' in front wearing ey ing to grab a piece ople wearing suits?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT:Yes M1:Yes M2:Yes GT:No M1:NoM2:Yes eglasses?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' of pretzel?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT:NoM1:Yes M2:Yes GT:Yes M1:No M2:Yes GT: Yes M1:Yes M2:Yes GT:YesM1:YesM2:No Is this face of ape Is there a person in Is there a black ted Is there a toaster i Does the person haveIs there a flying va rson?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' dy bear attached to n the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' a beard?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' n in the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT:NoM1:YesM2:No GT:NoM1:No M2:Yes the car door?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1:Yes M2:Yes GT:YesM1:Yes M2:Yes GT:Yes M1:No M2:No GT:YesM1:YesM2:Yes Does this bell peppe Is there an object h Is there a mirror in Is the person riding Is the van on the gr r look like a face?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' anging from the ceil Is there a mirror in the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' his bike?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' ound?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT:Yes M1:Yes M2:Yes ing?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2:Yes GT:NoM1:No M2:No GT: No M1:YesM2:No GT:Yes M1:YesM2:Yes GT:Yes M1:Yes M2:Yesaccperimage Baseline:median:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='625,mean:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='62 OFA:median:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='mean:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='75 300 OFA Baseline 200 Count 100 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='0#questionsperquestiontype 600 400 200 0# accuracy over questiontypes counting text spatial body parts direction time gaze direction fruits animals clothing OFA age actions Baseline gender emotions real/drawing vegetation color sky indoors/outdoors daytime/nighttime weather 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='0Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Failure cases of the OFA model over different categories of the BinaryVQA dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Impact of question length on accuracy Questions in VQA datasets have different levels of com- plexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Intuitively, a longer question may be harder to answer than a short one, since it involves unpacking and understanding the dependencies among words in the sen- tences and their corresponding objects in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' 7 shows the model accuracy as a func- tion of question length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Due to rarity, questions longer than 10 words are discarded (only 150 occurrences).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' As it can be noticed, accuracy decays as the question length grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The mean accuracy of the OFA model over questions less than 8 words is 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Its accuracy over questions longer than 8 words (and less than 10) is 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The corresponding num- bers for the baseline model in order are 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='3% and 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' This result corroborates previous findings over the VQA dataset and shows that models underperform over longer questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Since our dataset contains longer questions than the VQA dataset, it can better test this aspect of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Analysis of “yes” bias in models VQA datasets usually contain more questions with “yes” answers than questions with “no” answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' This is partially due to the tendency of annotators to query the existing con- tent in images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Consequently, a smart chance model that often produces positive answers may win over a sophisti- cated model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' One approach to combat this issue, as is done over the VQA v2 dataset, is to balance the distribution of positive and negative questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Here, we introduce a new score called “ShuffleAcc” to automatically address this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' A subset of 2n questions consisting of n positive and n nega- tive questions are randomly selected (here n = 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The average model accuracy over m such subsets is then com- puted (here m = 50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' A model that consistently generates a “yes” (or “no”) answer will achieve 50% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The same argument holds for a model that randomly chooses “yes” 50% of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The ShuffleAcc scores of OFA and baselines models in order are 75% and 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='4% which are about the same as their performance using the traditional accuracy score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' This entails that these models do not suffer from inherent biases towards positive answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Sensitivity to spelling and grammar errors Studies on understanding and evaluating VQA models have been primarily focused on the visual component of VQA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Less attention, however, has been paid to diagnosing errors in the NLP component, in particular the sensitivity of models to perturbations on asked questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' This is particu- larly important to study since we know humans are still able to correctly answer questions even in presence of significant spelling and grammar mistakes, so long the meaning of the question remains the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Here, we study three simple perturbations that are unlikely to change the answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Within-word character swap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Here, we first randomly select a word (with length > 3) in the question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Next, we randomly choose two characters in this word and swap them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' For example, the question “Is there a person in the image?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' will turn into “Is there a peosrn in the image?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' We then evaluate the OFA model by varying the number of words, from 1 to 3, for which we swap two characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' OFA accuracy drops to 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='4% with swap in one word, 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='5% with swaps in two words, and 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='1% with swaps in three words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' These results clearly show that spelling errors dras- tically hinder the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Humans often do not notice these changes during reading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' To test whether this result generalizes to other datasets, we repeated these experiments over the VQA-v2 test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The accuracy of the OFA model drops to 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' This num- ber drops to 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='7% with swap in one word, 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='3% with swaps in two words, and 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='5% with swaps in three words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Similar observations are made for the baseline model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Arethetwopeopleinfrontfighting?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='GT:Yes,Pred:No ActionsArethesepeoplerunning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='GT:Yes,Pred:NoIs thisakid?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='GT:Yes,Pred:No AgeIsthisanoldperson?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content="GT:No,Pred:YesIstheonion on person's left hand?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='GT:Yes,Pred:No BodyPartsIsthereahandintheimage?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='GT:Yes,Pred:NoIs this person wearing a suit?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='GT:No,Pred:Yes ClothingIsthispersonwearingearrings?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='GT:Yes,Pred:NaIsthisared fish?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content="GT:No,Pred:Yes ColorIstheswan'sheadred?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='GT:Yes,Pred:NoAremorethantwopeoplewearing suits?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='GT:No,Pred:Yes CountingAre thereten bananasintheimage?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='GT:No, Pred:YesFigure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Left: Performance of the OFA with and without faces masked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Sample images with faces masked are also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Right: Performance of the OFA model as a function of question length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Omission of the articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Here, all the articles (“the”, “a”, “an”) are removed from the question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' For instance, the question “Is the person on the right holding a camera?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' will be converted to “Is person on right holding camera?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The performance of the OFA model drops to 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='8% indicating that this model, similar to humans, is robust to the omission of the articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Negating the question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Questions in the BinaryVQA dataset are formulated positively without using the word “not”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Logically, if the question is negated the answer should also be negated6 For example, if the answer to the question “Is there a firefighter on the crane?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' is “yes”, then the answer to the question “Is there not a firefighter on the crane?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' should be “no”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' For this analysis, we focus only on “Is there” type questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Out of 1,841 such questions, the OFA model maintained its decision in 738 cases when the question was negated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' This amounts to about 40% of the cases, which is far above 0%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Ideally, the model should always reverse its decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Ablation analyses Following our interpretability analysis above, here we conduct two analyses which can be considered as sanity checks or baselines for models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Models can be right for wrong reasons, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' In the first analysis, we ask all the questions over a black image or a white noise image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The OFA model performs well below chance, about 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='4% and 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='89% over these images, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' This indicates that this model indeed requires the image to produce the right answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The second analysis investigates whether a model can consistently produce the “no” answer to questions for which we know the answer is surely “no”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' We asked 15 questions in the form of “Is there a/an X in the image?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' where X represents one of the following 6Of course there are exceptions in the conversational language, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Isn’t there a person in the room?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Answer: No!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' (assuming there are no people in the room).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Model Avg Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' ShuffledAcc Char Swap Article Question∗ Acc on (one word) Omission Negation (%) VQA v2+ Baseline 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='5 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='4 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='5 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='3 35 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='5 OFA 75 75 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='4 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='8 40 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='66 Pythia 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='1 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='2 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='8 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='4 46 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='7† Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Summary of model performance on BinaryVQA dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' ∗ = Percentage of questions for which the model retained its answer after negation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' + = Human perf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' is about 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='48 from https://visualqa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='org/roe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='html † = Pythia v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='1 the winning entry in 2018 VQA benchmark https://visualqa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' org/roe_2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='html objects ‘white orange’, ‘dragon’, ‘blue horse’, ‘backgammon board’, ‘parrot’, ‘boxer dog’, ‘ostrich’, ‘dinosaur egg’, ‘galaxy’, ‘mermaid’, ‘telescope’, ‘unicorn’, ‘centipede’, ‘yellow cow’, ‘yeti’ over all the 1024 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The mean accuracy of OFA model across all 15 × 1024 questions is 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='1% using original images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The breakdown per each of these questions is shown in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Interestingly, when we asked these questions on white noise images, the accuracy jumped to 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' These results again demonstrate that the OFA model indeed highly relies on the image content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Discussion and Conclusion Understanding complex questions in VQA is a big chal- lenge, so is the understanding of complex scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Our dataset is better suited to address the latter, whereas other datasets can address the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' It can be used to test mod- els that already perform above 95% on binary questions of VQA-v2 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Our dataset contains a lot of questions which are really challenging and need close examination of the image to be answered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Such questions ask about non-standard objects, surreal imagery, and/or other oddities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' an eagle with a banana for a beak, water spout wear- ing sneakers, an odd clothespin-like object on one side and spoon on the other, a face with multiple pairs of eyes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' We share a zip file containing images, questions, meta- data, and detailed documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' BinaryVQA is licensed under Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='0 (Appendix E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='0 OFA OFAFaceMasked OFA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='8 Baseline 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='8 acc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='6 directior fime direction fruits age gender real/drawing vegetation Ays weather ununco Kpoq 3 4 5 6 7 8 9 10 gaze question lengthReferences [1] Manoj Acharya, Kushal Kafle, and Christopher Kanan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Tal- lyqa: Answering complex counting questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' 33(01):8076– 8084, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' 2 [2] Aishwarya Agrawal, Dhruv Batra, and Devi Parikh.' 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of the baseline model (M2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is there an umbrella Are there two people Is the license plate Is this piece of a n Are there any trees Are there two people in the scene?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' wearing caps in the on the left side of ewspaper?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' in the scene?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' in the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: Yes image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' the table?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: No GT: Yes M1: Yes M2: Yes GT: Yes M1: Yes M2: No Is there a car in th GT:YesM1:YesM2:Yes GT:NoM1:NoM2:Yes Are there six slices Is there a person in Is there a male and e image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Does anyone have eyels there a manikin i of bread?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' a female person in t GT: Yes M1: Yes M2: Yes glasses?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' n the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: Yes GT: No M1: No M2: Yes he image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: No GT:YesM1:NoM2:Yes GT: Yes M1: Yes M2: Yes Are there two umbrelDoes the text on pil Are there five peopl Are there only two b Are there eleven peo Does the letter say las in the scene?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' low say "mustard"?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' e in the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' irds in the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' ple in the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' "U"?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: Yes GT: Yes M1: Yes M2: No GT: Yes M1: No M2: No GT: No M1: No M2: No GT: Yes M1: Yes M2: Yes GT: Yes M1: Yes M2: No Is there reflection Is the pillow white?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Are the three people Is the left bird try Are more than two pels the letter litten in the water?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' in front wearing ey ing to grab a piece ople wearing suits?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: Yes GT: No M1: No M2: Yes eglasses?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' of pretzel?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: No M1: Yes M2: Yes GT: Yes M1: No M2: Yes GT: Yes M1: Yes M2: Yes GT: Yes M1: Yes M2: No Is this face of a pe Is there a person in Is there a black ted Is there a toaster i Does the person have Is there a flying va rson?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' dy bear attached to n the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' a beard?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' n in the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: No M1: Yes M2: No GT: No M1: No M2: Yes the car door?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: Yes GT: Yes M1: Yes M2: Yes GT: Yes M1: No M2: No Does this bell peppe Is there an object h GT:Yes M1:YesM2:Yes Is there a mirror in Is the person riding Is the van on the gr r look like a face?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' anging from the ceil l Is there a mirror in the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' his bike?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' ipuno GT: Yes M1: Yes M2: Yes ing?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: Yes GT: No M1: No M2: No GT: No M1: Yes M2: No GT: Yes M1: Yes M2: Yes GT: Yes M1: Yes M2: Yes Is this a giraffe?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Does this object loo Is this a picture of Is there a car in th Is the person touchi Are these kids playi GT: Yes M1: Yes M2: No k like a computer mo a tattoo?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' e image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' ng the ground?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' ng basketball?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' use?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: No GT: Yes M1: Yes M2: Yes GT: No M1: No M2: No GT: Yes M1: Yes M2: No Are the eyes of gira ffe visible?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is the car fully vis Is the person touchi Are there only three GT: Yes M1: No M2: No Does this appear to ns visible?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' ible?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' ng the hoop?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' kids in the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' be a happy computer GT: No M1: No M2: Yes GT: No M1: No M2: No GT: Yes M1: Yes M2: Yes GT: Yes M1: Yes M2: No mouse?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: No M2: No Are there three peop Are these people shi A Are these people cro Does this picture ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is this person looki Is this a drawing of le in the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' rtless?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' ssing the street?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' pear to taken from i ng at the camera?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' face?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: No M1: Yes M2: No GT: Yes M1: Yes M2: Yes GT: Yes M1: Yes M2: Yes nside a car?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: No GT: Yes M1: Yes M2: No Is the laptop in the Are these people wea Do two people carryi GT:YesM1:Yes M2:Yes Is this picture take Is this face drawing image turned on?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' ring T shirts?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' ng umbrellas and theIs there a toy insid n in a kitchen?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' bald?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: No GT: No M1: No M2: No other two people ho e the car?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: Yes GT: Yes M1: Yes M2: No lding papers?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: YesFigure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Sample images along with the question, ground truth answer (GT), prediction of the OFA model (M1) and prediction of the baseline model (M2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Are there animals in Is there a person in Is there a person in Is this a castle?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is this a huge sewin Does this look like this image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT:Yes Ml:YesM2:Yes g machine?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' a computer mouse?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: No M2: Yes GT: Yes M1: Yes M2: Yes GT: Yes M1: Yes M2: Yes GT:Yes M1:Yes M2:Yes GT: Yes M1: No M2: Yes Are there a person c Are these animals re Is there a water bot Is the person fully arved on the tree?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Does this object see Does this look like al?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' tle in the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' visible?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: No M2: Yes m like a factory or a hand?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: No M1: No M2: No GT: Yes M1: Yes M2: Yes GT: No M1: Yes M2: No a house?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT:Yes M1:Yes M2:No GT: Yes M1: Factory M2: We 11 12 10 5 Is this a giant bear Is this a bottle?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Are all numbers fromAre these slippers?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is this closed up of Is there a dog in th GT:YesMi:NoM2:No 1 to 15 present in GT:YesM1:NoM2:No a pencil?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' e image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT:Yes M1:Yes M2: No this image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: No M2: Yes Is the bottle wrappe GT: Yes M1: Yes M2: Yes GT:No M1:Yes M2:No Are these slippers m Is this a real bear?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' d?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' ade from bread?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is the tip of the pe Are the colored dots GT: Yes M1: No M2: No Is there a"o"in th GT:Yes M1:Yes M2: No ncil touching the su in the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: No M1: No M2: No is image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' rface?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: No GT:YesM1:YesM2:Yes GT: No M1: No M2: Yes Has there been an in Are these bagels?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Are these birds?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Can this image be co Is there a sign with Is there a huge roos cident in this image GT:YesMi:No M2:No GT:YesM1:YesM2:No nsidered a piece of "Originals"written ter in this image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' art?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' on it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: No GT:Yes M1:Yes M2:Yes Are there 8 bagels s M1:Yes M2:Yes GT: Yes M1: No M2: Yes hown here?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' pen?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is this a red rooste Is anyone being inju GT: No M1: No M2: No GT: Yes M1: Yes M2: No Are there two pears Is the person on the r?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' red?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' on the floor?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' right with the bike GT: No M1: No M2: No GT: No M1: Yes M2: No GT: Yes M1: No M2: Yes wearing a helmet?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: YesMi:Yes M2: Yes Is this a kiwi?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is this a picture of Is this giant fish?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is this a handycam?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is there a single ca Are these earrings?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: No a banana attached t GT: Yes Mi: Yes M2: No GT:Yes M1:NoM2:No r in the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: Yes Is this a regular ki o a snake?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: No M1: No M2: Yes GT:Yes M1: Yes M2:Yes Is this a shoe?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is this handycam mac Do these earrings lo wi?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: No M1: No M2: Yes e of plastic?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Are the cars covered ok like butterfly?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: No M1: No M2: No Is there more than a GT: No M1: No M2: No by snow?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: No banana in the image GT: Yes M1: Yes M2:Yes GT: No M1: No M2: Yes Is this a rope coil Is this house upside Is this a sandwich?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is there a chimpanze Is this a hammer?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is this a hammer?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' that looks like coco down?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT:YesM1:YesM2:Yes e in this image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: YesMi:Yes M2:Yes GT: No M1: Yes M2: Yes nut?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: Yes GT: Yes M1: No M2: Yes GT:Yes M1:Yes M2:Yes Is this sandwich mad Is this a funny obje Is this a pipe wrenc Is this daytime?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' e of logo?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is the chimpanzee lo ct?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' h?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is this a real cocon GT:Yes M1:Yes M2:Yes GT: Yes M1: No M2: No oking at the person?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT:Yes Mi:No M2:Yes GT: Yes M1: Yes M2: No ut?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: No M1: Yes M2: No GT: No M1: No M2: NoFigure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Sample images along with the question, ground truth answer (GT), prediction of the OFA model (M1) and prediction of the baseline model (M2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Does this pigeon loo Is this an animal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is this a binder cli Is this a small golf Is this a wooden obj Is this a giant rat?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' k like a teapot?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' p?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' ball?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' ect?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT:Yes M1:Yes M2: Yes GT: Yes M1: Yes M2: No GT: No M1: No M2: No GT:Yes M1:Yes M2:Yes GT: Yes M1: No M2: No Is this a sheep?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is there a nest in t GT: Yes Mi:NoM2:No Is this a dragonfly?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is this a golf ball?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is there a face on t Is this inside a roo he image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' he wood?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' m?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1:Yes M2: Yes GT:Yes M1:Yes M2: No GT:Yes M1:Yes M2:Yes GT: Yes M1: Yes M2: Yes GT:Yes M1:No M2:No O Is this a watering c Is this a duck?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is there a person in Is this mousing bein Does this cloud look Is this person sitti an?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' g forced to take a b like a seahorse?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' ng on the road?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1:Yes M2: Yes Is there a hand in t GT: Yes M1: Yes M2: Yes ath?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: No M2: No GT: Yes M1: Yes M2: No GT:Yes M1:Yes M2:Yes Is this watering can he image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is this person weari Is this daytime?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is there a TV on the made of metal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: No M2: Yes ng a tie?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Isthemouth scratch GT:Yes Mi:Yes M2:Yes image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: No M1:Yes M2: Yes GT:Yes M1: Yes M2: Yes ing the wall?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: Yes GT: Yes M1: Yes M2: No Is this a horse?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Does this object loo Is there a car in th Does this rat seem t Is this a snake comi Are these mens paint GT:Yes M1: No M2: No k like a mosquito?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' e image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' o be working out?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' ng out of a tube?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' ing tennis balls?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Does this horse look GT: Yes M1:Yes M2:Yes GT: No M1: No M2: Yes GT: Yes M1: No M2: No GT: Yes M1: No M2: No GT: No M1: No M2: No like a rock?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Does this object loo Is this a huge clip?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is the rat looking t Is the snake yellow?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is there a mirror in GT: Yes M1: Yes M2: Yes k like a dragonfly?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' owards the camera?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: No M1: Yes M2: No GT: Yes M1: Yes M2: Yes GT:Yes M1:Yes M2: No GT: No M1: No M2: No GT: No M1: No M2: Yes Is this objectmade Does this object loo Is this a car?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is this a strange st Are there three isla Is this a spanner?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' of only two logos?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' k like a chimpanzee GT:YesM1:YesM2:No rawberry?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' nds in the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: Yes GT: No M1: No M2: No or a monkey?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: No GT: Yes M1: Yes M2: Yes GT: Yes Mi: Minecraft M2: Is there any number Is this a depiction Is there a yellow lo in the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is the strawberry in Do all these islands of a fox?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=" go piece in the imag Is this animal stand GT:No M1: No M2:No person's hand?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' look like fish?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: No e?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' ing up?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: No M2: Yes GT: No M1: No M2: No GT:Yes M1:Yes M2:Yes Is this a depiction Does this object loo Is this a piece of c Is there an animal i Is this a shoe that Is there a butterfly of a dog?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' k like a turtle?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' ake?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' n this image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' also looks like a ho in the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: No M1: No M2: No GT: Yes M1: Yes M2: No GT: Yes M1: Yes M2: No GT: Yes M1: No M2:Yes rse?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: Yes GT: Yes M1: Yes M2:Yes Is this a depiction Is the turtle made o Is there a candle be Is there a dog in th Does this butterfly of an eagle?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' f an orange?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' sides the cake?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' is image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is the shoe made of look like an elephan GT: Yes M1:Yes M2: No GT:Yes M1: Yes M2: No GT: No M1: No M2: Yes GT:Yes M1: No M2: Yes metal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: Yes GT: Yes M1: No M2: YesFigure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Sample images along with the question, ground truth answer (GT), prediction of the OFA model (M1) and prediction of the baseline model (M2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is this a picture of Is this a fish?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is there fire in thi Is there a fork in t Are these zebras?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is the person holdin a wall?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' s image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' his image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' g a garlic?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: Yes GT:YesMl:YesM2:Yes GT: No M1: No M2: Yes GT: No M1: No M2: No Is this a hen?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Does this image also Is there a cigar in GT: No M1: No M2: No Is this image taken Is this drawing of a show a lion face?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is the person holdin the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' daytime?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' spoon?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: No M2: No g an onion?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: No M2: Yes GT: No M1: No M2: Yes GT: Yes M1: No M2: No GT: Yes M1: Yes M2: No Does this object loo Is this an eagle?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Does this object loo Is this a fish?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is this a chair?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is this a strange fi k like an animal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT:YesMi:NoM2:No k like an ax?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT:YesM1:NoM2:No GT: YesMi:Yes M2:No sh?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=" GT: Yes M1: Yes M2: Yes GT: Yes M1: Yes M2: No GT: Yes M1: Yes M2: Yes Is the eagle's neck Does this fish look Does this object als Is this object made yellow?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is this a megaphone?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' like a person?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' o look like a coffee Is this fish in the of plastic?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: No GT: Yes M1: No M2: No saucer?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' water?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: No M1: No M2: No GT:Yes M1:Yes M2: No GT: Yes M1: Yes M2: No GT: No M1: No M2: No Is there a book in t Does this cloud look Does this object loo Is this a person?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is there a rabbit?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is this person weari his image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' an animal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' k like a hourglass?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT:Yes Ml:Yes M2: Yes GT: Yes Mi:NoM2:No ng shoes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1:Yes M2: Yes GT:Yes M1: No M2: No GT: Yes M1: Yes M2: Yes GT: Yes M1: No M2: No Is this a real panda Is the rabbit runnin Is there an object l Does this cloud look Are there pills insi g?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is she running?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' ooking like a fox in a dog?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' de the inside the ho GT: No Mi: No M2: No GT: No M1: No M2: No GT: Yes M1: No M2: No this image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT:Yes M1: No M2: No urglass?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes MI: Yes M2: Yes GT: Yes M1: No M2: No 8008 Is this an airplane?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Does this object loo Does this image look Is this a hen?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is there any person Is this a tree?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' k like a cherry?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' like a fight scene?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT:Yes M1:YesM2:No in the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT:Yes M1:Yes M2: No GT: Yes M1:Yes M2: No GT: Yes M1: No M2: No GT: No M1: No M2: No GT:Yes Mi:No M2:No Is this hen made of Is there a butterfly Are there more than Is there only one di egg shells?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is this a model of c on the tree?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' two coffee cups?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' ce in the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is there a pickle in GT:Yes M1:Yes M2: No olosseum?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: No M2: Yes GT: No M1: No M2: No GT: No M1: No M2: No the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1:Yes M2:No GT: Yes M1: Yes M2: Yes Is this butterfly?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is there a dog in th Is this a dog?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is this a sandwich?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is the a picture of Is this a deer?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: No M2: No e image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' the cloudy sky?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is this a toy?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: No M2: Yes GT: Yes M1: Yes M2: No Is this picture of a Is this sandwich mad Is the whole body of GT: Yes M1:Yes M2: Yes Is there a mountain?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' real dog?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' e of bread and paper Does the cloud in th the deer visible?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: No S e center look a dog?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT:No Mi:No M2:No GT: Yes M1: Yes M2: Yes GT: Yes M1: Yes M2: No GT: No M1: No M2: NoFigure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Sample images along with the question, ground truth answer (GT), prediction of the OFA model (M1) and prediction of the baseline model (M2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Are these people fig Is there a crab?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is there a person in Is there a text in t Is this a gym?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is this a camera?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' hting?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT:Yes Mi:Yes M2:Yes the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' he image saying "LIL GT: Yes MI: No M2: No GT: Yes M1: Yes M2: Yes GT: No M1: No M2: No GT: Yes M1: No M2: Yes Y"?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is there water?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT:Yes M1:No M2:Yes Is there a person on Is this a real camer Are all of these peo GT:YesMi:YesM2:Yes Is there an ice crea the treadmill?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' a?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' ple standing?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' m?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is there an adult in GT:Yes M1:No M2:Yes GT: No M1: No M2: No GT: No M1: No M2: No GT:Yes M1: No M2:No this image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: No M1: No M2: No Are there some animels this a deer?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is there a mirror onI Is this a monkey?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is there a chair in Does this object loo Is in this scene?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: No the wall?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' k like a boat?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: Yes GT: Yes M1: Yes M2: Yes GT: Yes M1:Yes M2:Yes GT: Yes M1: No M2: Yes Is this nighttime?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is this a dog?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Isthere onlyone anGT:No M1:No M2:No Is there a reflectio GT: No M1: No M2: Yes Is this a bird?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is this a melon crus imal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' n on the mirror?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: No M1: No M2: No t?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: No M1: No M2: No GT: Yes M1: Yes M2: Yes GT:Yes M1:Yes M2: No Is this a dog?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is there an elephant Is this a house?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is there a huge golf Is this a stool?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is this a pair of sc GT: No M1: No M2: No in the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT:YesM1:YesM2:Yes ball in the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' issors?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: No Does this house look GT: Yes M1: Yes M2: Yes GT: Yes M1: Yes M2: Yes Is this a rabbit?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is this a person?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: No M2: No Is there a pillow on like a fish?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is this daytime?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: No M1: Yes M2: Yes Are two people kissi the chair?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: Yes GT:Yes M1:Yes M2: Yes ng in this image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT:Yes M1:Yes M2:Yes GT: Yes M1: No M2: No Is this a paint brus Is this a hand?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is this a banana?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is there a dog in th Is there a text in t Is this a chair?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' h?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT:YesM1:YesM2:No e image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' his image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: Yes GT: Yes M1: Yes M2: No GT: Yes M1: No M2: Yes GT: Yes M1: Yes M2: Yes Is this a giraffe?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is this a real banan Does this chair look Is there a hole in t GT:YesMi:YesM2:Yes a?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is the there a blank Is there a road in t like a dog?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' he paint brush?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: No M1: Yes M2: No et in the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' he image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: No M1: Yes M2: Yes GT: Yes M1:Yes M2: Yes GT:Yes M1:Yes M2:Yes GT: Yes M1: Yes M2:Yes Is this a chair?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Does this object loo Is there a fish in t Is there a fish in t Is there a fish in t Is there a fish in t GT:Yes M1:Yes M2: No k like a dress?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' his image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' his image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' his image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' his image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Can someone sit on t GT: Yes M1: Yes M2: No GT: Yes M1: No M2: No GT: Yes M1: No M2: No GT: Yes M1: Yes M2: No GT: Yes Mi: Yes M2: No his object comfortab Is this object hangi Is this a white fish Is this a white fish Is this a red fish?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is there only one fi ly?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' ng?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: No M1: Yes M2: No sh in this image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: No M2: Yes GT: Yes M1: Yes M2: Yes GT: No M1: NoM2: No GT: No M1: No M2: No GT: No M1: No M2: NoFigure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Sample images along with the question, ground truth answer (GT), prediction of the OFA model (M1) and prediction of the baseline model (M2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is there a fish in t Are there two forks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is this a globe?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Does this object loo Is the fish made of Does the letter look his image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT:YesM1:YesM2:No k like a pair of sho stone?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' like"w"?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT:Yes M1:Yes M2: No GT: Yes M1: Yes M2: No Is this globe made o es?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT:Yes M1:Yes M2:Yes GT: Yes M1: Yes M2: Yes GT:Yes M1:Yes M2:No Is this a big fish?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Are the shadows of f f an orange?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is there a kid on th Is the French fries GT: Yes M1: Yes M2: No orks visible?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: Yes Is this object yello e left side of the f in the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: Yes w?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' ish?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: Yes GT: No M1: No M2: Yes Is there a frog in t Is there milk?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Are there some coins Is this an apple?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Are these people wea Is this a toy?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' his image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT:Yes Mi:Yes M2:Yes in this image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: No ring the same shoes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: Yes GT: Yes M1: No M2: Yes Is there some milk s GT: Yes M1: Yes M2: Yes Is this apple uncut?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT:Yes M1:Yes M2:No Is this a teddy bear Does the frog look1 pilt?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Are there six elepha ike leaves?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: No nts in this image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: No M1: Yes M2: No Are these shoes red?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes Mi: Yes M2: Yes GT:Yes M1:No M2: No GT: Yes M1: No M2: No GT:YesMi:YesM2:Yes Is this a shadow?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is this a wine opene Is this a giant bird Are these slippers?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is this an eagle?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is there a car in th GT: Yes M1: Yes M2: Yes r?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1:Yes M2:No e pool?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=" GT: Yes M1: No M2: Yes GT: Yes M1: Yes M2: No GT: Yes M1: Yes M2: No Is this shadow of a Do these slippers ha Is the eagle's beak person?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Does this object loo Is this a white bird ve a butterfly patte made of a banana?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is the car fully und GT: Yes M1:Yes M2: No k like a person?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' rn?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: Yes erwater?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT:Yes M1: No M2: No GT: Yes M1: No M2: No GT:Yes M1:Yes M2: No GT: Yes M1: No M2: No Is this a hand?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Are these earrings?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Are these bananas?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Does this object loo Is there a lemon in Are these eyeglasses GT: Yes M1: Yes M2: Yes GT:YesM1:YesM2:No k like a bread?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: No GT: Yes M1: Yes M2: Yes GT: Yes M1: Yes M2: No Is this hand holding Are these earrings m Does these bananas l a person?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' ade of lego?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' ook like ducks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is the bread sliced?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is the lemon sliced?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is there a shadow in GT: Yes M1: Yes M2: No GT: Yes M1: Yes M2: Yes GT: Yes M1: Yes M2: No the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: No M2: No GT: No M1: No M2: No GT: Yes M1: Yes M2: Yes Is there a bird in t Is this an strange e Does this object loo Is this a snake?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is this a rooster?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is there an apple an he image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' yeglass?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' k like an octopus?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=" GT:YesM1:NoM2:No GT:YesMi:NoM2:No d a pear in this ima GT: Yes M1: No M2: Yes GT:Yes M1: No M2: No GT: Yes M1: Yes M2: No Is the snake's head Is there a black leg ge?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT:Yes M1: No M2:Yes Is there a fruit in Does this eyeglass h Is the octopus holdi i on the right side of o piece in the image the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' ave two lenses?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' ng a cup?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Are these fruits mad GT: Yes M1:Yes M2:Yes GT: No M1: Yes M2: Yes GT: Yes M1: No M2: Yes GT: Yes M1: No M2: No GT: No M1: No M2: Yes e of cloth?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' GT: Yes M1: Yes M2: NoB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Samples images from the BinaryVQA dataset Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Sample clock images with Roman numerals (left) and English numerals (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Sample clock images with (left) and without eye glasses (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' 2CN rosesFigure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Additional images from the BinaryVQA dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' art materialperception sentiment pop out illusions transformations face illusions counting fake vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' real drawings perceptual grouping I scene congruency humor MANAC!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' rC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Breakdown over the words in ablation study The accuracy of the OFA model over questions asking about existence of non-existing objects in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Object Accuracy over the original image Accuracy over the white noise image white orange 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='702 1 dragon 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='916 1 blue horse 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='958 1 backgammon board 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='953 1 parrot 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='983 1 boxer dog 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='965 1 ostrich 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='990 1 dinosaur egg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='985 1 galaxy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='863 1 mermaid 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='956 1 telescope 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='900 1 unicorn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='983 1 centipede 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='981 1 yellow cow 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='933 1 yeti 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='891 1 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' performance of the OFA model over questions of the type “Is there a/an X in the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Replace X with the object name in the first column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Data collection We adopt the following high-level process to collect the images and (question,answer) pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' First, we generated some phrases and then searched Flickr or Google search to find matching images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' We limited the search results to only those images that had the creative commons licences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Some sample search queries include: “A couple of kids watch- ing TV in a room while sitting on the floor?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=', “A woman looking at the camera while eating a burger?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=', “A couple of people in a meeting room?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=', “Two people fighting”, “A cat in the clouds”, “A sheep made of lego”, “A man with blond hair”, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' We then formulated some questions on these images along with answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The (question,answer) pairs were presented to three AMT workers for further ver- ification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Few questions for which AMT workers did not agree were then corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Our AMT interface for collecting the verification of our answers to the questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Workers were paid 25 cents per question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The experiment took 30 hours per participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Our AMT interface for collecting the verification of our answers to the questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' We have 17 images (from 0700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='jpeg to 0716.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='jpeg) that have blue rectangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' 25 questions were asked on these rect- angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' These questions either asked about an object or a person inside the rectangle (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Is there a spatula inside the blue rectangle?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=') or something about the rectangle itself (Is the blue rectangle on the bottom right corner of the image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Instructions:Youarepresentedwithan image,a questionaskedaboutit,and ananswerto the question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='Pleaseverifythe answertothe question and choose"Correct"if the answeris correct and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Question:Aremorethantwopeoplewearingsuits?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Answer:No Correct Incorrect prev nextE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Dataset License BinaryVQA dataset is free to use only for research and academic purposes (not commercial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' It is licensed un- der Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='0 with three additional clauses: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' BinaryVQA may never be used to tune the parameters of any model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The images containing people should not to be posted anywhere unless the people in the images are appro- priately de-identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Even in this case, written agree- ment from dataset creators is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' This is to check whether all the clauses are properly followed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' To stop or limit the misuse of our BinaryVQA by bad ac- tors, we have made a dataset request form7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' We review the requests that we receive and allow access for a legitimate use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The dataset we share contains images and questions is a zip file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' The package also contains the detailed documen- tation with all relevant metadata specified to users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' 7https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='ly/3bDY0MS F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Experimental details and evaluation setup We have used the validation set of the balanced real scens from the VQAv2 dataset from https://visualqa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' org/download.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' We are only using the binary questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Images are resized and normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' A question- mark is added to the questions if it is missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' BOS and EOS tokens are also added to the question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Model parame- ters for each of the tested models are listed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content=' Parameter settings for VQA baseline: VGG 16 model 4096 D feature vector for the image representation Image size 224 x 224 Each word in the question is a Glove vector 300D OFA model: Checkpoint: ofa large 384.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='pt Images are resized to and normalized A questionmark is added if missing BOS and EOS tokens are added Pythia model: TARGET IMAGE SIZE = [448, 448] CHANNEL MEAN = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='485, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='456, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='406] CHANNEL STD = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='229, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFLT4oBgHgl3EQfQS-o/content/2301.12032v1.pdf'} +page_content='224, 0.' 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https://git-lfs.github.com/spec/v1 +oid sha256:77665305a68875167e50da0b873bc61e06422266a0e6a774018ce80468d4acf0 +size 104478 diff --git a/W9E5T4oBgHgl3EQfcg_y/content/tmp_files/2301.05605v1.pdf.txt b/W9E5T4oBgHgl3EQfcg_y/content/tmp_files/2301.05605v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..cc5fbce9649ef48a08c3680faf659323f0174400 --- /dev/null +++ b/W9E5T4oBgHgl3EQfcg_y/content/tmp_files/2301.05605v1.pdf.txt @@ -0,0 +1,3769 @@ +arXiv:2301.05605v1 [cs.DS] 13 Jan 2023 +Differentially Private Continual Releases of Streaming +Frequency Moment Estimations +Alessandro Epasto , +Jieming Mao , +Andres Munoz Medina , +Vahab Mirrokni , +Sergei Vassilvitskii +Peilin Zhong +April 2022 +Abstract +The streaming model of computation is a popular approach for working with large-scale data. +In this setting, there is a stream of items and the goal is to compute the desired quantities +(usually data statistics) while making a single pass through the stream and using as little space +as possible. +Motivated by the importance of data privacy, we develop differentially private streaming +algorithms under the continual release setting, where the union of outputs of the algorithm +at every timestamp must be differentially private. Specifically, we study the fundamental ℓp +(p ∈ [0, +∞)) frequency moment estimation problem under this setting, and give an ε-DP +algorithm that achieves (1 + η)-relative approximation (∀η ∈ (0, 1)) with poly log(T n) additive +error and uses poly log(T n) · max(1, n1−2/p) space, where T is the length of the stream and n +is the size of the universe of elements. Our space is near optimal up to poly-logarithmic factors +even in the non-private setting. +To obtain our results, we first reduce several primitives under the differentially private +continual release model, such as counting distinct elements, heavy hitters and counting low +frequency elements, to the simpler, counting/summing problems in the same setting. Based +on these primitives, we develop a differentially private continual release level set estimation +approach to address the ℓp frequency moment estimation problem. +We also provide a simple extension of our results to the harder sliding window model, where +the statistics must be maintained over the past W data items. +1 +Introduction +Data privacy is a central concern in the deployment of real-world computational systems. In the vast +literature on privacy in computation Hsu et al. [2021], the notion of differential privacy (DP) Dwork +[2008], Dwork et al. [2014] has remained the de facto standard for more than a decade. The classical +formulation of differential privacy assumes that the data is static Dwork [2008], and that the data +curator is interested obtaining answers to a predetermined number of queries on the dataset. +1 + +Real world applications, however, often require the analysis of user datasets that are organi- +cally and rapidly growing. This is illustrated by the popular streaming model of computation Sr. +[1978], Alon et al. [1996], where data arrives over time, and at each update, a new solution is out- +put by the algorithm. In such streaming applications, the continual release model of differential +privacy Dwork et al. [2010a], Hubert Chan et al. [2010] promises a rigours guarantee of privacy: an +observer of all the outputs of the algorithm is information-theoretically bounded in the ability to +learn about the existence of an individual data point in the stream. +In this paper, we focus on two fundamental challenges in the field of private streaming algorithms: +the insertion only, or streaming, model and the sliding window model. In the former model, a data +curator receives a stream of data a1, a2, . . . and at each time t releases a statistical query depending +on all data received up to that point. In the latter, the computation depends only on the last W +items observed by the data curator. +The sliding window model may be generally more practically relevant compared to the streaming +model as it allows to account for information freshness and in some cases it can be a legal or privacy +requirement as well. For instance, in some situations, data privacy laws such as the General Data +Protection Regulation (GDPR)1 do not allow unlimited retention of user data. +Our main contribution is to provide private algorithms for a series of foundational streaming +problems under both the streaming and sliding window model. +Motivating example: Privacy Sandbox. We present a concrete practical application of our +results. As part of the Privacy Sandbox initiative, Chrome has developed a series of APIs to reduce +cross site tracking while supporting the digital advertising ecosystem. A key part of one of the +proposals is a k-Anonymity Server.2 The server ensures that each ad creative that is reported to +advertisers has won its respective auction at least k times over a particular time window. Abstract- +ing the specifics, this problem requires computing the number of distinct elements over a sliding +window. +Moreover, to further strengthen privacy protections, the computation itself should be +made differentially private, which is precisely the setting we consider in this work. +The previous example elucidates a concrete motivation for the study of sliding window algo- +rithms for counting distinct element problems with differential privacy in the continual release +setting. The rest of the paper proceeds by formalizing this model and our results in this space. +In particular, we study a more general class of statistics of the input data than the problem +of counting distinct elements: the ℓp frequency moments problem. The ℓp frequency moment is +the sum of the p-th power of the frequencies of the elements. The number of distinct elements +is a special case for p = 0. The ℓp frequency moment problem is one of the most fundamental +problems in the streaming literature. The first non-trivial algorithm for p = 1 is Morris [1978]. +Later Flajolet and Martin [1985] is the first to study the case for p = 0. Alon et al. [1996] initiates +the study for p = 2 and other p ∈ [0, ∞). After the developments over several decades, the spaces +of the current best ℓp frequency moment estimation algorithms are near optimal for all p ∈ [0, ∞), +i.e. they almost match the proven space lower bounds (see e.g., Ganguly [2011], Kane et al. [2011, +2010], Flajolet et al. [2007], Li and Woodruff [2013]). +However the landscape of DP streaming ℓp frequency moment is mysterious even in the non- +continual release setting. Most existing work only studied for p = 0, 1, 2 (see more discussion in +Section 1.4). The work of Wang et al. [2022] considered general p ∈ (0, 1]. A recent independent +work Blocki et al. [2022] studied all p ∈ [0, ∞). But none of Blocki et al. [2022], Wang et al. [2022] +1https://gdpr-info.eu/art-17-gdpr/ +2https://github.com/WICG/turtledove/blob/main/FLEDGE_k_anonymity_server.md +2 + +considered continual release setting. In the DP continual release setting, Bolot et al. [2013] studied +the count of distinct elements but not in the low space streaming setting. In the DP streaming +continual release setting, existing work Dwork [2008], Hubert Chan et al. [2010] only studied the +case for p = 1. No previous algorithm for p ̸= 1 is known in the DP low space streaming continual +release setting. +For ℓp frequency moment in the sliding window model, there are known techniques Datar et al. +[2002], Braverman and Ostrovsky [2007] which can convert the streaming algorithm into sliding +window algorithm using some additional small space. We show how to extend these techniques to +convert our DP streaming continual release streaming algorithms to DP sliding window continual +release algorithms. +1.1 +Computational Model +In this paper, we consider a streaming setting with T timestamps. At each timestamp t ∈ [T ], we +get an input at ∈ U ∪ {⊥}, where U represents the universe of all possible input elements, and ⊥ +represents empty. We sometimes also consider an input stream of integers, i.e., at each timestamp +t ∈ [T ], we get an input at ∈ Z. The goal is to compute some function g(·) based on the inputs. +Instead of only computing g at the end of the stream, we consider the continual release model +throughout the paper: +• Continual Release Model: At every timestamp t ∈ [T ], we want to output g(·) based on +the data a1, a2, · · · , at. +We consider two different streaming models depending on the range of inputs that g is based +on. +• (Insertion-only) Streaming Model: In this model, g depends on all past inputs, i.e. at +timestamp t ∈ [T ], we want to compute g(a1, ..., at). +Unless otherwise specified, we use +streaming model to refer to insertion-only streaming model throughout the paper. +• Sliding Window Model: +In this model, we have a parameter W ∈ Z≥1 for window +size. +g depends on the last W inputs, i.e. +at timestamp t ∈ [T ], we want to compute +g(amax(t−W+1,1), ..., at). +In this paper, we are particularly interested in algorithms that use space sub-linear in T . For +a (sub-)stream S, we use ∥S∥p +p to denote the ℓp frequency moment of S. In particular, for p = 0, +∥S∥0 denotes the number of distinct elements. For p = 1, ∥S∥1 denotes the number of non-empty +elements. We refer readers to preliminaries section (Section 2) for detailed notation and definitions. +1.2 +Our Results and Comparison to Prior Work +In this section, we give a brief overview of our results and comparison with prior work. We use +n to denote the size of the universe U. We use (α, γ)-approximation to specify the approximation +guarantee with multiplicative factor α and additive error γ. In all of our results, we use ε ≥ 0 for +the DP parameter, and use η ∈ (0, 0.5) in the relative error. +3 + +1.2.1 +Differentially Private Streaming Continual Release Algorithms +We developed a series of DP algorithms for solving frequency moments estimation and its related +problems in the streaming continual release model. +ℓp Frequency moment estimation (p ∈ [0, ∞)). +Our main result is a general ℓp frequency +moment estimation algorithm which works for all p ∈ [0, ∞). +Theorem 1.1 (ℓp Frequency moment, informal version of Theorem 5.31). There is an ε-DP al- +gorithm in the streaming continual release model such that with probability at least 0.9, it always +outputs an +� +1 + η, +� +log(T n) +ηε +�O(max(1,p))� +-approximation to ∥S∥p +p for every timestamp t, where S de- +notes the stream up to timestamp t. The algorithm uses space max(1, n1−2/p)· +� +log(T n) +ηε +�O(max(1,p)) +. +To the best of our knowledge, we are the first to study the general ℓp frequency moment estima- +tion problem in the differentially private streaming continual release setting. Dwork et al. [2010a] +and Hubert Chan et al. [2010] studies the summing problem in the same setting, where the sum- +ming problem can be seen as a special case for p = 1. Wang et al. [2022] studies the streaming +ℓp frequency moment estimation for p ∈ (0, 1] based on the p-stable distribution, and a concurrent +independent work Blocki et al. [2022] studies the case for p ∈ (0, 1], but it is not clear how to +generalize their techniques to the continual release model, i.e., their approach only provides the +differential privacy guarantee of the output at the end of the stream. In addition, the approach of +Wang et al. [2022] does not achieve ε-DP for an arbitrarily small ε > 0, and they also mention that +their technique might not be easily extended to the case for p > 1. +Our space usage is near optimal up to poly-logarithmic factors even when comparing with the +non-private streaming ℓp frequency moment estimation algorithms: for p ≤ 2, the space needed +for both our algorithm and previous non-private algorithm (see e.g., Kane et al. [2010]) is poly- +logarithmic, for p > 2, the space needed for both our algorithm and previous non-private algo- +rithm Indyk and Woodruff [2005] is ˜O(n1−2/p)3. +Note that Ω(n1−2/p) space is a proven lower +bound for p > 2 even in the non-private case Saks and Sun [2002], Bar-Yossef et al. [2004]. +Summing (ℓ1 frequency moment estimation). The easiest problem that is related to the ℓp +frequency moment estimation problem would be the summing problem: the goal is to compute the +summation of the input numbers. Note that ℓ1 frequency moment estimation is a special case of +the summing of a binary stream, i.e., we regard ⊥ as 0 and all other elements as 1. +Theorem 1.2 (Summing of a non-negative stream, informal version of Theorem 3.3). There is +an ε-DP algorithm for summing problem in the streaming continual release model. If the input +numbers are guaranteed to be non-negative, with probability at least 0.9, the output is always a +(1 + η, Oε,η(log T ))-approximation to the sum of all input numbers at any timestamp t ∈ [T ]. The +algorithm uses space O(1). +The summing problem was studied by Dwork et al. [2010a], Hubert Chan et al. [2010] in the +differentially private streaming continual release model. Their approximation has O(log2.5 T ) addi- +tive error. In our work, we show that if we allow (1 + η) relative error and work on the stream with +non-negative numbers only, we can reduce the additive error to O(log T ). This is useful when we +3We use ˜O(g) to denote g · poly(log(g)) +4 + +cannot avoid the relative error for some problem (such as the number of distinct elements) in the +streaming model but we still need summing as a subroutine. +Counting distinct elements (ℓ0 frequency moment estimation). Counting distinct elements +if one of the fundamental problems in the streaming literature. The goal is to estimate the number +of distinct elements that appeared in the stream. We provide a DP streaming continual release +algorithm for counting distinct elements. +Theorem 1.3 (Number of distinct elements, informal version of Corollary 4.11). There is an ε-DP +algorithm for the number of distinct elements in the streaming continual release model. With proba- +bility at least 0.9, the output is always a +� +1 + η, Oε,η +� +log2(T ) +� +) +� +-approximation for every timestamp +t ∈ [T ]. The algorithm uses poly +� +log(T ) +η min(ε,1) +� +space. +In the non-private streaming setting, counting distinct element can be solved via sketching +algorithms of Flajolet and Martin [1985] and its variants e.g., Flajolet et al. [2007]. Some recent +work Choi et al. [2020], Smith et al. [2020] extends these sketching techniques for counting distinct +element in a DP streaming setting. However, it is not clear how to extend these techniques to the +continual release setting. Continual release of counts of distinct elements is studied by Bolot et al. +[2013]. However, Bolot et al. [2013] is not in the low space streaming setting. +Estimation of frequencies and ℓ2 frequency moments. The goal of ℓ2 frequency moment +estimation is to estimate the sum of square of frequencies of elements. We present a DP streaming +continual release CountSketch Charikar et al. [2002] algorithm and use it for estimating ℓ2 frequency +moments and the frequency of each element. +Theorem 1.4 (Frequency and ℓ2 frequency moments, informal version of Theorem 5.4). There is +an ε-DP algorithm in the streaming continual release model such that with probability at least 0.9, +it always outputs for every timestamp t ∈ [T ]: +1. ˆfa for every a ∈ U such that |fa− ˆfa| ≤ η∥S∥2+ ˜Oε,η +� +log3.5(T n) +� +, where S denotes the stream +up to timestamp t and fa denotes the frequency of a in S, +2. ˆF2 such that | ˆF2 − ∥S∥2 +2| ≤ η∥S∥2 +2 + ˜Oε,η +� +log7(T n) +� +The algorithm uses O +� +log(T n) +η2 +· log(T ) +� +space. +Although DP ℓ2 frequency moment was studied by a line of work (see e.g., Blocki et al. [2012], +Sheffet [2017], Bu et al. [2021]), none of them considers the streaming continual release setting, and +it is not clear how to extend previous techniques to the the continual release setting. +ℓp Heavy hitters. In the ℓp heavy hitters problem, we are given a parameter k, and the goal +is to find elements whose frequency to the p-th power is at least 1/k fraction of the ℓp frequency +moment. By extending our DP streaming continual release CountSketch algorithm, we obtain a +DP streaming continual release ℓp heavy hitters algorithm. +Theorem 1.5 (ℓp Heavy hitters for all p ∈ [0, ∞), informal version of Theorem 5.10). There is an +ε-DP algorithm in the streaming continual release model such that with probability at least 0.9, it +always outputs a set H ⊆ U and a function ˆf : H → R for every timestamp t ∈ [T ] satisfying +1. ∀a ∈ H, ˆf(a) ∈ (1 ± η) · fa where fa is the frequency of a in the stream S up to timestamp t, +5 + +2. ∀a ∈ U, if fa ≥ +1 +εη · poly +� +log +� +T ·k·n +η +�� +and f p +a ≥ ∥S∥p +p/k then a ∈ H, +3. The size of H is at most O (log(T n) · 2p · k). +The algorithm uses max(1, n1−2/p) · k3 +η2 · poly (log (T · k · n)) space. +To the best of our knowledge, though DP streaming continual release ℓ1 heavy hitters problem is +studied by Chan et al. [2012], ℓp (for p ̸= 1) heavy hitters problem has not been studied in the DP +streaming continual release setting before. Note that Ω(n1−2/p) for p > 2 is a lower bound of space +needed for ℓp heavy hitters even in the non-private setting Saks and Sun [2002], Bar-Yossef et al. +[2004]. +1.2.2 +Differentially Private Sliding Window Continual Release Algorithms +Smooth histogram Braverman and Ostrovsky [2007] is a general algorithmic framework which can +convert a relative-approximate streaming algorithm into a relative-approximate sliding window +algorithm if the objective function that we want to compute has some nice properties. +We generalize the smooth histogram to make it support converting an approximate streaming +algorithm with both relative error and additive error into an approximate sliding window algorithm +with both relative error and additive error if the objective function has good properties. In addition, +we show that if the streaming algorithm is DP in the continual release setting, then the converted +sliding window algorithm is also DP in the continual release setting. +By applying our generalized smooth histogram approach and paying a poly +� +log T +η +� +more factor +than our DP streaming continual release algorithms in both additive error and space usage, we +show DP sliding window continual release algorithms for +1. ℓp Moment estimation (see Corollary 6.10), +2. Summing (see Corollary 6.7), +3. Counting distinct elements (see Corollary 6.8), +4. ℓ2 Moment estimation (see Corollary 6.9). +1.3 +Our Techniques +In this section, we briefly discuss the high level ideas of our algorithms. +We present a set of +techniques to reduce almost all problems that we considered in the DP streaming continual release +setting to the summing problem in the DP streaming continual release setting. +Summing with better additive error via grouping. +To illustrate the intuition of using +grouping for differentially private streaming continual release algorithms, we start with the following +simple problem: given a stream of numbers c1, c2, · · · , cT where each ci is at least 10 · ln(10 · T )/ε, +the goal is to output a (1 ± 0.1)-approximation to �t +j=1 cj for every prefix t with probability at +least 0.9, and we want the set of all outputs to be ε-DP, i.e., the continual released results to +be ε-DP. A simple way to solve the above problem is that we release a stream of noisy numbers +ˆc1, ˆc2, · · · , ˆcT where ∀t ∈ [T ], ˆct = ct + Lap(1/ε), and we report �t +j=1 ˆcj for every prefix t ∈ [T ]. It +is easy to see that (ˆc1, ˆc2, · · · , ˆcT ) is ε-DP. Since the reported approximate prefix sums only depend +on (ˆc1, ˆc2, · · · , ˆcT ), the continual released results are ε-DP. Furthermore, with probability at least +6 + +0.9, ∀t ∈ [T ], |ct − ˆct| ≤ ln(10 · T )/ε. Since ct is at least 10 ln(10 · N)/ε, we have 0.9ct ≤ ˆct ≤ 1.1ct +which implies that every reported approximate prefix sum is a (1 ± 0.1)-approximation. +To generalize the above idea, we propose a grouping approach in to group the consecutive +numbers in the stream in a differentially private way such that the total count of each group is large +enough. To implement grouping, we need to apply the sparse vector technique (see e.g., Dwork et al. +[2014]) iteratively. +The similar idea also appeared in Dwork et al. [2015] which shows a better +additive error guarantee for the summing problem than Dwork et al. [2010a] when the stream is +sparse. In contrast, our additive error guarantee is always better than Dwork et al. [2010a] and +Dwork et al. [2015] while we allow an additional (1 + ε) relative approximation. +Counting distinct elements. +We explain how to reduce counting distinct elements problem +to the summing problem. Suppose the element universe is small, we are able to track the set of +elements that already appeared during the stream. Then, we can create a binary stream of {0, 1} +where 1 denotes that we see a new element and 0 denotes that the input element already appeared +or it is empty. Therefore, the sum of the binary stream at timestamp t is exactly the number of +distinct elements. Furthermore, if we change an element in the input stream from a to b, there are +only constant number of positions of the binary stream will flip: consider the change a →⊥→ b. +If a is not its first appearance in the input stream, changing a to ⊥ does not cause any change in +the binary stream. If a is its first appearance in the input stream, changing a to ⊥ will make the +corresponding 1 in the binary stream be 0 and make the 0 corresponding to the original second +appearance of a in the input stream to be 1. Thus, it will affect at most 2 entries of the binary +stream. Similarly, changing ⊥ to b will cause the change of at most 2 entries of the binary stream. +Thus, the binary stream has low sensitivity which implies that a DP streaming continual release +summing algorithm gives a good approximation to the number of distinct elements with a small +additive error. Next, we discuss how to handle the large universe. For large universe, we can try +different sampling rate 1/2, 1/4, 1/8 · · · , 1/T . There should be a sampling rate such that (1) if we +hash the sampled elements into hashing buckets, there is no collision with a good probability, (2) +the number of samples is much larger than the additive error caused by the summing subroutine +so we can have a good relative approximation of the number of distinct sampled elements. Then +we can use the number of distinct sampled elements to estimate the number of distinct elements in +the input stream. +CountSketch and ℓp heavy hitters. Let h : U → [k] be a hash function which uniformly hash +elements into k hash buckets. Let g : U → {−1, 1} randomly map each element to −1 or 1 with +equal probability. The CountSketch is a tuple of k numbers (z1, z2, · · · , zk) where zi is the sum of +weighted frequencies of elements hashed to the bucket i, and the weight of the frequency of element +a is g(a). Changing a to b in the input stream will change at most 2 buckets: the bucket i contains +a and the bucket j contains b. Since |g(a)| ≤ 1, zi and zj will be changed by at most 1. We can +use DP streaming continual release summing algorithm of Dwork et al. [2010a], Hubert Chan et al. +[2010] to estimate (z1, z2, · · · , zk) such that each estimation ˆzi of zi only has poly(log T ) additive +error, and (ˆz1, ˆz2, · · · , ˆzk) is DP under the streaming continual release model. +Suppose the ℓ2 +frequency moment is much larger than poly(log T ), then the additive error becomes the relative +error, and we can use ˆz1, ˆz2, · · · , ˆzk to obtain a good relative approximation of the ℓ2 frequency +moment. Similarly, if an element has frequency much larger than poly(log(T )), then poly(log(T )) +becomes small relative error of the frequency and we are able to check whether it is an ℓ2 heavy +hitter by the standard analysis of CountSketch. Thus, we can use this DP streaming continual +release CountSketch to estimate ℓ2 frequency moment with (1 + η)-relative error and poly(log T )- +additive error, and we can use such CountSketch to find all elements which are at least poly(log T ) +7 + +and are ℓ2 heavy hitters. +Note that for p ≤ 2, if a has the largest frequency and it is an (1/k)-ℓp heavy hitter, then a +must be an (1/k)-ℓ2 heavy hitter. For p > 2, if a is an 1/k-ℓp heavy hitter, than a must be an +1/(kn1−2/p)-ℓ2 heavy hitter. Therefore, by some hashing technique, we can use ℓ2 heavy hitters +algorithm to construct ℓp heavy hitters algorithm. But since ℓ2 heavy hitters can only report the +elements with frequency larger than poly(log T ), the obtained ℓp heavy hitters algorithm can only +report the elements with frequency larger than poly(log T ) as well. +ℓp Frequency moment estimation. In high level we want to simulate the level set estimation +idea of Indyk and Woodruff [2005] in the DP streaming continual release setting. In particular, let +α = 1 + η, let fa denote the frequency of a and let Gi = {a | fa ∈ (αi, αi+1]}. Then � +i |Gi| · (αi)p +is a good approximation to the ℓp frequency moment. We say Gi is contributing, if |Gi| · (αi)p is +at least Ωα(1/ log(T )) fraction of the ℓp moment. Since non-contributing elements only contributes +a small total amount to the ℓp frequency moment, it is easy to see that � +contributing Gi |Gi| · (αi)p +is still a good approximation to the ℓp frequency moment. Thus, we only need to estimate the +size of each contributing Gi. Due to the definition of contributing, it is easy to see that if Gi is +contributing, either αi is large or |Gi| is large. In fact, as observed by Indyk and Woodruff [2005], +for each contributing level set Gi, there must be a proper sampling probability such that after +sampling, there are at least poly(log T ) elements from Gi sampled and all of the sampled elements +from Gi are at least 1/poly(log T )-ℓp heavy hitters among the set of all sampled elements from +the universe U. Ideally, we can try different sampling rate 1, 1/2, 1/4, 1/8, · · · , 1/T and use our ℓp +heavy hitters algorithm to report the heavy hitters and estimate |Gi| for each i. However, for i with +αi ≪ poly(log T ), our DP streaming continual release ℓp heavy hitters algorithm does not report +any element from Gi. We must find another way to estimate |Gi| instead of using heavy hitters. +Similar to counting distinct elements, let us start with the case that the universe size is small so +we can track the sets A1, A2, · · · , Ak for some k = poly(log T ), where Ai is the set of all elements +whose frequency is exactly i. We can construct streams S1, S2, · · · , Sk with numbers in {−1, 0, 1}. +During the stream, when we see an input element a, and if a is in the set Ai, then we move a to +the set Ai+1 due to the increase of the frequency of a. At the same time, we append −1 to the +stream Si, append 1 to the stream Si+1 and append 0 to Sj for j ̸= i, i + 1. It is easy to check that +the sum of Sl is always the same as |Al|, i.e., the number of elements which have frequency exactly +l. Furthermore, similar to the analysis for counting distinct elements, if we change an element +in the input stream, each Sl might be affected by at most 4 entries. Thus, the total sensitivity +of (S1, S2, · · · , Sk) is at most O(k). Therefore, we can use the DP continual release summing to +estimate the sum of each Sl with additive error poly(log T ). Thus, we can estimate |Gi| for each i +with αi ≪ poly(log T ) with additive error poly(log T ) and will only introduce at most poly(log T ) +additive error in approximating the ℓp frequency moment. +Now let us go back to the case that the size of the universe is large. In this case, we can use +the similar hashing and subsampling technique discussed for counting distinct elements to estimate +|Al| for each l ∈ [k]. +1.4 +Related Work +Dwork et al. [2010a] and Hubert Chan et al. [2010] initiated the study of differential privacy in +the continual release model, and proposed the binary tree mechanism for computing summations. +Bolot et al. [2013] and Perrier et al. [2019] generalized their results to decayed summations, counting +distinct elements without space constraints and summations with real-valued data. +Song et al. +8 + +[2018], Fichtenberger et al. [2021] studied graph problems under the differentially private continual +release model. +Jain et al. [2012], Smith and Thakurta [2013], Agarwal and Singh [2017] studied +differentially private online learning. +Jain et al. [2021] gave the first polynomial separation in +terms of error between the continual release model and the batch model under differential privacy. +Upadhyay [2019] studied heavy hitters in the differentially private sliding window model. +Differentially private frequency moment estimation for p = 0, 1, 2 (without continual releases) +has been well-studied Mir et al. [2011], Dwork et al. [2010b], Blocki et al. [2012], Sheffet [2017], +Choi et al. [2020], Smith et al. [2020], Bu et al. [2021]. Wang et al. [2022] studied frequency moment +estimation (without continual releases) for p ∈ (0, 1] with low space complexity. Recent concurrent +independent work Blocki et al. [2022] studies p ∈ [0, ∞) with low space complexity but not in +continual release setting as well. +The differentially private ℓ1 heavy hitters problem is studied +by Mir et al. [2011], Dwork et al. [2010b] in the low space streaming setting but not in the continual +release setting. Chan et al. [2012] studied differentially private ℓ1 heavy hitters problem in the low +space continual release streaming setting. But it is not clear how to extend their techniques to lp +case for p ̸= 1. +ℓp Frequency moment estimation and ℓp heavy hitters are heavily studied in the non-private +streaming literature. For ℓp frequency moment estimation, the problem can be solved by e.g. Flajolet and Martin +[1985], Flajolet et al. [2007], Durand and Flajolet [2003] for p = 0, Alon et al. [1996], Charikar et al. +[2002], Thorup and Zhang [2004] for p = 2, Kane et al. [2010], Indyk [2006], Li [2008], Kane et al. +[2011] for p ∈ (0, 2) and Indyk and Woodruff [2005], Andoni et al. [2011], Andoni [2017] for p > 2. +For ℓp heavy hitters, the problem can be solved by e.g., Cormode and Muthukrishnan [2005], +Misra and Gries [1982] for p = 1, +Charikar et al. [2002] for p = 2, +Jowhari et al. [2011] for +p ∈ (0, 2), and Indyk and Woodruff [2005], Andoni et al. [2011] for p > 2. +2 +Preliminaries +2.1 +Notation +In this paper, for n ≥ 1, we use [n] to denote the set {1, 2, · · · , n}. If there is no ambiguity, for +i ≤ j ∈ Z, we sometimes use [i, j] to denote the set of integers {i, i + 1, · · · , j} instead of the +set of real numbers {a ∈ R | i ≤ a ≤ j}. We use fa(a1, a2, · · · , ak) to denote the frequency of +a in the sequence (a1, a2, · · · , ak), i.e., fa(a1, a2, · · · , ak) = |{i ∈ [k] | ai = a}|. If the sequence +(a1, a2, · · · , ak) is clear in the context, we use fa to denote the frequency of a for short. We use 1E +to denote a indicator, i.e., 1E = 1 if condition E holds and 1E = 0 if condition E does not hold. +For α ≥ 1, γ ≥ 0, if +1 +α · x − γ ≤ y ≤ α · x + γ, then y is an (α, γ)-approximation to x. If y is +a (1, γ)-approximation to x, we say y is an approximation to x with additive error γ. If y is an +(α, 0)-approximation to x, we say y is an α-approximation to x. We use a ± b to denote the real +number interval [a−|b|, a+|b|]. For a set S of real numbers, we use S±b to denote the set � +a∈S a±b, +and use S · c to denote the set � +a∈S a · c. We use Lap(b) to denote the Laplace distribution with +scale b, i.e., Lap(b) has density function given by +1 +2b exp(|x|/b) +2.2 +Functions to Compute +We study several fundamental functions in the streaming literature. When the inputs are integers, +we consider the summing problem over a (sub-)stream (ai, · · · , aj): +• Sum of numbers: Sum(ai, · · · , aj) := �j +k=i ak +9 + +When the inputs are from U ∪ {⊥}, we consider the functions g(ai, · · · , aj) that are based on +the frequencies of the elements in a (sub-)stream (ai, · · · , aj). +• Count of non-empty elements: ∥(ai, ..., aj)∥1 := � +a∈U fa(ai, ..., aj). +• The number of distinct elements: ∥(ai, ..., aj)∥0 := � +a∈U 1fa(ai,...,aj)>0. +• ℓp-Frequency moment: ∥(ai, ..., aj)∥p +p := � +a∈U fa(ai, ..., aj)p. +• ℓp-Heavy hitters: (1/k)-ℓp-HH(ai, ..., aj) := {a ∈ U | fa(ai, · · · , aj)p ≥ ∥(ai, · · · , aj)∥p +p/k}. +Note that ∥(ai, ..., aj)∥1 is a special case of Sum(ai, · · · , aj) with binary inputs. +2.3 +Differential Privacy +Neighboring streams: Consider two streams S = (a1, a2, · · · , aT ) and S′ = (a′ +1, a′ +2, · · · , a′ +T ). If +there is at most one timestamp t ∈ [T ] such that (1). |at − a′ +t| ≤ 1 (only required when the inputs +are treated as integers) (2). ∀i ̸= t, ai = a′ +i, then we say S and S′ are neighboring streams. +Definition 2.1 (Differential Privacy). +We say algorithm A is ε-DP, if for any two neighboring +streams S, S′, and any output set O, +Pr[A(S) ∈ O] ≤ eε · Pr[A(S′) ∈ O]. +Note that in the continual release model, the output A(S) mentioned in Definition 2.1 is the +entire output history of the algorithm A over stream S at every timestamp. +Definition 2.2 (Distance between streams). Consider two streams S and S′. If d is the minimum +number such that there exists a sequence of streams S0, S1, · · · , Sd where S0 = S, Sd = S′ and +∀i ∈ [d], Si and Si−1 are neighboring streams, then the distance between S and S′ is dist(S, S′) = d. +Definition 2.3 (Sensitivity of a stream mapping). Let F be a mapping which maps a given input +stream S to a tuple of streams (F1(S), F2(S), · · · , Fk(S)). The sensitivity of F is the minimum +value s such that for any two neighboring streams S and S′, � +i∈[k] dist(Fi(S), Fi(S′)) ≤ s. +Theorem 2.4 (Composition Dwork et al. [2014]). Let F be a mapping which maps a given input +stream S to a tuple of streams (F1(S), F2(S), · · · , Fk(S)). +Let A1, A2, · · · , Ak be k ε-DP algo- +rithms. Let A be an algorithm such that A(S) = M(A1(F1(S)), A2(F2(S)), · · · , Ak(Fk(S))) for +some function M(·). Then the algorithm A is (sε)-DP. +Example usage of Theorem 2.4: Some example usage of Theorem 2.4 in our paper are presented +as the following: +• Composition of multiple algorithms over the input stream: Suppose each of A1, A2, · · · , Ak +is ε-DP, then for any function M(·), M(A1(S), A2(S), · · · , Ak(S)) is (kε)-DP. +• Composition of algorithms on disjoint sub-streams: For an input stream S = (a1, a2, · · · , aT ), +we partition S into S1 = (a1,1, a1,2, · · · , a1,T ), S2 = (a2,1, a2,2, · · · , a2,T ), · · · , Sk = (ak,1, ak,2, · · · , ak,T ), +i.e., ∀t ∈ [T ], there is only one i ∈ [k] such that ai,t = at, and ∀i′ ̸= i, ai,t =⊥ (or 0). It +is clear that such partitioning has sensitivity 1. Thus, if each of A1, A2, · · · , Ak is an ε-DP +algorithms, then for any function M(·), M(A1(S1), A2(S2), · · · , Ak(Sk)) is ε-DP. +10 + +2.4 +Streaming Continual Release Summing and Counting +For summing problem in the streaming continual release model, the binary tree mechanism was +proposed in Dwork et al. [2010a], Hubert Chan et al. [2010]. It gets poly-logarithmic additive error +and uses logarithmic space. Furthermore, it can handle negative numbers in the stream. +Theorem 2.5 (Dwork et al. [2010a], Hubert Chan et al. [2010]). Let ε ≥ 0, ξ ∈ (0, 0.5), there is an +ε-DP algorithm for summing in the streaming continual release model. With probability 1 − ξ, the +additive error of the output for every timestamp t ∈ [T ] is always at most O +� +1 +ε log2.5(T ) log +� +1 +ξ +�� +. +The algorithm uses O(log(T )) space. +2.5 +Probability Tools +Lemma 2.6 (Bellare and Rompel [1994]). Let λ ≥ 4 be an even integer. Let X be the sum of n +λ-wise independent random variables which take values in [0, 1]. Let µ = E[X] and A > 0. Then +we have +Pr [|X − µ| > A] ≤ 8 · +�λµ + λ2 +A2 +�λ/2 +Lemma 2.7 (Median trick to boost success probability). Suppose X is a random estimator of value +v such that with probability at least 2/3, X is an (α, γ)-approximation to v. Then, for ξ ∈ (0, 0.5), +if we draw k = ⌈50 log(1/ξ)⌉ independent copies X1, X2, · · · , Xk of X, with probability at least 1−ξ, +the median of X1, X2, · · · , Xk is an (α, γ)-approximation to v. +Proof. We say Xi is good if Xi is an (α, γ)-approximation to v. If the median is not good, then +� +i∈[k] 1Xi is good < k +2. By Chernoff bound, Pr[median is not good] ≤ Pr[� +i∈[k] 1Xi is good < k +2] ≤ +Pr[E[� +i∈[k] 1Xi is good] − k +2 > k +6] ≤ e−k/48 ≤ ξ. +3 +Continual Released Summing with Better Additive Error +In this section, we show that if we allow a relative approximation and the input stream only contains +non-negative numbers, then we can have a continual released summing algorithm with additive error +11 + +better than Theorem 2.5. +Algorithm 1: Grouping Stream of Counts. +Input: A stream of non-negative numbers c1, c2, · · · , cT DP parameter ε > 0, +approximation parameter η ∈ (0, 0.5), and failure probability ξ ∈ (0, 1). +Output: A stream of groups with grouped noisy counts (ˆc1, ˆc2, · · · , ˆcT ) +Let ε0 ← ε/2. +Initialize a group index i ← 1, current group G1 ← ∅, and threshold +τ1 ← +� +1 +η + 1 +� +· 7 +ε0 · ln (3 · T/γ) + Lap(2/ε0). +//Gi is used for analysis only. +for t = 1 to T do +Gi ← Gi ∪ {t}. +Let νt ← Lap(4/ε0). +if νt + � +j∈Gi cj ≥ τi then +ˆct ← Lap(1/ε0) + � +j∈Gi cj. +i ← i + 1. +τi ← +� +1 +η + 1 +� +· 7 +ε0 · ln (3 · T/γ) + Lap(2/ε0). +Gi ← ∅. +else +ˆct ← 0. +end +end +Lemma 3.1. The output stream ˆc1, ˆc2, · · · , ˆcT of Algorithm 1 is ε-DP. +The proof idea is to iteratively apply sparse vector technique. +We put the proof into Ap- +pendix A.1. +Lemma 3.2. Let ˆc1, ˆc2, · · · , ˆcT be the output stream of Algorithm 1. Then with probability at least +1 − ξ, ∀l, r satisfying 1 ≤ l ≤ r ≤ T , +(1 − η) +r +� +j=l +cj − +�1 +η + 4 +� +· 7 +ε0 +· ln (3 · T/ξ) ≤ +r +� +j=l +ˆcj ≤ (1 + η) +r +� +j=l +cj + +�1 +η + 4 +� +· 7 +ε0 +· ln (3 · T/ξ) . +We put the proof of Lemma 3.2 into Appendix A.2 +Theorem 3.3 (Summing of a non-negative stream). Let ε ≥ 0, ξ ∈ (0, 0.5), there is an ε-DP +algorithm for summing in the streaming continual release model. If the input numbers are guaranteed +to be non-negative, with probability at least 1 − ξ, the output is always a +� +1 + η, O +� +log(T/ξ) +εη +�� +- +approximation to the summing problem at any timestamp t ∈ [T ]. The algorithm uses space O(1). +Proof. According to Lemma 3.1, the output stream of Algorithm 1 is ε-DP, thus, we only need +to solve non-private summing over (ˆc1, ˆc2, · · · , ˆcT ). +The approximation guarantee is given by +Lemma 3.2. Note that we do not need to store Gi, we only need to maintain the sum of num- +bers in Gi at any timestamp. Thus, the total space needed is O(1). +4 +Continual Released Number of Distinct Elements +In this section, we show how to use ε-DP streaming continual release summing to solve ε-DP +streaming continual release number of distinct elements. In Section 4.1, we show how to estimate +12 + +the number of distinct elements if the universe is small. In Section 4.2, we reduce the number +of distinct elements of a large universe to the number of distinct elements of a small universe via +subsampling. +4.1 +Number of Distinct Elements for Small Universe +Algorithm 2: Number of Distinct Elements for Small Universe +Input: A stream S of elements a1, a2, · · · , aT ∈ U ∪ {⊥} with guarantee that |U| ≤ m. +Parameters : Relative approximation factor α ≥ 1 and additive approximation factor γ ≥ 0 +depending on the streaming continual release summing algorithm. +//See Theorem 2.5. +Output: Estimation of the number of distinct elements at every timestamp t. +Initialize an empty stream C. +Let S ← ∅. +for each at in the stream S do +if at ̸∈ S and at ̸=⊥ then +S ← S ∪ {at}. +Append 1 to the end of the stream C. +end +else +Append 0 to the end of the stream C. +end +Output an (α, γ)-approximation to the total counts of C. +end +Lemma 4.1. At the end of any time t ∈ [T ], the output of Algorithm 2 is an (α, γ)-approximation +to the number of distinct elements. +Proof. Since we append 1 to the stream C if and only if we see a new non-empty element, the +total counts in C is always equal to the number of distinct elements at the end of any time t ∈ [T ]. +Thus, an (α, γ)-approximation to the total counts of C is an (α, γ)-approximation to the number of +distinct elements. +Lemma 4.2. If the algorithm to continually release the approximate total counts of C in Algorithm 2 +is ε-DP, Algorithm 2 is 5ε-DP in the continual release model. +Proof. Consider two neighboring stream S = (a1, a2, · · · , aT ) and S′ = (a′ +1, a′ +2, · · · , a′ +T ) of elements +in U where they only differ at timestamp t, i.e., at ̸= a′ +t. Let us consider the difference between the +generated count stream C = (c1, c2, · · · , cT ) and C′ = (c′ +1, c′ +2, · · · , c′ +T ). +Consider any timestamp i ∈ [T ], if ai ̸= at and ai ̸= a′ +t, it is easy to verify that ci = c′ +i. Suppose +i ̸= t. If ai = at = u ∈ U but ai is at least the third appearance of u in S, then a′ +i = ai = u is at +least the second appearance of u in S′ which implies that ci = c′ +i = 0. Similarly, if ai = a′ +t = u ∈ U +is at least third appearance of u in S, we can show ci = c′ +i = 0 as well. Thus the sensitivity of the +stream C is at most 5: only when i = t or ai is the first/second appearance of at, at′, ci might be +different from c′ +i. Therefore, if we use an ε-DP algorithm to continually release the total counts of +C, the continually released output of Algorithm 2 is (5 · ε)-DP. +13 + +Theorem 4.3. Let ε ≥ 0, ξ ∈ (0, 0.5), suppose there is an ε-DP streaming continual release sum- +ming algorithm (for stream of non-negative numbers) which uses space J and with probability at least +1−ξ always outputs an (α, γ)-approximation for every timestamp. There is a (5ε)-DP algorithm for +the number of distinct elements of streams with universe size at most m in the streaming continual +release model. With probability at least 1 − ξ, the algorithm always outputs an (α, γ)-approximation +for every timestamp t ∈ [T ]. The algorithm uses O(m + J) space. +Proof. Consider Algorithm 2. The approximation guarantee is proven by Lemma 4.1. The DP +guarantee is proven by Lemma 4.2. In the remaining of the proof, we only need to prove the space +usage. Since |U| ≤ m, the space needed to maintain set S is at most m. The space needed to +continually release an (α, γ)-approximation to the summing problem over C is at most J . Thus, +the total space needed is at most O(m + J). +By combining the above theorem with Theorem 2.5, we obtain the following corollary. +Corollary 4.4 (Streaming continual release distinct elements for small universe). There is an +ε-DP algorithm for the number of distinct elements of streams with universe size at most m in +the streaming continual release model. +With probability at least 1 − ξ, the additive error of the +output is always at most O +� +1 +ε log2.5(T ) log +� +1 +ξ +�� +for every timestamp t ∈ [T ]. The algorithm uses +O(m + log(T )) space. +By combining Theorem 4.3 with Theorem 3.3, we obtain the following corollary: +Corollary 4.5 (Streaming continual release distinct elements for small universe, better additive +error). There is an ε-DP algorithm for the number of distinct elements of streams with universe size +at most m in the streaming continual release model. With probability at least 1−ξ, the additive error +of the output is always an +� +1 + η, O +� +log(T/ξ) +εη +�� +-approximation to the number of distinct elements +for every timestamp t ∈ [T ]. The algorithm uses O(m) space. +14 + +4.2 +Number of Distinct Elements for General Universe +Algorithm 3: Number of Distinct Elements via Subsampling +Input: A stream S of elements a1, a2, · · · , aT ∈ U ∪ {⊥}, and a error parameter η ∈ (0, 0.5). +Parameters : Relative approximation factor α ≥ 1 and additive approximation factor γ ≥ 0 +depending on the streaming continual release algorithm for number of distinct +elements of streams with small universe of elements. +//See Theorem 4.3. +Output: Estimation of the number of distinct elements ∥S∥0. +L ← ⌈log min(|U|, T )⌉, λ ← 2 log(1000L), m ← 100L · +� +16α max +� +γ/η, 32αλ/η2��2. +Let h : U → [m] be a pairwise independent hash function. +//Here we treat [m] as a universe of elements with size m instead of a set of integers. +Let g : U → [L] ∪ {⊥} be a λ-wise independent hash function and +∀a ∈ U, i ∈ [L], Pr[g(a) = i] = 2−i, Pr[g(a) =⊥] = 2−L. +Initialize empty streams S1, S2, · · · , SL. +for each at in the stream S do +for i ∈ [L] do +if at ̸=⊥ and g(at) = i then +Append h(at) to the end of the stream Si. +end +else +Append ⊥ to the end of the stream Si. +end +end +∀i ∈ [L], compute ˆsi which is an (α, γ)-approximation to ∥Si∥0. +Find the largest i ∈ [L] such that ˆsi ≥ max +� +γ/η, 32αλ/η2� +, and output ˆsi · 2i. +If such i does not exist, output 0. +end +Lemma 4.6. Consider any timestamp t ∈ [T ]. Let v be the output of Algorithm 3. With probability +at least 0.9, v is a ((1 + O(η))α, O(α2 max(γ/η, α log(L)/η2)))-approximation to ∥(a1, a2, · · · , at)∥0. +To prove Lemma 4.6, we need following intermediate statements. We consider a timestamp +t ∈ [T ]. Let S denote the input stream at timestamp t, i.e., S = (a1, a2, · · · , at). Let Gi = {aj | +g(aj) = i, j ≤ t} for i ∈ [L]. +Claim 4.7. ∀i ∈ [L], if ∥S∥0 ≥ 2i · 4λ/η2, Pr[|Gi| ∈ (1 ± η) · ∥S∥0/2i] ≥ 1 − 0.01/L. Otherwise, +Pr[||Gi| − ∥S∥0/2i| ≤ 4λ/η] ≥ 1 − 0.01/L. +Proof. Suppose ∥S∥0 ≥ 2i · 4λ/η2. Due to Lemma 2.6, we have: +Pr +���|Gi| − ∥S∥0/2i�� > η · ∥S∥0/2i� +≤8 · +� +λ · ∥S∥0/2i + λ2 +(η · ∥S∥0/2i)2 +�λ/2 +≤0.01/L, +where the last inequality follows from that λ = 2 · log(1000L) and ∥S∥0 ≥ 2i · 4λ/η2. +Suppose ∥S∥0 ≤ 2i · 4λ/η2, By applying Lemma 2.6 again, we have: +Pr +���|Gi| − ∥S∥0/2i�� > 4λ/η +� +15 + +≤8 · +�λ · ∥S∥0/2i + λ2 +(4λ/η)2 +�λ/2 +≤0.01/L, +where the last inequality follows from ∥S∥0/2i ≤ 4λ/η2 and λ = 2 · log(1000L). +Claim 4.8. For i ∈ [L], conditioning on |Gi| ≤ 16α max +� +γ/η, 32αλ/η2� +, the probability that +|Gi| = ∥Si∥0 is at least 1 − 0.01/L. +Proof. Since h is pairwise independent, ∀a, b ∈ Gi, Pr[h(a) = h(b)] = 1/m. Since m = 100L · +� +16α max +� +γ/η, 32αλ/η2��2, Pr[∃a ̸= b ∈ Gi, h(a) = h(b)] ≤ |Gi|2/m ≤ 0.01/L by a union bound. +Let E be the event that both of the following hold: +1. ∀i ∈ [L] with 2i · 4λ/η2 ≤ ∥S∥0, |Gi| ∈ (1 ± η) · ∥S∥0/2i. +2. ∀i ∈ [L] with 2i · 4λ/η2 > ∥S∥0, |Gi| ∈ ∥S∥0/2i ± 4λ/η. +According to Claim 4.7, E happens with probability at least 0.99. Let E′ be the event that ∀i ∈ [L] +with |Gi| ≤ 16α max(γ/η, 32αλ/η2), |Gi| = ∥Si∥0. +According to Claim 4.8, E′ happens with +probability at least 0.99. +Next, we are going to prove Lemma 4.6. +Proof of Lemma 4.6. In this proof, we condition on both events E and E′. Note that the probability +that both E and E′ happen is at least 0.98. +Consider the case that ∥S∥0 ≥ 8α · max +� +γ/η, 32αλ/η2� +. +Let i∗ ∈ [L] be the largest value +such that ∥S∥0/2i∗ ≥ 4α · max(γ/η, 32αλ/η2). +According to event E, we have |Gi∗| ∈ (1 ± +η) · ∥S∥0/2i∗. +Due to our choice of i∗, we have ∥S∥0/2i∗ ≤ 8α · max +� +γ/η, 32αλ/η2� +. +Thus, +|Gi∗| ≤ 16α max +� +γ/η, 32αλ/η2� +. +According to event E′, we have ∥Si∗∥0 = |Gi∗|. +Therefore, +we have ˆsi∗ ≥ ∥Si∗∥0/α − γ ≥ |Gi∗|/α − γ. Since |Gi∗| ≥ 2α · max +� +γ/η, 32αλ/η2� +, we have +ˆsi∗ ≥ max +� +γ/η, 32αλ/η2� +. +Therefore, Algorithm 3 will output ˆsi′ · 2i′ for some i′ ≥ i∗. +Due +to event E, we know |Gi′| ≤ max(2 · ∥S∥0/2i∗, ∥S∥0/2i∗ + 4λ/η) ≤ 16α max(γ/η, 32αλ/η2). Ac- +cording to event E′, we have ∥Si′∥0 = |Gi′|. Since the algorithm outputs ˆsi′ · 2i′, we know that +ˆsi′ ≥ max(γ/η, 32αλ/η2) which implies that +|Gi′| = ∥Si′∥0 +≥ (ˆsi′ − γ)/α +≥ (1 − η)ˆsi′/α +≥ 16λ/η2. +According to event E, we have |Gi′| ∈ (1 ± η) · ∥S∥0/2i′. Thus, we have +ˆsi′ ≤ α∥Si′∥0 + γ += α|Gi′| + γ +≤ +α +1 − η · |Gi′| +16 + +≤ 1 + η +1 − η · α · ∥S∥0/2i′ +≤ (1 + 4η)α∥S∥0/2i′, +where the second inequality follows from ˆsi′ ≥ γ/η and the last inequality follows from η ≤ 0.5. +Similarly, we have: +ˆsi′ ≥ ∥Si′∥0/α − γ += |Gi′|/α − γ +≥ |Gi′|/((1 + η)α) +≥ 1 − η +1 + η · 1 +α · ∥S∥0/2i′ +≥ (1 − 4η)/α · ∥S∥0/2i′, +where the second inequality follows from ˆsi′ ≥ γ/η. +Next, consider the case that ∥S∥0 < 8α · max(γ/η, 32αλ/η2). Algorithm 3 either outputs 0 or +outputs ˆsi′ · 2i′ for some i′ ∈ [L]. Suppose it outputs ˆsi′ · 2i′. We have ˆsi′ ≥ max +� +γ/η, 32αλ/η2� +, +which implies that +|Gi′| ≥ ∥Si′∥0 ≥ (ˆsi′ − γ)/α ≥ (1 − η)ˆsi′/α ≥ 16λ/η2. +According to event E, we have |Gi′| ≤ 2 · ∥S∥0/2i′. Therefore +ˆsi′ · 2i′ ≤ (α∥Si′∥0 + γ) · 2i′ +≤ +α +1 − η · ∥Si′∥0 · 2i′ +≤ +α +1 − η · |Gi′| · 2i′ +≤ 4α∥S∥0 +≤ 32α2 max(γ/η, 32αλ/η2). +Theorem 4.9. Let ε ≥ 0, ξ, ξ′ ∈ (0, 0.5), η ∈ (0, 0.5), suppose there is an ε-DP algorithm for +the number of distinct elements of streams with element universe size at most 100 log(min(|U|, T )) · +(16α max(γ/η, 32α·2 log(1000 log(min(|U|, T )))/η2))2 in the streaming continual release model which +uses space J and with probability at least 1 − ξ always outputs an (α, γ)-approximation for ev- +ery timestamp. There is an (ε′ = ⌈50 log(T/ξ′)⌉ε)-DP algorithm for the number of distinct ele- +ments of streams with element universe U in the streaming continual release model. With prob- +ability at least 1 − ξ′ − log(min(|U|, T )) · ⌈50 log(T/ξ′)⌉ · ξ, the algorithm always outputs an ((1 + +O(η))α, O(α2 max(γ/η, α log log(min(|U|, T ))/η2)))-approximation for every timestamp t ∈ [T ]. The +algorithm uses O(J · log(min(|U|, T )) · log(T/ξ′)) space. +Proof. According to our construction of S1, S2, · · · , SL and the definition of the sensitivity (Defi- +nition 2.3), (S1, S2, · · · , SL) has sensitivity 1. Since the algorithm to report ˆsi for each i ∈ [L] is +17 + +ε-DP, Algorithm 3 is ε-DP. According to Lemma 4.6, for any t ∈ [T ], condition on that ˆsi is an (α, γ)- +approximation to ∥S∥0, with probability at least 0.9, the output is a ((1+O(η))α, O(α2 max(γ/η, α log log(min(|U|, T ))/η2)))- +approximation. To boost the probability to 1 − ξ′ such that ∀t ∈ [T ] the approximation guarantee +always holds, we need to run ⌈50 log(T/ξ′)⌉ independent copies of Algorithm 3 and take the median +of the outputs at every timestamp. Thus, the overall algorithm is (⌈50 log(T/ξ′)⌉ · ε)-DP. +Next, consider the success probability that every ˆsi is an (α, β)-approximation to ∥Si∥0. By +taking a union bound over all i and all independent copies of Algorithm 3, the success probability +is at least 1 − log(min(|U|, T )) · ⌈50 log(T/ξ′)⌉ · ξ. +Finally, consider the space usage. Consider each running copy of Algorithm 3. Hashing function +h(·) takes O(1) space. +Hashing function g(·) takes O(λ) = O(log log(min(|U|, T ))) space. +To +continually release ˆsi for all i, we need to use O(J · log(min(|U|, T ))) space. Thus, the total space +needed for all copies is at most O(J · log(min(|U|, T )) · log(T/ξ′)). +By plugging Corollary 4.4 into above theorem with ξ = +ξ′/2 +log(min(|U|,T ))·⌈50 log(2T/ξ′)⌉, ε = +ε′ +⌈50 log(2T/ξ′)⌉, +α = 1, γ = O( 1 +ε log2.5(T ) log(1/ξ)) and J = O(log(T ) + 100 log(min(|U|, T )) · (16α max(γ/η, 32α · +2 log(1000 log(min(|U|, T )))/η2))2), we get the following corollary. +Corollary 4.10 (Streaming continual release distinct elements). For η ∈ (0, 0.5), there is an +ε-DP algorithm for the number of distinct elements of streams with element universe U in the +streaming continual release model. +With probability at least 1 − ξ, the output is always a (1 + +η, O +� +max +� +log(T/ξ) log2.5(T ) log(1/ξ) log log(T/ξ) +ηε +, log log T +η2 +�� +)-approximation for every timestamp t ∈ [T ]. +The algorithm uses poly +� +log(T/ξ) +η min(ε,1) +� +space. +By plugging Corollary 4.5 into Theorem 4.9 with ξ = +ξ′/2 +log(min(|U|,T ))·⌈50 log(2T/ξ′)⌉, ε = +ε′ +⌈50 log(2T/ξ′)⌉, +α = 1 + η, γ = O +� +log(T/ξ) +εη +� +and +J = O(100 log(min(|U|, T )) · (16α max(γ/η, 32α · 2 log(1000 log(min(|U|, T )))/η2))2), +we get the following corollary. +Corollary 4.11 (Streaming continual release distinct elements, better dependence in log(T )). For +η ∈ (0, 0.5), there is an ε-DP algorithm for the number of distinct elements of streams with element +universe U in the streaming continual release model. With probability at least 1 − ξ, the output is +always a (1 + O(η), O +� +log2(T/ξ) +η2ε +� +)-approximation for every timestamp t ∈ [T ]. The algorithm uses +poly +� +log(T/ξ) +η min(ε,1) +� +space. +5 +Continual Released ℓp Heavy Hitters and Frequency Mo- +ment Estimation +In this section, we present ε-DP streaming continual release algorithms for ℓp heavy hitters and fre- +quency moment estimation. In Section 5.1, we present an algorithm for ε-DP CountSketch Charikar et al. +[2002] in the streaming continual release model. The CountSketch is used for ℓ2 heavy hitters and +ℓ2 moment estimation. In Section 5.2, we show how to use ℓ2 heavy hitters to solve ℓp heavy hitters. +In Section 5.3, we show how to estimate the number of elements which have low frequencies. In +18 + +Section 5.4, we show how to use ℓp heavy hitters and the estimator of low frequency elements to +estimate the ℓp frequency moment. +5.1 +Continual Released CountSketch +Algorithm 4: Continual Released CountSketch +Input: A stream S of elements a1, a2, · · · , aT ∈ U ∪ {⊥}, a parameter k ∈ Z≥1. +Parameters : Relative approximation factor α ≥ 1 and additive approximation factor γ ≥ 0 +depending on the streaming continual release summing algorithm. +//See Theorem 2.5. +Output: A tuple (z1, z2, · · · , zk) at every timestamp t. +Let h : U → [k] be a 4-wise independent hash function, s.t., ∀a ∈ U, i ∈ [k], Pr[h(a) = i] = 1 +k. +Let g : U → {−1, 1} be a 4-wise independent hash function, s.t., ∀a ∈ U, Pr[g(a) = 1] = 1 +2. +Initialize empty streams S1, S2, · · · , Sk. +for each at in the stream S do +if at =⊥ then +Append 0 to the end of every stream S1, S2, · · · , Sk. +end +else +Append g(at) to the end of the stream Sh(at) and append 0 to the end of every stream Si +for i ̸= h(at). +end +Output a tuple (z1, z2, · · · , zk) where zi is an estimation of the total counts of Si with additive +error at most γ. +end +Lemma 5.1 (DP guarantee). If the subroutine of continually releasing the approximate total counts +of Si for every i ∈ [k] in Algorithm 4 is ε-DP, Algorithm 4 is 2ε-DP. +Proof. Consider two neighboring streams S and S′ where the only difference is the t-th element, +i.e., at ̸= a′ +t. Consider their corresponding streams S1, S2, · · · , Sk and S′ +1, S′ +2, · · · , S′ +k in Algorithm 4. +For i ∈ [k] and j ̸= t, the j-th number in Si should be the same as the j-th number in S′ +i. Thus, +we only need to consider the t-th number in Si and S′ +i for every i ∈ [k]. Since at can only make the +t-th number of Sh(at) be non-zero, a′ +t can only make the t-th number of S′ +h(a′ +t) be non-zero, and the +non-zero number can be only ±1, the sensitivity is at most 2. Thus, if we use ε-DP algorithm to +continually release the total counts of S1, S2, · · · , Sk, the continually released output of Algorithm 4 +is 2ε-DP. +Lemma 5.2 (Good approximation for frequent elements). Consider any a ∈ U and any timestamp +t ∈ [T ]. Let fa be the frequency of a in a1, a2, · · · , at. Let (z1, z2, · · · , zk) be the output of Algorithm 4 +at timestamp t. +Then ∀η ∈ (0, 0.5), with probability at least 1 − 1/(kη2), |fa − g(a) · zh(a)| ≤ +η · +�� +b∈U f 2 +b + γ. +Proof. Let ˆzh(a) be the true total counts of stream Sh(a) at timestamp t. According to the original +CountSketch Charikar et al. [2002], we have Pr +� +|fa − g(a) · ˆzh(a)| ≤ η +�� +b∈U f 2 +b +� +≥ 1 − 1/(kη2). +Since +|fa − g(a) · zh(a)| ≤ |fa − g(a) · ˆzh(a)| + |g(a)| · |zh(a) − ˆzh(a)| ≤ |fa − g(a) · ˆzh(a)| + γ, +19 + +we have with probability at least 1 − 1/(kη2), |fa − g(a) · zh(a)| ≤ η · +�� +b∈U f 2 +b + γ. +Lemma 5.3 (ℓ2 Frequency moment estimation). Consider any timestamp t ∈ [T ]. For a ∈ U, +let fa be the frequency of a in a1, a2, · · · , at. Let (z1, z2, · · · , zk) be the output of Algorithm 4 at +timestamp t. Then ∀η ∈ (0, 0.5), with probability at least 1 − 100/(kη2), | �k +i=1 z2 +i − � +a∈U f 2 +a| ≤ +η � +a∈U f 2 +a + 4kγ2/η +Proof. Let F2 = � +a∈U f 2 +a. Let Z = � +i∈[k] z2 +i . For i ∈ [k], let ˆzi be the true total counts of stream Si +at timestamp t. Let ˆZ = � +i∈[k] ˆz2 +i . According to the analysis of CountSketch Charikar et al. [2002], +Thorup and Zhang [2004], We have Pr +� +|F2 − ˆZ| ≤ η/4 · F2 +� +≥ 1 − 100/(kη2). In the following, we +condition on |F2 − ˆZ| ≤ η/4 · F2. +We have +|F2 − Z| +≤|F2 − ˆZ| + | ˆZ − Z| +≤η/4 · F2 + +k +� +i=1 +|z2 +i − ˆz2 +i |. +Denote zi = ˆzi + vi for i ∈ [k]. We have |vi| ≤ γ. Due to convexity, we have: +(1 − η/4)ˆz2 +i − 4v2 +i /η ≤ (ˆzi + vi)2 ≤ ˆz2 +i /(1 − η/4) + 4v2 +i /η. +Since η ∈ (0, 0.5), 1/(1 − η/4) ≤ (1 + η/2). Therefore, |z2 +i − ˆz2 +i | ≤ η/2 · ˆz2 +i + 4v2 +i /η. We have: +|F2 − Z| ≤ η/4 · F2 + η/2 · ˆZ + 4kγ2/η ≤ η/4 · F2 + η/2 · 3/2 · F2 + 4kγ2/η ≤ ηF2 + 4kγ2/η. +Theorem 5.4 (Streaming continual release ℓ2 frequency estimators). Let ε > 0, η ∈ (0, 0.5), ξ ∈ +(0, 0.5). There is an ε-DP algorithm in the streaming continual release model such that with proba- +bility at least 1 − ξ, it always outputs for every timestamp t ∈ [T ]: +1. ˆfa for every a ∈ U such that |fa− ˆfa| ≤ η∥S∥2+O +� +log(T/ξ)+log(|U|) +ε +· log2.5(T ) · log +� +log(T/ξ)+log(|U|) +ξη +�� +, +where S denotes the stream (a1, a2, · · · , at) and fa denotes the frequency of a in S, +2. ˆF2 such that | ˆF2 − ∥S∥2 +2| ≤ η∥S∥2 +2 + O +� +(log(T/ξ)+log(|U|))2 +ε2η3 +· log5(T ) · log2 � +log(T/ξ)+log(|U|) +ξη +�� +The algorithm uses O +� +log(T/ξ)+log(|U|) +η2 +· log(T ) +� +space. +Proof. Suppose we set k = 400/η2. Due to Lemma 5.2 and Lemma 5.3, the approximation guar- +antees hold with probability at least 2/3 for each particular timestamp t ∈ [T ] and a ∈ U. To +boost the success probability to 1 − ξ/2 for the approximation guarantees and simultaneously for +all t ∈ [T ] and all a ∈ |U|, according to Lemma 2.7, we run ⌈50(log(2T/ξ) + log(|U|))⌉ copies of +Algorithm 4 and take the median of each estimator. +We apply Theorem 2.5 for the summing problem of Si for each i ∈ [k]. Since we run ⌈50(log(2T/ξ)+ +log(|U|))⌉ copies of Algorithm 4, if we desire ε-DP algorithm in the end, we need each summing +20 + +subroutine to be ε/(2 · ⌈50(log(2T/ξ) + log(|U|))⌉)-DP according to Lemma 5.1. +To simultane- +ously make the call of each run of the summing subroutine succeeds with probability at least +1 − ξ/2, we need to apply union bound over all calls of summing and thus each run of the sum- +ming subroutine should success with probability at least 1 − ξ/(2 · ⌈50(log(2T/ξ) + log(|U|))⌉ · k) = +1 − ξ/(2 · ⌈50(log(2T/ξ) + log(|U|))⌉ · 400/η2). Thus, according to Theorem 2.5, we have α = 1 and +γ = O +� +log(T/ξ)+log(|U|) +ε +· log2.5(T ) · log +� +log(T/ξ)+log(|U|) +ξη2 +�� +. +Finally, let us consider the total space usage, since we call (⌈50(log(2T/ξ) + log(|U|))⌉ · 400/η2) +times of summing subroutine, the space needed for executing them is O +� +log(T/ξ)+log(|U|) +η2 +· log(T ) +� +. +Additional space needed is O(1). Thus, the total space required is O +� +log(T/ξ)+log(|U|) +η2 +· log(T ) +� +5.2 +Continual Released ℓp Heavy Hitters +By applying the CountSketch, we are able to develop ℓp heavy hitters. +Algorithm 5: Continual Released ℓp Heavy Hitters (p ∈ [0, ∞)) +Input: A stream S of elements a1, a2, · · · , aT ∈ U ∪ {⊥}, a parameter k ∈ Z≥1, an error +parameter η ∈ (0, 0.5). +Parameters : Additive error parameters γ1, γ2 ≥ 0 depending on the streaming continual release +CountSketch algorithm. +//See Theorem 5.4. +Output: A set H ⊆ U of elements and their estimated frequencies ˆf : H → R≥0 at every +timestamp t. +Let φ ≥ max +� +|U|1−2/p, 1 +� +. Let m = 10k2. +Let h : U → [m] be a pairwise independent hash function where +∀a ∈ U, i ∈ [m], Pr[h(a) = i] = 1/m. +Initialize empty streams S1, S2, · · · , Sm. for each at in the stream S do +if at =⊥ then +Append ⊥ to the end of every stream S1, S2, · · · , Sm. +end +else +Append at to the end of the stream Sh(at) and append ⊥ to the end of every stream Si for +i ̸= h(at). +end +For i ∈ [m], compute ˆF2,i which is a (1.1, γ1)-approximation to ∥Si∥2 +2. +For a ∈ U, compute ˆfa which is a (1, (η/16)/(10√φk) · ∥Sh(a)∥2 + γ2)-approximation to fa, the +frequency of a in S (or equivalently in Sh(a)). +For a ∈ U, if ˆf 2 +a ≥ +ˆ +F2,h(a)+γ1 +25φk ++ +512γ2 +2 +η2 +, add a into ˆH. +Let H ⊆ ˆH only keep the elements a such that ˆfa is one the top- +�� +1+η +1−η +�p +· k +� +values among +{ ˆfb | b ∈ ˆH}. For each a ∈ H, report ˆf(a) ← ˆfa. +end +Lemma 5.5 (DP guarantee). If ∀i ∈ [m] the subroutine in Algorithm 5 of continually releasing +ˆF2,i and ˆfa for all a satisfying h(a) = i is ε-DP, Algorithm 5 is 2ε-DP. +Proof. Consider two neighboring streams S and S′ where the only difference is the t-th element, +i.e., at ̸= a′ +t. If at ̸=⊥, it only causes the difference of at most one element between Sh(at) and +S′ +h(at). Similarly, if a′ +t ̸=⊥, it only causes the difference of at most one element between Sh(a′ +t) and +21 + +S′ +h(a′ +t). Thus if for each i ∈ [m], the continual release algorithm which releases ˆF2,i and ˆfa for every +a ∈ U with h(a) = i is ε-DP, the overall algorithm is 2ε-DP. +Lemma 5.6. At any timestamp t ∈ [T ], ∀a ∈ U, if a ∈ ˆH, (1 − η)f 2 +a ≤ ˆf 2(a) ≤ (1 + η)f 2 +a where fa +is the frequency of a in a1, a2, · · · , at. +Proof. Since a ∈ ˆH, we have: +ˆf 2 +a ≥ +ˆF2,h(a) + γ1 +25φk ++ 512γ2 +2 +η2 +. +Thus: +η +16 · ˆf 2 +a ≥ 16 +η · +�2 · (η/16)2 +100φk +· +� +2 ˆF2,h(a) + 2γ1 +� ++ 2γ2 +2 +� +≥ 16 +η · +�2 · (η/16)2 +100φk +· ∥Sh(a)∥2 +2 + 2γ2 +2 +� += 16 +η · +� +2 · +� η/16 +10√φk · ∥Sh(a)∥2 +�2 ++ 2γ2 +2 +� +≥ 16 +η · +� η/16 +10√φk · ∥Sh(a)∥2 + γ2 +�2 +By convexity: +ˆf 2 +a ≥ (1 − η/16) · f 2 +a − 16 +η · +� η/16 +10√φk · ∥Sh(a)∥2 + γ2 +�2 +≥ (1 − η/16) · f 2 +a − η/16 · ˆf 2 +a +and +ˆf 2 +a ≤ 1/(1 − η/16) · f 2 +a + 16 +η · +� η/16 +10√φk · ∥Sh(a)∥2 + γ2 +�2 +≤ 1/(1 − η/16) · f 2 +a + η/16 · ˆf 2 +a. +Since η ∈ (0, 0.5), we have (1 − η)f 2 +a ≤ ˆf 2 +a ≤ (1 + η)f 2 +a. +Lemma 5.7. At any timestamp t, the output H of Algorithm 5 has size at most +� +1+η +1−η +�p +· k. +Proof. Note that H only keeps the top- +�� +1+η +1−η +�p +· k +� +values from ˆH. +Lemma 5.8. At any timestamp t, consider any a ∈ U. Let fa be the frequency of a in a1, a2, · · · , at. +If fa ≥ 4 +� +γ1/(φk) + 512γ2 +2/η2 and f p +a ≥ ∥S∥p +p/k, with probability at least 0.9, a ∈ ˆH. +Proof. In this proof, we consider all streams and variables at the timestamp t. Suppose f p +a ≥ ∥S∥p +p/k. +Let B = {b ∈ U | f p +b ≥ ∥S∥p +p/k}. Then with probability at least 0.9, ∀b ∈ B \ {a}, h(a) ̸= h(b). In +the remaining of the proof, we condition on ∀b ∈ B \ {a}, h(a) ̸= h(b). +Case 1, p ≤ 2: In this case, we have φ = 1. ∀x ∈ U with h(x) = h(a), we have fx ≤ fa which +implies that f p−2 +x +≥ f p−2 +a +since p ≤ 2. Since ∀x ∈ U, f p +x +f 2 +x ≤ 1, we have +∥Sh(a)∥p +p +∥Sh(a)∥2 +2 += +� +x∈U:h(x)=h(a) f p +x +� +x∈U:h(x)=h(a) f 2x +≥ +min +x∈U:h(x)=h(a) +f p +x +f 2x += f p +a +f 2a +, +22 + +which implies that f 2 +a/∥Sh(a)∥2 +2 ≥ f p +a/∥Sh(a)∥p +p ≥ f p +a/∥S∥p +p ≥ 1/k and thus f 2 +a ≥ ∥Sh(a)∥2 +2/(φk). +Case 2, p > 2: In this case, we have φ = |U|1−2/p. Since f p +a ≥ ∥S∥p +p/k, we have: +fa ≥ ∥S∥p/k1/p ≥ ∥S∥p/k1/2 ≥ ∥S∥2/(k1/2 · |U|1/2−1/p), +where the second inequality follows from k1/2 ≥ k1/p for p > 2, and the third inequality follows +from Holder’s inequality that ∥S∥2 ≤ |U|1/2−1/p · ∥S∥p. Therefore, f 2 +a ≥ ∥S∥2 +2/(φk). +Therefore, in both above cases, we always have f 2 +a ≥ ∥S∥2 +2/(φk). +By convexity, we have +ˆf 2 +a ≥ (1 − η/16) · f 2 +a − 16 +η · +� η/16 +10√φk · ∥Sh(a)∥2 + γ2 +�2 +≥ (1 − η/16) · f 2 +a − 16 +η · +�2(η/16)2 +100φk +· ∥Sh(a)∥2 +2 + 2γ2 +2 +� +Thus we have: +ˆf 2 +a ≥ 1 +2 · ∥Sh(a)∥2 +2/(φk) − 16 +η · +�2(η/16)2 +100φk +· ∥Sh(a)∥2 +2 + 2γ2 +2 +� +≥ 1 +2 · 1 +2 · ( ˆF2,h(a) − γ1)/(φk) − 16 +η · +�2(η/16)2 +100φk +· 2 · ( ˆF2,h(a) + γ1) + 2γ2 +2 +� += +�1 +4 − η/16 +25 +� +· +ˆF2,h(a) +φk +− +�1 +4 + η/16 +25 +� +· γ1 +φk − 32 +η · γ2 +2 +≥ 1 +5 · +ˆF2,h(a) +φk +− 1 +3 · γ1 +φk − 32 +η2 · γ2 +2 +≥ 2 · +� ˆF2,h(a) + γ1 +25φk ++ 512γ2 +2 +η2 +� +− +�2048γ2 +2 +η2 ++ 2 +3 · γ1 +φk − +3 +25φk · ˆF2,h(a) +� +(1) +On the other hand, by convexity, we have: +ˆf 2 +a ≥ (1 − η/16)f 2 +a − 16 +η · +� η/16 +10√φk ∥Sh(a)∥2 + γ2 +�2 +≥ (1 − η/16)f 2 +a − 16 +η · +�2(η/16)2 +100φk ∥Sh(a)∥2 +2 + 2γ2 +2 +� +and thus +ˆf 2 +a ≥ 1 +2 · 16 · (γ1/(φk) + 512γ2 +2/η2) − 16 +η · +�2(η/16)2 +100φk +· 2 · ( ˆF2,h(a) + γ1) + 2γ2 +2 +� += +� +8 − η/16 +25k +� +· γ1 +φk + +�4096 +η2 +− 32 +η +� +γ2 +2 − η/16 +25φk · ˆF2,h(a) +≥ 2 +3 · γ1 +φk + 2048 +η2 +· γ2 +2 − +3 +25φk · ˆF2,h(a) +(2) +By looking at Equation (1) + Equation (2), we have ˆf 2 +a ≥ +ˆ +F2,h(a)+γ1 +25φk ++ 512γ2 +2 +η2 . Thus, a ∈ ˆH. +23 + +Lemma 5.9. At any timestamp t, if f p +a ≥ ∥S∥p +p/k and a ∈ ˆH, then a ∈ H. +Proof. We prove the statement by contradiction. Suppose a ̸∈ H, there is a subset Q ⊆ H with +|Q| ≥ +� +1+η +1−η +�p +· k such that ∀b ∈ Q, ˆfb ≥ ˆfa. According to Lemma 5.6, we have ∀b ∈ Q, f p +b ≥ +� +1−η +1+η +�p +·f p +a ≥ +� +1−η +1+η +�p +·∥S∥p +p/k. Then we have � +b∈Q f p +b ≥ ∥S∥p +p which leads to a contradiction. +Theorem 5.10 (ℓp Heavy hitters for all p ∈ [0, ∞)). Let ε > 0, η ∈ (0, 0.5), k ≥ 1, ξ ∈ (0, 0.5). Let +φ = max(1, |U|1−2/p). There is an ε-DP algorithm in the streaming continual release model such +that with probability at least 1 − ξ, it always outputs a set H ⊆ U and a function ˆf : H → R for +every timestamp t ∈ [T ]: such that +1. ∀a ∈ H, ˆf(a) ∈ (1 ± η) · fa where fa is the frequency of a in the stream S = (a1, a2, · · · , at), +2. ∀a ∈ U, if fa ≥ +1 +εη ·logC � +T ·k·|U| +ξη +� +for some sufficiently large constant C > 0 and f p +a ≥ ∥S∥p +p/k +then a ∈ H, +3. The size of H is at most O +� +(log(T/ξ) + log(|U|)) · +� +1+η +1−η +�p +· k +� +. +The algorithm uses φk3 +η2 · poly +� +log +� +T ·k·|U| +ξ +�� +space. +Proof. The first property follows from Lemma 5.6. +According to Lemma 5.8 and Lemma 5.9, the guarantee holds with probability 0.9 for any +particular t ∈ [T ] and a ∈ U. To boost the probability to make the guarantee holds simultane- +ously for all t ∈ [T ] and a ∈ U with probability at least 1 − ξ/2, we need to repeat Algorithm 5 +⌈50(log(2T/ξ) + log(|U|))⌉ times, and let final H at timestamp t be the union of all output H at +timestamp t, and let final ˆf(a) at timestamp t be any output ˆf(a) at timestamp t. According to +Lemma 5.7, the output H of a single running copy of Algorithm 5 is at most +� +1+η +1−η +�p +· k. Thus, +the size of the final output H is at most O +� +(log(T/ξ) + log(|U|)) · +� +1+η +1−η +�p +· k +� +which proves the +second property. +We apply Theorem 5.4 for the frequency and ℓ2 frequency moment estimators of Si for each +i ∈ [m]. +Since we run ⌈50(log(2T/ξ) + log(|U|))⌉ copies of Algorithm 5, if we desire ε-DP al- +gorithm in the end, we need each frequency and ℓ2 frequency moment subroutine to be ε/(4 · +⌈50(log(2T/ξ) + log(|U|))⌉)-DP according to Lemma 5.5. +To simultaneously make the call of +each run of the frequency and ℓ2 frequency moment subroutine succeeds with probability at least +1 − ξ/2, we need to apply union bound over all calls of the subroutines and thus each run of +the subroutine should succeed with probability at least 1 − ξ/(4 · ⌈50(log(T/ξ) + log(|U|))⌉ · m) = +1 − ξ/(4 · ⌈50(log(T/ξ) + log(|U|))⌉ · 10k2). Notice that we also need to re-scale η in Theorem 5.4 +to be (η/16)/(10√φk) used in Algorithm 5 for estimation of the frequency of each element and +set η = 0.01 in Theorem 5.4 for the estimation of the ℓ2 frequency moment. Thus, according to +Theorem 5.4, we have +γ1 = 1 +ε2 · poly +� +log +�T · k · |U| +ξ +�� +24 + +and +γ2 = 1 +ε · poly +� +log +�T · k · |U| +ξη +�� +Therefore, the second property follows from Lemma 5.8 and Lemma 5.9 and our probability boosting +argument. +Finally, let us consider the space usage. +Since we run O(log(T/ξ) + log(|U|)) copies of Al- +gorithm 5, and each copy calls O(k2) frequency estimator and ℓ2 frequency moment estimator +using the parameters discussed above. +Due to Theorem 5.4, the total space usage is at most +φk3 +η2 · poly +� +log +� +T ·k·|U| +ξ +�� +. +5.3 +Differentially Private Continual Released Counting of Low Frequency +Elements +In this section, we show a differentially private continual released algorithm for counting the number +of elements that have a certain (low) frequency. Similar to our counting distinct elements algorithm, +we first consider the case where the universe of the elements is small. +5.3.1 +Number of Low Frequency Elements for Small Universe +Algorithm 6: Number of Low Frequency Elements for Small Universe +Input: A stream S of elements a1, a2, · · · , aT ∈ U ∪ {⊥} with gaurantee that |U| ≤ m, and a +target frequency k. +Parameters : Relative approximation factor α ≥ 1 and additive approximation factor γ ≥ 0 +depending on the streaming continual release summing algorithm. +//See Theorem 2.5. +Output: Estimation of the number of elements with frequency exactly i for each i ∈ [k] at every +timestamp t. +Initialize empty streams C1, C2, · · · , Ck. +For each a ∈ U, initialize frequency f(a) ← 0 +for each at in the stream S do +If at ̸=⊥, f(at) ← f(at) + 1. +for each i ∈ [k] do +if at ̸=⊥ and f(at) = i + 1 then +Append −1 at the end of Ci. +end +else if at ̸=⊥ and f(at) = i then +Append 1 at the end of Ci. +end +else +Append 0 at the end of Ci. +end +end +For each i ∈ [k], output ˆsi which is an (α, γ)-approximation to the total counts of Ci. +end +Lemma 5.11. At the end of any time t ∈ [T ], ∀i ∈ [k], the output ˆsi of Algorithm 6 is an (α, γ)- +approximation to |{a ∈ U | fa = i}|, the size of the set of elements of which the frequency is exact +i. +25 + +Proof. It is easy to observe that Algorithm 6 maintains f(a) such that f(a) = fa for any a ∈ U. +For i ∈ [k], if before adding at the frequency of at is i, we append −1 to Ci. If after adding at the +frequency of at becomes i, we append 1 to Ci. Note that if the frequency of at has the value which +is not in above two cases, it does not affect the number of elements with frequency i. Therefore, +the total counts of Ci is always the number of elements of which frequency is i. Thus, ˆsi is an +(α, γ)-approximation to the number of elements of which frequency is i. +Lemma 5.12. If the algorithm that continually release the approximate total counts of Ci for every +i ∈ [k] is ε-DP, Algorithm 6 is (8kε)-DP in the continual release model. +Proof. Consider two neighboring stream S = (a1, a2, · · · , aT ) and S′ = (a′ +1, a′ +2, · · · , a′ +T ) of elements +in U where they only differ at timestamp t, i.e., at ̸= a′ +t. Fix any i ∈ [k], let us consider the +differences between the corresponding count streams Ci = (c1, c2, · · · , cT ) and C′ +i = (c′ +1, c′ +2, · · · , c′ +T ). +Consider an intermediate neighboring stream S′′ = (a1, a2, · · · , at−1, ⊥, at+1, · · · , aT ). Let C′′ +i = +(c′′ +1, c′′ +2, · · · , c′′ +T ) be the count stream corresponding to S′′. The total difference between Ci and C′ +i is +bounded by the sum of the total difference between Ci and C′′ and the total difference between C′ +i +and C′′ +i . +Consider the difference between Ci and C′′ +i . If at =⊥, C′′ +i is exactly the same as Ci. Suppose +at = a ∈ U is the j-th appearance of a in S. If j > i, change at to ⊥ does not affect Ci. Suppose +j ≤ i. Let at1, at2, at3 be the i-th, (i + 1)-th, (i + 2)-th appearances of a in S respectively. Then +it is easy to verify that ct1 = 1, c′′ +t1 = 0, ct2 = −1, c′′ +t2 = 1, ct3 = 0, c′′ +t3 = −1, and for any other +p ̸= t1, t2, t3, we have cp = c′′ +p. Thus, the total difference between Ci and C′′ +i is at most 4. +Similarly, the total difference between C′ +i and C′′ +i is at most 4. Thus, the total difference between +Ci and C′ +i is at most 8. Therefore, the total sensitivity of C1, C2, · · · , Ck is at most 8k. If we use +an ε-DP algorithm to continually release the total counts of Ci for every i ∈ [k], the continually +released outputs of Algorithm 6 is (8kε)-DP. +Theorem 5.13 (Streaming continual release count of low frequency elements for small universe). +Let k ≥ 1, ε ≥ 0, ξ ∈ (0, 0.5). Suppose the universe U has size at most m. There is an ε-DP +algorithm in the streaming continual release model such that with probability at least 1−ξ, it always +outputs k numbers ˆs1, ˆs2, · · · , ˆsk for every timestamp t, such that ∀i ∈ [k], ˆsi is an approximation to +|{a ∈ U | fa = i}| with additive error O +� +k +ε · log2.5(T ) log +� +k +ξ +�� +. The algorithm uses O(m+k log(T )) +space. +Proof. Consider Algorithm 6, we use Theorem 2.5 as the summing subroutine. +According to +Lemma 5.12, if we want the final algorithm to be ε-DP, then each subroutine must be (ε/(8k))-DP. +Furthermore, if we want the over success probability to be 1 − ξ, the success probability of each +summing subroutine should be at least 1 − ξ/k. By applying Theorem 2.5, we have α = 1 and +γ = O +� +k +ε · log2.5(T ) log +� +k +ξ +�� +. +We need O(m) space to maintain the frequency of the elements. We need O(k log(T )) space to +run k summing subroutines. Therefore, the total space usage is O(m + k log(T )). +26 + +5.3.2 +Number of Low Frequency Elements for General Universe +Algorithm 7: Number of Low Frequency Elements via Subsampling +Input: A stream S of elements a1, a2, · · · , aT ∈ U ∪ {⊥}, an error parameter η ∈ (0, 0.5), and a +parameter k ≥ 1. +Parameters : Additive approximation factor γ1 depending on the streaming continual release +number of distinct elements, relative approximation factor α2 ≥ 1 and additive +approximation factor γ2 ≥ 0 depending on the streaming continual release count of +low frequency elements for small universe. +//See Corollary 4.10 and Theorem 5.13. +Output: Estimation of the number of elements with frequency exactly i for each i ∈ [k] at every +timestamp t. +L ← ⌈log min(|U|, T )⌉, λ ← 2 log(1000k), m ← 100 · (25600λ/η2)2. +Let h : U → [m] be a pairwise independent hash function. +Let g : U → [L] ∪ {⊥} be a λ-wise independent hash function and +∀a ∈ U, i ∈ [L], Pr[g(a) = i] = 2−i, Pr[g(a) =⊥] = 2−L. +Initialize empty streams S1, S2, · · · , SL. +for each at in the stream S do +for i ∈ L do +if at ̸=⊥ and g(at) = i then +Append h(at) to the end of the stream Si. +end +else +Append ⊥ to the end of the stream Si. +end +end +Compute ˆd which is a (1.1, γ1)-approximation to the number of distinct elements in S. +For i ∈ [L], compute ˆsi,1, ˆsi,2, · · · , ˆsi,k where ˆsi,j is an (α2, γ2)-approximation to the number of +elements in Si where each element has frequency exact j. +If ˆd ≤ max(3γ1, 64λ/η2), set ˆs1 = ˆs2 = · · · = ˆsk = 0. +Otherwise, find the largest i∗ ∈ [L] such that 2i∗ · (64λ/η2) ≤ ˆd, and ∀j ∈ [k], set ˆsj = ˆsi∗,j · 2i. +Output ˆs1, ˆs2, · · · , ˆsk. +end +Lemma 5.14. If the subroutine of continually releasing ˆd in Algorithm 7 is ε-DP and the subroutine +of continually releasing {ˆsi,1, ˆsi,2, · · · , ˆsi,k} for each i ∈ [k] is also ε-DP, Algorithm 7 is 3ε-DP. +Proof. Consider two neighboring streams S and S′ where the only difference is the t-th element, i.e., +at ̸= a′ +t. Consider their corresponding streams S1, S2, · · · , SL and S′ +1, S′ +2, · · · , S′ +L in Algorithm 7. +For i ∈ [L] and j ̸= t, the j-th element in Si should be the same as the j-th element in S′ +i. Since +at can only make the t-th element of Sh(at) be different from the t-th element of S′ +h(at), and a′ +t +can only make the t-th element of Sh(a′ +t) be different from the t-th element of S′ +h(a′ +t), the total +sensitivity of {S1, S2, · · · , SL} is at most 2. Thus, if the continual release of ˆd is ε-DP and for every +i ∈ [L], the continual release of ˆsi,1, ˆsi,2, · · · , ˆsi,k is also ε-DP, the continual release of ˆs1, ˆs2, · · · , ˆsk +is (3ε)-DP. +Lemma 5.15. Consider an arbitrary timestamp t ∈ [T ]. ∀a ∈ U, let fa denote the frequency +of a in a1, a2, · · · , at. With probability at least 0.9, ∀j ∈ [k], the output ˆsj of Algorithm 7 is a +27 + +� +α2, +� +α2η + η2γ2 +32λ +� +· ∥S∥0 + 6γ1 + 128λ/η2� +-approximation to |{a ∈ U | fa = j}|, where ∥S∥0 is the +number of distinct elements in a1, a2, · · · , at. +To prove Lemma 5.15, we need following intermediate statements. Consider timestamp t ∈ [T ]. +∀a ∈ U, let fa denote the frequency of a in a1, a2, · · · , at. If i∗ exists in Algorithm 7, we define +G = {aj | g(aj) = i∗, j ∈ [t]}. Let ∥S∥0 denote the number of distinct elements in a1, a2, · · · , at. +For l ∈ [k], define Gl = {a ∈ G | fa = l}. +Claim 5.16. With probability at least 0.98, ∀a ̸= a′ ∈ G, h(a) ̸= h(a′). +Proof. Since E [|G|] = ∥S∥0/2i∗, we have |G| ≤ 100 · ∥S∥0/2i∗ with probability at least 0.99 by +Markov’s inequality. Since i∗ can be found by Algorithm 7, we have ∥S∥0 ≤ 1.1 · ( ˆd + γ1) ≤ 2 ˆd. +By the choice of i∗, we have 2i∗ ≥ ˆd/(128λ/η2). Therefore, ∥S∥0/2i∗ ≤ 256λ/η2. With probability +at least 0.99, |G| ≤ 25600λ/η2. By union bound and Markov’s inequality, Pr[∀a ̸= a′ ∈ G, h(a) ̸= +h(a′) | |G| ≤ 25600λ/η2] ≥ 1 − |G|2/m ≥ 0.99. +Claim 5.17. ∀l ∈ [k], Pr[2i∗ · |Gl| ∈ |{a ∈ U | fa = l}| ± η∥S∥0] ≥ 1 − 0.01/k. +Proof. Let sl = |{a ∈ U | fa = l}|. Since i∗ can be found, we have ˆd ≥ 3γ1. Note that sl ≤ ∥S∥0 ≤ +1.1( ˆd + γ1) ≤ 2 ˆd ≤ 2i∗ · 4 · 64λ/η2. By applying Lemma 2.6, ∀l ∈ [k] we have: +Pr +����|Gl| − sl/2i∗��� > 32λ/η +� +≤8 · +�λ · sl/2i∗ + λ2 +(32λ/η)2 +�λ/2 +≤0.01/k, +where the last inequality follows from sl/2i∗ ≤ 4 · 64λ/η2 and λ = 2 · log(1000k). Since ∥S∥0 ≥ +( ˆd − γ1)/1.1 ≥ ˆd/2 ≥ 2i∗ · 32λ/η2, we have +Pr +����|Gl| − sl/2i∗��� > η∥S∥0/2i∗� +≤ 0.01/k. +Next we show the proof of Lemma 5.15 +Proof of Lemma 5.15. Let E denote the event that ∀a ̸= a′ ∈ G, h(a) ̸= h(a′). Let ∀l ∈ [k], sl = +|{a ∈ U | fa = l}|. Let E′ denote the event that ∀l ∈ [k], 2i∗ · |Gl| ∈ sl ± η∥S∥0. According to +Claim 5.16 and Claim 5.17, the probability that both E and E′ happen is at least 0.97. In the +remaining of the proof, we condition on both events E and E′. +First, consider the case that ˆd < max(3γ1, 64λ/η2). Since ˆd is a (1.1, γ1)-approximation to ∥S∥0, +we have ∥S∥0 ≤ 6γ1 + 128λ/η2. In this case, ∀l ∈ [k], ˆsl = 0 and |{a ∈ U | fa = l}| ≤ ∥S∥0 ≤ +6γ1 + 128λ/η2. +Next, consider the case that ˆd ≥ max(3γ1, 64λ/η2). +For each j ∈ [k], let si∗,l denote the +number of elements in Sl where each element has frequency exact j. According to event E, we have +∀l ∈ [k], si∗,l = |Gl|. According to event E′, ∀l ∈ [k], we have: +ˆsl ≥ 2i∗ · +� 1 +α2 +· |Gl| − γ2 +� +≥ 1 +α2 +· (sl − η∥S∥0) − 2i∗ · γ2 +28 + +≥ 1 +α2 +· sl − +� +η + η2γ2 +32λ +� +· ∥S∥0, +where the first inequality follows from that ˆsi∗,l is an (α2, γ2)-approximation to si∗,l and si∗,l = |Gl|, +the second inequality follows from event E′, and the third inequality follows from α2 ≥ 1 and +∥S∥0 ≥ ( ˆd − γ1)/1.1 ≥ ˆd/2 ≥ 2i∗ · 32λ/η2. Similarly, ∀l ∈ [k], we have: +ˆsl ≤ 2i∗ · (α2 · |Gl| + γ2) ≤ α2 · (sl + η∥S∥0) + 2i∗ · γ2 +≤ α2 · sl + +� +α2η + η2γ2 +32λ +� +· ∥S∥0. +Theorem 5.18 (Streaming continual release of count of low frequency elements). Let k ≥ 1, ε ≥ +0, ξ ∈ (0, 0.5), η ∈ (0, 0.5). There is an ε-DP algorithm in the streaming continual release model such +that with probability at least 1 − ξ, it always outputs k numbers ˆs1, ˆs2, · · · , ˆsk for every timestamp +t such that ∀i ∈ [k], ˆsi is an approximation to |{a ∈ U | fa = i}| with additive error: +� +η + η2 · k +ε · poly +� +log +�T k +ξ +��� +· ∥S∥0 + 1 +ε · poly +� +log +�T +ξ +�� ++ O +�log k +η2 +� +The algorithm uses +1 +η4 · poly +� +log(T ·k/ξ) +min(ε,1) +� +space. +Proof. According to Lemma 5.15, the approximation guarantee holds with probability at least 0.9. +To boost the success probability to 1 − ξ/3 for all t ∈ [T ], we need to run ⌈50 log(3T/ξ)⌉ indepen- +dent copies of Algorithm 7. Since we run ⌈50 log(3T/ξ)⌉ copies of Algorithm 7 and according to +Lemma 5.14, if we want the final algorithm to be ε-DP, we need each subroutine of the streaming +continual release of {ˆsi,1, ˆsi,2, · · · , ˆsi,k} for each i ∈ [k] to be (ε/(3·⌈50 log(2T/ξ))⌉)-DP and we need +the subroutine of the streaming continual release of ˆd to be also (ε/(3·⌈50 log(3T/ξ))⌉)-DP. To simul- +taneously make the call of each subroutine of the streaming continual release of {ˆsi,1, ˆsi,2, · · · , ˆsi,k} +over all independent copies of Algorithm 7 satisfy the desired (α2, γ2)-approximation with prob- +ability at least 1 − ξ/3, we need to make {ˆsi,1, ˆsi,2, · · · , ˆsi,k} satisfy the approximation guaran- +tee for each particular i ∈ [L] and a particular copy of Algorithm 7 with probability at least +1 − ξ/(3 · ⌈50 log(3T/ξ))⌉ · L). To simultaneously make the call of each subroutine of the stream- +ing continual release of ˆd over all independent copies of Algorithm 7 satisfy the desired (1.1, γ1)- +approximation with probability at least 1 − ξ/3, we need to make ˆd satisfy the approximation guar- +antee for each particular copy of Algorithm 7 with probability at least 1 − ξ/(3 · ⌈50 log(3T/ξ))⌉). +Then, according to Corollary 4.10, we have γ1 = 1 +ε · poly +� +log +� +T +ξ +�� +. According to Theorem 5.13, +we have α2 = 1 and γ2 = k +ε · poly +� +log +� +T k +ξ +�� +. By plugging above parameters into Lemma 5.15, we +have ∀i ∈ [k], ˆsi is an approximation to |{a ∈ U | fa = i}| with additive error: +� +η + η2 · k +ε · poly +� +log +�T k +ξ +��� +· ∥S∥0 + 1 +ε · poly +� +log +�T +ξ +�� ++ O +�log k +η2 +� +. +Next, consider the space usage. According to Corollary 4.10, the total space needed to release +ˆd for all copies of Algorithm 7 needs poly +� +log(T/ξ) +min(ε,1) +� +. According to Theorem 5.13, the total space +needed to release {ˆsi,1, ˆsi,2, · · · , ˆsi,k} for all i ∈ [L] over all copies of Algorithm 7 needs O(log(T/ξ)· +L·(m+log(T ))) = +1 +η4 ·poly +� +log +� +T ·k +ξ +�� +. Thus, the overall space is at most +1 +η4 ·poly +� +log(T ·k/ξ) +min(ε,1) +� +. +29 + +5.4 +ℓp Moment Estimation +In this section, we show how to use our ℓp heavy hitters and the estimator of number of low frequency +elements to solve ℓp moment estimation problem. +Algorithm 8: ℓp Frequency Moment Estimation +Input: A stream S of elements a1, a2, · · · , aT ∈ U ∪ {⊥}, an error parameter η ∈ (0, 0.5). +Parameters : A threshold parameter τ depending on the heavy hitter algorithm, a relative +approximation factor α and an additive approximation factor γ depending on the +algorithm of estimating count of low frequency elements. //See Theorem 5.10 and +Theorem 5.18. +Output: Estimation of the ℓp frequency moment at every timestamp t. +Let β′ be drawn uniformly at random from [1/2, 1] and let β ∈ β′ ± (η/T )C for some sufficiently +large constant C > 0. +//Thus β can be represented by Θ(log(T/η)) bits. +Let q∗ +1 be the smallest integer that β(1 + η)q∗ +1 > τ and let q∗ +2 be the smallest integer that +β(1 + η)q∗ +2 +1 ≥ T , and for any q ∈ [q∗ +1, q∗ +2], define the interval Iq = (β(1 + η)q, β(1 + η)q+1]. +⌈L ← log(|U|)⌉, λ ← 2 · log(1000(L + 1) log(4T )/η). +Let k be the largest integer such that k ≤ β(1 + η)q∗ +1 . Let B ← +� +log(4T ) +η +� +· 100(L + 1) · 32λ +η3 · (1 + η)p. +Let g : U → [L] ∪ {⊥} be a λ-wise independent hash function and +∀a ∈ U, i ∈ [L], Pr[g(a) = i] = 2−i, Pr[g(a) =⊥] = 2−L. +Initialize empty streams S0, S1, S2, · · · , SL. +for each at in the stream S do +if +at ̸=⊥ then +Append at at the end of S0. +For i ∈ [L], if g(at) = i, append at to Si, otherwise append ⊥ to Si. +end +else +For i ∈ [L] ∪ {0}, append ⊥ at the end of Si. +end +For each i ∈ [L] ∪ {0}, compute a set Hi ⊆ U together with a function ˆfi : Hi → R≥0 satisfying: +1. ∀a ∈ Hi, (1 − η′) · fa ≤ ˆfi(a) ≤ (1 + η′) · fa, where fa is the frequency of a in a1, a2, · · · , at, and η′ +satisfies η′ ≤ +η +10000(L+1)|Hi|. +2. ∀a ∈ U that appears in Si, if fa ≥ τ and f p +a ≥ ∥Si∥p +p/B, a ∈ Hi. +Compute ˆs1, ˆs2, · · · , ˆsk where ∀l ∈ [k], ˆsl is an (α, γ)-approximation to |{a ∈ U | fa = l}|. +for q ∈ [q∗ +1, q∗ +2] do +Initialize ˆzq = 0. +for i ∈ [L] ∪ {0} do +if |{a ∈ Hi | ˆfi(a) ∈ Iq}| ≥ 8λ/η2 or i = 0 then +ˆzq ← max(ˆzq, |{a ∈ Hi | ˆfi(a) ∈ Iq}| · 2i). +end +end +end +Output ˆFp = � +l∈[k] ˆsl · lp + � +q∈[q∗ +1 ,q∗ +2 ] ˆzq · (β(1 + η)q)p +end +Lemma 5.19. Consider the subroutines in Algorithm 8. If the algorithm that continually release +ˆs1, ˆs2, · · · , ˆsk is ε-DP and for every i ∈ [L] ∪ {0} the algorithm that continually release (Hi, ˆfi) is +ε-DP. Algorithm 8 is 4ε-DP in the continual release model. +30 + +Proof. Consider two neighboring stream S = (a1, a2, · · · , aT ) and S′ = (a′ +1, a′ +2, · · · , a′ +T ) where +they only differ at timestamp t, i.e., at ̸= a′ +t. Consider the corresponding streams S0, S1, · · · , SL +and S′ +0, S′ +1, · · · , S′ +L. ∀i ∈ [L] ∪ {0}, j ̸= t, the j-th element of Si should be the same as the j- +th element of S′ +i. +Furthermore, ∀i ∈ [L] with i ̸= g(at), i ̸= g(a′ +t), the t-th element of Si is +also the same as the t-th element of S′ +i. Thus, at most 3 streams S0, Sg(at) and Sg(a′ +t) might be +different from S′ +0, S′ +g(at) and S′ +g(a′ +t) respectively. Thus, the output {(H0, ˆf0), (H1, ˆf1), · · · , (HL, ˆfL)} +is 3ε-DP. Since (ˆs1, ˆs2, · · · , ˆsk) is ε-DP and the final output only depends on (ˆs1, ˆs2, · · · , ˆsk) and +{(H0, ˆf0), (H1, ˆf1), · · · , (HL, ˆfL)}, the final output is 4ε-DP. +In the remaining of the section, let us analyze the approximation guarantee of Algorithm 8. Let +us consider a timestamp t ∈ [T ] and ∀a ∈ U, let fa denote the frequency of a in a1, a2, · · · , at. Let +S, S0, S1, · · · , SL denote the streams up to timestamp t. Let I = {[1, 1], [2, 2], · · · , [k, k], Iq∗ +1 , Iq∗ +1 +1, · · · , Iq∗ +2 }. +By our choice of k, q∗ +1, q∗ +2, we have the following observation: +Observation 5.20. � +I∈I +� +a∈U:fa∈I f p +a = ∥S∥p +p. +Definition 5.21. For any I ∈ I, if � +a∈U:fa∈I f p +a ≥ η∥S∥p +p/(q∗ +2 − q∗ +1 + 1) or I is {i} for some +i ∈ [k], then interval I is contributing. +5.4.1 +Analysis of High Frequency Elements +In this section, we show that � +q∈[q∗ +1 ,q∗ +2 ] ˆzq·(β(1+η)q)p is a good approximation to � +q∈[q∗ +1 ,q∗ +2 ] +� +a∈U:fa∈Iq f p +a +and � +q∈[q∗ +1 ,q∗ +2 ]:Iq is contributing +� +a∈U:fa∈Iq f p +a. +The following lemma says that the frequency of any particular sampled element should be +sufficiently far away from the boundary of intervals Iq∗ +1 , Iq∗ +1 +1, · · · , Iq∗ +2 with a good probability. +Lemma 5.22 (Indyk and Woodruff [2005]). Consider β in Algorithm 8. Consider any f ∈ [T ] and +any r ≥ 2(η/T )C−1. +Pr +β′ +� +min +q∈{q∗ +1 ,q∗ +1 +1,··· ,q∗ +2 ,q∗ +2 +1} |f − β(1 + η)q| < r +� +≤ 100r +η · f +By applying above lemma, we show that with high probability, we can use ˆfi(a) to correctly +classify a into right interval in Iq∗ +1 , Iq∗ +1 +1, Iq∗ +1 +2, · · · , Iq∗ +2 . +Lemma 5.23. With probability at least 0.99, ∀i ∈ [L] ∪ {0}, ∀a ∈ Hi, and ∀q ∈ [q∗ +1, q∗ +2], ˆfi(a) ∈ Iq +if and only if fa ∈ Iq. +Proof. According to Lemma 5.22, for any i ∈ [L] ∪ {0} and any a ∈ Hi, we have: +Pr +� +min +q∈{q∗ +1 ,q∗ +1 +1,··· ,q∗ +2 +1} +|fa − β(1 + η)q| < η′fa +� +≤ +1 +100 · (L + 1) · |Hi|, +where the inequality follows from η′ ≤ +η +10000(L+1)|Hi|. By taking a union bound over all i ∈ [L]∪{0} +and all a ∈ Hi, with probability at least 0.99, the following event happens: ∀i ∈ [L] ∪ {0}, ∀a ∈ +Hi, ∀q ∈ {q∗ +1, q∗ +1 + 1, · · · , q∗ +2 + 1}, +1. if fa ≤ β(1 + η)q, then ˆfi(a) ≤ fa + η′fa ≤ β(1 + η)q, +31 + +2. if fa > β(1 + η)q, then ˆfi(a) ≥ fa − η′fa > β(1 + η)q. +Therefore, with probability at least 0.99, ∀i ∈ [L] ∪ {0}, ∀a ∈ Hi, and ∀q ∈ [q∗ +1, q∗ +2], ˆfi(a) ∈ Iq if +and only if fa ∈ Iq. +Let Gi denote the set of elements that appears in the stream Si and let Gi,q denote the set of +elements that appear in the stream Si and whose frequency is in the interval Iq. Formally, ∀i ∈ [L], +let Gi = {aj | g(aj) = i, j ≤ t}, G0 = {aj | j ≤ t}, and ∀q ∈ {q∗ +1, q∗ +1 + 1, · · · , q∗ +2}, ∀i ∈ [L] ∪ {0}, let +Gi,q = {a ∈ Gi | fa ∈ Iq}. For q ∈ [q∗ +1, q∗ +2], let zq = |G0,q|, i.e., the number of elements that are in +the stream S and have frequency in the range Iq. +Lemma 5.24. ∀i ∈ [L]∪{0}, q ∈ [q∗ +1, q∗ +2], if i = 0 or zq ≥ 2i ·4λ/η2, Pr[|Gi,q| ∈ (1±η)·zq/2i] ≥ 1− +0.01/ ((L + 1) · log(4T )/η). Otherwise, Pr[||Gi,q| − zq/2i| ≤ 4λ/η] ≥ 1 − 0.01/ ((L + 1) · log(4T )/η). +Proof. Consider any i ∈ [L] ∪ {0} and q ∈ [q∗ +1, q∗ +2]. If i = 0, by definition zq = |G0,q|. +Suppose zq ≥ 2i · 4λ/η2. Due to Lemma 2.6, we have: +Pr +���|Gi,q| − zq/2i�� > η · zq/2i� +≤8 · +� +λ · zq/2i + λ2 +(η · zq/2i)2 +�λ/2 +≤0.01/ ((L + 1) · log(4T )/η) , +where the last inequality follows from that λ = 2 · log(1000(L + 1) log(4T )/η) and zq ≥ 2i · 4λ/η2. +Suppose zq ≤ 2i · 4λ/η2, By applying Lemma 2.6 again, we have: +Pr +���|Gi,q| − zq/2i�� > 4λ/η +� +≤8 · +�λ · zq/2i + λ2 +(4λ/η)2 +�λ/2 +≤0.01/ ((L + 1) · log(4T )/η) , +where the last inequality follows from zq/2i ≤ 4λ/η2 and λ = 2 · log(1000(L + 1) log(4T )/η). +We define events E1 and E2 as the following: E1 denotes the event ∀i ∈ [L] ∪ {0}, ∀a ∈ Hi, ∀q ∈ +[q∗ +1, q∗ +2], ˆfi(a) ∈ Iq if and only if fa ∈ Iq. E2 denotes the event: +1. ∀i ∈ [L] ∪ {0}, ∀q ∈ [q∗ +1, q∗ +2], if zq ≥ 2i · 4λ/η2 or i = 0, |Gi,q| · 2i ∈ (1 ± η)zq. +2. ∀i ∈ [L], ∀q ∈ [q∗ +1, q∗ +2], if zq < 2i · 4λ/η2, |Gi,q| ∈ zq/2i ± 4λ/η. +According to Lemma 5.22, E1 happens with probability at least 0.99. According to Lemma 5.24, +since ∀q ∈ [q∗ +1, q∗ +2], zq = |G0,q| and q∗ +2 − q∗ +1 + 1 ≤ log(4T )/η, by taking a union bound over all +i ∈ [L] ∪ {0} and all q ∈ [q∗ +1, q∗ +2], E2 happens with probability at least 0.99. +Lemma 5.25 (Upper bound of estimation of high frequency moment). Condition on E1 and E2, +� +q∈[q∗ +1 ,q∗ +2 ] +ˆzq · (β(1 + η)q)p ≤ (1 + η) · +� +q∈[q∗ +1 ,q∗ +2 ] +� +a∈U,fa∈Iq +f p +a +32 + +Proof. Due to event E1, we have ∀i ∈ [L]∪{0}, ∀q ∈ [q∗ +1, q∗ +2], {a ∈ Hi | ˆfi(a) ∈ Iq} ⊆ Gi,q. Note that +for q ∈ [q∗ +1, q∗ +2], ˆzq is either 0, or ˆzq = 2i′ · |{a ∈ Hi′ | ˆfi′(a) ∈ Iq}| for some i′ satisfying i′ = 0 or +|{a ∈ Hi′ | ˆfi′(a) ∈ Iq}| ≥ 8λ/η2. Since ∀q ∈ [q∗ +1, q∗ +2], ∀i ∈ [L] ∪ {0}, |{a ∈ Hi | ˆfi(a) ∈ Iq}| ≤ |Gi,q|, +∀q ∈ [q∗ +1, q∗ +2], if ˆzq ̸= 0, there is some i′ ∈ [L] ∪ {0} such that: +ˆzq = 2i′ · |{a ∈ Hi′ | ˆfi′(a) ∈ Iq}| +≤ 2i′ · |Gi′,q| +≤ (1 + η) · zq, +where the second inequality follows from that |Gi′,q| ≥ 8λ/η2 which implies that |Gi′,q|·2i′ ∈ (1±η)zq +according to event E2. +Therefore, +� +q∈[q∗ +1 ,q∗ +2 ] +ˆzq · (β(1 + η)q)p +≤(1 + η) +� +q∈[q∗ +1 ,q∗ +2 ] +zq · (β(1 + η)q)p +≤(1 + η) +� +q∈[q∗ +1 ,q∗ +2 ] +� +a∈U:fa∈Iq +(β(1 + η)q)p +≤(1 + η) +� +q∈[q∗ +1 ,q∗ +2 ] +� +a∈U:fa∈Iq +f p +a, +where the first inequality follows from ˆzq ≤ (1+η)zq, the second inequality follows from the definition +of zq, i.e., zq = |{a ∈ U | fa ∈ Iq}|, and the last inequality follows from that fa ≥ β(1 + η)q if +fa ∈ Iq. +Define the event E3 as: ∀i ∈ [L] ∩ {0}, ∥Si∥p +p ≤ 100(L + 1) · ∥S∥p +p/2i. +Lemma 5.26. E3 happens with probability at least 0.99. +Proof. Consider i ∈ [L]∪{0}. We have E +� +∥Si∥p +p +� += � +a∈U Pr[g(a) = i]·f p +a = ∥S∥p +p/2i. By Markov’s +inequality, with probability at least 1 − 1/(100(L + 1)), ∥Si∥p +p ≤ 100(L + 1) · ∥S∥p +p/2i. By taking a +union bound over all i ∈ [L] ∪ {0}, E3 happens with probability at least 0.99. +In the remaining of the analysis, we condition on E3 as well. +Lemma 5.27 (Lower bound of estimation of contributing high frequency moment). Condition on +E1, E2, E3. +� +q∈[q∗ +1 ,q∗ +2 ] +ˆzq · (β(1 + η)q)p ≥ (1 − η)p+1 · +� +q∈[q∗ +1 ,q∗ +2 ]:Iq is contributing +� +a∈U,fa∈Iq +f p +a +Proof. Consider any contributing q ∈ [q∗ +1, q∗ +2]. +Case 1: zq ≤ 16λ/η2. According to Definition 5.21, we have � +a∈U:fa∈Iq f p +a ≥ η∥S∥p +p/(q∗ +2 − q∗ +1 + 1) +which implies that η∥S0∥p +p/(q∗ +2 − q∗ +1 + 1) ≤ zq · (β(1 + η)q+1)p ≤ 16λ/η2 · (1 + η)p · (β(1 + η)q)p. +33 + +Since B ≥ (q∗ +2 − q∗ +1 + 1) · 16λ · (1 + η)p/η3, ∀a ∈ U with fa ∈ Iq, f p +a ≥ ∥S0∥p +p/B . Therefore, ∀a ∈ U +with fa ∈ Iq, we have a ∈ H0. According to event E1, we have |{a ∈ H0 | ˆf0(a) ∈ Iq}| = |G0,q| = zq. +Thus, we have ˆzq ≥ |{a ∈ H0 | ˆf0(a) ∈ Iq}| ≥ zq. +Case 2: zq > 16λ/η2. Let i∗ ∈ {0}∪[L] be the largest value such that zq/2i∗ ≥ 16λ/η2. According +to event E2, we have |Gi∗,q| ≥ 8λ/η2. +Since q is contributing, we have: +zq · (β(1 + η)q+1)p +≥ +� +a∈U:fa∈Iq +f p +a +≥η∥S∥p +p/(q∗ +2 − q∗ +1 + 1) +≥2i∗ · η∥Si∗∥p +p/((q∗ +2 − q∗ +1 + 1) · 100(L + 1)), +where the last inequality follows from event E3. Therefore, we have +(β(1 + η)q)p +≥η∥Si∗∥p +p/((q∗ +2 − q∗ +1 + 1) · 100(L + 1) · (zq/2i∗) · (1 + η)p) +≥η∥Si∗∥p +p/((q∗ +2 − q∗ +1 + 1) · 100(L + 1) · (32λ/η2) · (1 + η)p), +where the last inequality follows from zq/2i∗ ≤ 32λ/η2. Since B ≥ ((q∗ +2 − q∗ +1 + 1) · 100(L + 1) · +(32λ/η2) · (1 + η)p)/η, we have ∀a ∈ U with a ∈ Gi∗,q, f p +a ≥ ∥Si∗∥p +p/B and fa > β(1 + η)q > τ +which implies that a ∈ Hi∗. According to event E1, we have {a ∈ Hi∗ | ˆfi∗(a) ∈ Iq} = Gi∗,q. +Therefore, |{a ∈ Hi∗ | ˆfi∗(a) ∈ Iq}| ≥ 8λ/η2. Finally, in addition, according to event E2, we have +ˆzq ≥ |{a ∈ Hi∗ | ˆfi∗(a) ∈ Iq}| · 2i∗ ≥ (1 − η)zq. +Therefore, in any case, we have ˆzq ≥ (1 − η)zq, and we have: +� +q∈[q∗ +1 ,q∗ +2 ] +ˆzq · (β(1 + η)q)p +≥ +� +q∈[q∗ +1 ,q∗ +2 ]:Iq is contributing +ˆzq · (β(1 + η)q)p +≥ +� +q∈[q∗ +1 ,q∗ +2 ]:Iq is contributing +(1 − η)zq · (β(1 + η)q)p += 1 − η +(1 + η)p +� +q∈[q∗ +1 ,q∗ +2 ]:Iq is contributing +� +a∈U:fa∈Iq +(β(1 + η)q+1)p +≥ 1 − η +(1 + η)p +� +q∈[q∗ +1 ,q∗ +2 ]:Iq is contributing +� +a∈U:fa∈Iq +f p +a +≥(1 − η)p+1 +� +q∈[q∗ +1 ,q∗ +2 ]:Iq is contributing +� +a∈U:fa∈Iq +f p +a, +where the equality follows from the definition that zq = |{a ∈ U | fa ∈ Iq}|. +34 + +5.4.2 +Analysis of Low Frequency Elements +In this section, we show that � +l∈[k] ˆsl · lp is a good approximation to � +l∈[k] +� +a∈U:fa=l f p +a. +Lemma 5.28 (Approximation of low frequency moments). � +l∈[k] ˆsl · lp is a (α, γ · (2τ)p+1)- +approximation to � +l∈[k] +� +a∈U:fa=l f p +a. +Proof. For l ∈ [k], let sl = |{a ∈ U | fa = l}|. We have: +� +l∈[k] +ˆsl · lp +≥ +� +l∈[k] +( 1 +α · sl − γ) · lp +≥ 1 +α · +� +l∈[k] +� +a∈U:fa=l +f p +a − k · γ · kp +≥ 1 +α · +� +l∈[k] +� +a∈U:fa=l +f p +a − γ · (2τ)p+1, +where the last inequality follows from that our choice of k implies that k ≤ 2τ. Similarly, we have: +� +l∈[k] +ˆsl · lp +≤ +� +l∈[k] +(α · sl + γ) · lp +≤α · +� +l∈[k] +� +a∈U:fa=l +f p +a − γ · (2τ)p+1. +5.4.3 +Putting High Frequency Moments and Low Frequency Moments Together +Lemma 5.29. � +contributing I∈I +� +a∈U:fa∈I f p +a ≥ (1 − η) · ∥S∥p +p. +Proof. +� +contributing I∈I +� +a∈U:fa∈I +f p +a += +� +I∈I +� +a∈U:fa∈I +f p +a − +� +non-contributing I∈I +� +a∈U:fa∈I +f p +a +≥∥S∥p +p − (q∗ +2 − q∗ +1 + 1) · η · ∥S∥p +p/(q∗ +2 − q∗ +1 + 1) +≥(1 − η) · ∥S∥p +p. +35 + +Lemma 5.30. Consider any timestamp t ∈ [T ]. +∀a ∈ U, let fa denote the frequency of a in +a1, a2, · · · , at. With probability at least 0.9, the output ˆFp of Algorithm 8 is a +� +max +� +α +1−η , (1 + 2η)p+2� +, γ · (2τ)p+1� +- +approximation to ∥S∥p +p. +Proof. According to Lemma 5.25 and Lemma 5.27, we have: +(1 − η)p+1 +� +q∈[q∗ +1 ,q∗ +2 ]:Iq is contributing +� +a∈U,fa∈Iq +f p +a ≤ +� +q∈[q∗ +1 ,q∗ +2 ] +ˆzq · (β(1 + η)q)p ≤ (1 + η) +� +q∈[q∗ +1 ,q∗ +2] +� +a∈U,fa∈Iq +f p +a. +According to Lemma 5.28, we have: +1 +α · +� +l∈[k] +� +a∈U:fa=l +f p +a − γ · (2τ)p+1 ≤ +� +l∈[k] +ˆsl · lp ≤ α · +� +l∈[k] +� +a∈U:fa=l +f p +a + γ · (2τ)p+1 +Therefore, we have: +ˆFp ≥ min +� 1 +α, (1 − η)p+1 +� +� +I∈I:I is contributing +� +a∈U:fa∈I +f p +a − γ · (2τ)p+1 +≥ min +�1 − η +α +, (1 − η)p+2 +� +∥S∥p +p − γ · (2τ)p+1 +≥ min +�1 − η +α +, +1 +(1 + 2η)p+2 +� +∥S∥p +p − γ · (2τ)p+1 +where the second step follows from Lemma 5.29. Similarly, we have: +ˆFp ≤ max(α, 1 + η) +� +I∈I +� +a∈U:fa∈I +f p +a + γ · (2τ)p+1 += max(α, 1 + η) · ∥S∥p +p + γ · (2τ)p+1 +Theorem 5.31 (Streaming continual release ℓp frequency moment estimation). Let p > 0, ε ≥ 0, ξ ∈ +(0, 0.5), η ∈ (0, 0.5). There is an ε-DP algorithm in the streaming continual release model such that +with probability at least 1 − ξ, it always outputs an +� +1 + η, +� +log(T |U|/ξ) +ηε +�O(max(1,p))� +-approximation +to ∥S∥p +p. The algorithm uses space at most +φ · +�log(T |U|/ξ) +ηε +�O(max(1,p)) +, +where φ = max(1, |U|1−2/p). +Proof. To boost the probability of the approximation guarantee of Lemma 5.30 to 1 − ξ/3 and +simultaneously for all timestamps t ∈ T , we run ⌈50 log(3T/ξ)⌉ independent copies of Algorithm 8 +and take the median of the outputs. Since we run ⌈50 log(3T/ξ)⌉ independent copies of Algorithm 8 +and according to Lemma 5.19, if we want the final algorithm to be ε-DP, we need each subroutine +of the streaming continual release of (Hi, ˆfi) for i ∈ [L] ∪ {0} to be (ε/(4 · ⌈50 log(3T/ξ)⌉))-DP +36 + +and we need the subroutine of the streaming continual release of {ˆs1, ˆs2, · · · , ˆsk} to be (ε/(4 · +⌈50 log(3T/ξ)⌉))-DP as well. To simultaneously make the call of each subroutine of the streaming +continual release of (Hi, ˆfi) over all independent copies of Algorithm 8 satisfy the desired properties +stated in Algorithm 8 with probability at least 1 − ξ/3, we need to make (Hi, ˆfi) satisfy the +property for each particular i ∈ [L] ∪ {0} and a particular copy of Algorithm 8 with probability +at least 1 − ξ/(4 · ⌈50 log(3T/ξ)⌉ · (L + 1)). To simultaneously make the call of each subroutine +of the streaming continual release of {ˆs1, ˆs2, · · · , ˆsk} over all independent copies of Algorithm 8 +satisfy the desired property stated in Algorithm 8 with probability at least 1 − ξ/3, we need to +make {ˆs1, ˆs2, · · · , ˆsk} satisfy the desired property for each particular i ∈ [L] ∪ {0} and a particular +copy of Algorithm 8 with probability at least 1 − ξ/(3 · ⌈50 log(3T/ξ)⌉). According to Algorithm 8, +we have +B = Θ +�log(T ) log(|U|) log(log(T |U|)/η) · (1 + η)p +η4 +� +. +Thus, according to Theorem 5.10, the size of |Hi| in Algorithm 8 for i ∈ [L] ∪ {0} is at most +poly +� +log(T |U|/ξ) +η +� +· 2O(p). Thus, we choose η′ = 1/ +� +poly +� +log(T |U|/ξ) +η +� +· 2O(p)� +. Then according to +Theorem 5.10, we have +τ = 1 +ε · poly +�log(T · |U|/ξ) +η +� +· 2O(p). +According to Theorem 5.18, we have α = 1, and we choose γ to be +� +η′′ + η′′2 · τ +ε · poly +� +log +�T τ +ξ +��� +· ∥S∥0 + 1 +ε · poly +� +log +�T +ξ +�� ++ O +�log τ +η′′2 +� +, +where we choose η′′ to be +εη +τ O(max(1,p))poly +� +log(T |U|/ξ) +η +� = +1 +� +log(T |U|/ξ) +ηε +�O(max(1,p)) +such that +γ · (2τ)p+1 ≤ η∥S∥0 + +�log(T |U|/ξ) +ηε +�O(max(1,p)) +≤ η∥S∥p +p + +�log(T |U|/ξ) +ηε +�O(max(1,p)) +. +Note that η∥S∥p +p becomes the relative error. Thus, according to Lemma 5.30, the output ˆFp is an +� +(1 + η)O(max(1,p)), +� +log(T |U|/ξ) +ηε +�O(max(1,p))� +-approximation. +Next, consider the space usage. According to Theorem 5.10, the total space needed to run all +heavy hitters subroutines is at most +φ · 2O(max(1,p)) · poly +�log(T |U|/ξ) +η +� +. +37 + +According to Theorem 5.18, the total space needed for computing {ˆs1, ˆs2, · · · , ˆsk} for all running +copies of Algorithm 8 is at most +�log(T |U|/ξ) +ηε +�O(max(1,p)) +. +Therefore, the overall space needed is at most +φ · +�log(T |U|/ξ) +ηε +�O(max(1,p)) +. +6 +Extension to Sliding Window Continual Release Algorithms +In this section, we briefly review the smooth histogram Braverman and Ostrovsky [2007] technique +which converts any (non-private) streaming algorithm into (non-private) sliding window algorithm. +The original framework only supports the approximation algorithm which only has relative error +and no additive error. In this section, we show how to extend it to support the additive error as +well. +Suppose there are two streams A = (a1, a2, · · · , at1) and B = (b1, b2, · · · , bt2). We use A ∪ B +to denote the concatenation of two streams, i.e., A ∪ B = (a1, a2, · · · , at1, b1, b2, · · · , bt2). If B is a +suffix of A, i.e., ∃i ∈ [t1] such that ai = b1, ai+1 = b2, · · · , at1 = bt2, then we denote it as B ⊆r A. +Definition 6.1 (Smooth function Braverman and Ostrovsky [2007]). Let g(·) be a function over +streams. Function g(·) is (ζ, β)-smooth if: +1. ∀A, 0 ≤ g(A) ≤ poly(T ). +2. ∀A, B with B ⊆r A, g(A) ≥ g(B) +3. For any η ∈ (0, 1), there exists ζ(η, g) and β(η, g) such that +(a) 0 < β ≤ ζ < 1. +(b) If B ⊆r A and (1 − β)g(A) ≤ g(B) then (1 − ζ)g(A ∪ C) ≤ g(B ∪ C) for any stream C. +Lemma 6.2 (Smoothness of frequency moments Braverman and Ostrovsky [2007]). For p > 1, +∥S∥p +p is +� +η, +� +η +p +�p� +-smooth. For 0 < p ≤ 1, ∥S∥p +p is (η, η)-smooth. ∥S∥0 is (η, η)-smooth. +Lemma 6.3. Let 0 ≤ Z ≤ poly(T ). If g(S) := ∥S∥p +p + Z, then g(·) is +� +η, +� +η +p +�p� +-smooth if p > 1 +and g(·) is (η, η)-smooth if 0 < p ≤ 1. If g(S) = ∥S∥0 + Z, g(·) is (η, η)-smooth. +Proof. For g(·) stated in the lemma statement, the first two requirements in Definition 6.1 are +satisfied obviously. Therefore, we only need to justify the third requirement of Definition 6.1. +Let us consider A and B such that B ⊆r A and (1−β)g(A) ≤ g(B). Consider any stream C. We +are going to prove (1 − ζ)g(A ∪ C) ≤ g(B ∪ C). Consider the case g(S) = ∥S∥p +p + Z. We construct +an auxiliary stream X such that the elements of X do not appear in any of A, B, C and ∥X∥p +p = Z. +Then, we have (1 − β)∥A ∪ X∥p +p = (1 − β)g(A) ≤ g(B) = ∥B ∪ X∥p +p. Due to the smoothness of ∥S∥p +p +38 + +(Lemma 6.2), we have (1 − ζ)g(A ∪ C) = (1 − ζ)∥A ∪ X ∪ C∥p +p ≤ ∥B ∪ X ∪ C∥p +p = g(B ∪ C). According +to Lemma 6.2, g(·) is +� +η, +� +η +p +�p� +-smooth if p > 1 and g(·) is (η, η)-smooth if 0 < p ≤ 1. +Similarly, using the similar argument by constructing ∥X∥0 = Z, we can show that g(S) = +∥S∥0 + Z is (η, η)-smooth. +Lemma 6.4. Let α ≥ 1, γ ≥ 0. +If g′ is an (α, γ)-approximation to g, then g′ + Z is an α- +approximation to g + Z if Z ≥ +α +α−1 · γ. +Proof. We have: +g′ + Z ≥ 1 +α · g − γ + Z +≥ 1 +α · g − α − 1 +α +· Z + Z +≥ 1 +α · (g + Z). +On the other hand, +g′ + Z ≤ α · g + γ + Z +≤ α · g + (α − 1)Z + Z +≤ α · (g + Z). +Theorem 6.5 (Smooth histogram algorithmic framework Braverman and Ostrovsky [2007]). Let +η ∈ (0, 0.5). Let g(·) be an (ζ, β)-smooth function. If there exists a streaming algorithm Λ which +maintains an ( +1 +1−η)-approximation of g(·) simultaneously for all timestamps t ∈ [T ] with probability +at least 1 − ξ, using space h(η, ξ), then there is a sliding window algorithm Λ′ that maintains a +� +1 +1−η−ζ +� +-approximation of g(·) over sliding windows simultaneously for all timestamps t ∈ [T ] with +probability at least 1 − ξ and uses space O +� +log T +β +· h(η, ξβ/ log(T )) +� +. +Furthermore, at any timestamp t ∈ [T ], Λ′ starts a new instance of Λ which regards the t-th +element in the stream as the beginning of the stream, and Λ′ only keeps at most O(log(T )/β) past +instances of Λ (started from different timestamps). The output of Λ′ at any timestamp t only depends +on the outputs of its maintained instances Λ, and the decision of whether keeping an instance Λ to +timestamp t + 1 only depends on the outputs of its maintained instances Λ at timestamp t as well. +We are able to extend the above smooth histogram framework to the differentially private +continual release setting. +Theorem 6.6 (Smooth histogram for differentially private continual release model). Let g(·) be +an (ζ, β)-smooth function. If there exists a ε′-DP streaming continual release algorithm Λ which +maintains an +� +1 +1−η +� +-approximation of g(·) simultaneously for all timestamps t ∈ [T ] with probability +at least 1 − ξ, using space h(η, ξ), then there is a ε-DP sliding window continual release algorithm +Λ′ with ε = O(ε′β/ log(T )) which maintains a +� +1 +1−η−ζ +� +-approximation of g(·) over sliding windows +for all timestamps t ∈ [T ] with probability at least 1 − ξ and uses space O +� +log T +β +· h(η, ξβ/ log(T )) +� +. +39 + +Proof. The approximation guarantee, space guarantee and success probability follows from Theo- +rem 6.5 directly. In the remaining of the proof, we prove the DP guarantee. +Let Λ1, Λ2, · · · , ΛT be instances of Λ where Λt is started at timestamp t. Let ot,1, ot,2, · · · , ot,T +be the outputs of Λt, if at timestamp j, Λt is not started yet or is already kicked out by Λ′, then +ot,j =⊥. According to Theorem 6.5, the outputs of Λ′ over all timestamps t ∈ [T ] is determined +by {oi,j | i, j ∈ [T ]}. Thus, we only need to show that {oi,j | i, j ∈ [T ]} is ε-DP. Consider two +neighboring streams S and S′ where only the r-th elements are different. Let us fix a possible +configuration {oi,j | i, j ∈ [T ]}. According to Theorem 6.5, there are at most O(log(T )/β) different +i ∈ [T ] such that oi,r ̸=⊥. Let such set of i to be I. Since each Λi for i ∈ I is ε′-DP in the streaming +continual releasing setting, we have: +Pr [∀i ∈ I, j ∈ [T ], oi,j(S) = oi,j] ≤ exp(ε′ · |I|) · Pr [∀i ∈ I, j ∈ [T ], oi,j(S′) = oi,j] +≤ exp(ε) · Pr [∀i ∈ I, j ∈ [T ], oi,j(S′) = oi,j] . +On the other hand, we have: +Pr [∀i ̸∈ I, j ∈ [T ], oi,j(S) = oi,j | oi,j(S) = oi,j∀i ∈ I, j ∈ [T ]] += Pr [∀i ̸∈ I, j ∈ [T ], oi,j(S′) = oi,j | oi,j(S′) = oi,j∀i ∈ I, j ∈ [T ]] +Therefore, the algorithm is ε-DP. +By combining Theorem 3.3 with Lemma 6.3, Lemma 6.4 and Theorem 6.6, we are able to obtain +the following sliding window continual release algorithm for summing non-negative numbers. +Corollary 6.7 (Sliding window summing of a non-negative numbers). Let η ∈ (0, 0.5), ε ≥ 0, ξ ∈ +(0, 0.5), there is an ε-DP algorithm for summing in the sliding window continual release model. If +the input numbers are guaranteed to be non-negative, with probability at least 1 − ξ, the output is +always a +� +1 + η, O +� +log(T/(ηξ)) log(T ) +εη3 +�� +-approximation to the summing problem at any timestamp +t ∈ [T ]. The algorithm uses space O(log(T )/η). +By combining Corollary 4.11 with Lemma 6.3, Lemma 6.4 and Theorem 6.6, we are able to +obtain the following sliding window continual release algorithm for number of distinct elements. +Corollary 6.8 (Sliding window continual release distinct elements). For η ∈ (0, 0.5), ε ≥ 0, ξ ∈ +(0, 0.5) there is an ε-DP algorithm for the number of distinct elements of streams with element +universe U in the sliding window continual release model. With probability at least 1 − ξ, the output +is always a (1+η, O +� +log2(T/(ηξ)) log(T ) +η4ε +� +)-approximation for every timestamp t ∈ [T ]. The algorithm +uses poly +� +log(T/ξ) +η min(ε,1) +� +space. +By combining Theorem 5.4 with Lemma 6.3, Lemma 6.4 and Theorem 6.6, we are able to obtain +the following sliding window continual release algorithm for ℓ2 frequency moment. +Corollary 6.9 (Sliding window continual release ℓ2 frequency moments). Let ε > 0, η ∈ (0, 0.5), ξ ∈ +(0, 0.5). There is an ε-DP algorithm in the sliding window continual release model such that with +probability at least 1 − ξ, it always outputs ˆF2 for every timestamp t ∈ [T ] such that | ˆF2 − ∥S∥2 +2| ≤ +η∥S∥2 +2 + O +� +(log(T/(ξη))+log(|U|))2 log2(T ) +ε2η8 +· log5(T ) · log2 � +log(T/ξ)+log(|U|) +ξη +�� +, where S denotes the sub- +stream corresponding to the latest W elements at timestamp t. The algorithm uses O +� +log(T/(ξη))+log(|U|) +η4 +· log2(T ) +� +space. +40 + +By combining Theorem 5.31 with Lemma 6.3, Lemma 6.4 and Theorem 6.6, we are able to +obtain the following sliding window continual release algorithm for ℓp frequency moment. +Corollary 6.10 (Sliding window continual release ℓp frequency moments). Let p > 0, ε ≥ 0, ξ ∈ +(0, 0.5), η ∈ (0, 0.5). There is an ε-DP algorithm in the sliding window continual release model such +that with probability at least 1−ξ, it always outputs an +� +1 + η, +� +log(T |U|/ξ) +ηε +�O(p)� +-approximation to +∥S∥p +p, where S denotes the sub-stream corresponding to the latest W elements at timestamp t. The +algorithm uses space at most +φ · +�log(T |U|/ξ) +ηε +�O(p) +, +where φ = max(1, |U|1−2/p). +References +Naman Agarwal and Karan Singh. The price of differential privacy for online learning. 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OpenReview.net, 2022. URL +https://openreview.net/forum?id=7I8LPkcx8V. +A +Missing Details of Section 3 +A.1 +Proof of Lemma 3.1 +Proof. We first show that the output groups G1, G2, · · · , Gm is ε0-DP. Let O1, O2, · · · , Om be any +fixed grouping. Let c′ +1, c′ +2, · · · , c′ +T be any neighboring stream, i.e., ∃q ∈ [T ] such that |cq − c′ +q| ≤ +1 and ∀j ̸= q, cj = c′ +j. +Let G′ +1, G′ +2, · · · , G′ +m′ be the output groups of the neighboring stream. +Suppose q ∈ Or for some r ∈ [m]. Let us consider Pr [(G1, G2, · · · , Gm) = (O1, O2, · · · , Om)] and +Pr [(G′ +1, G′ +2, · · · , G′ +m′) = (O1, O2, · · · , Om)] (in this case m′ = m). We have: +Pr [(G1, G2, · · · , Gm) = (O1, O2, · · · , Om)] +Pr [(G′ +1, G′ +2, · · · , G′m) = (O1, O2, · · · , Om)] += Pr [(G1, G2, · · · , Gr−1) = (O1, O2, · · · , Or−1)] +Pr +� +(G′ +1, G′ +2, · · · , G′ +r−1) = (O1, O2, · · · , Or−1) +� · Pr [Gr = Or | (G1, G2, · · · , Gr−1) = (O1, O2, · · · , Or−1)] +Pr +� +G′r = Or | (G′ +1, G′ +2, · · · , G′ +r−1) = (O1, O2, · · · , Or−1) +� +· Pr [(Gr+1, Gr+2, · · · , Gm) = (Or+1, Or+2, · · · , Om) | (G1, G2, · · · , Gr) = (O1, O2, · · · , Or)] +Pr +� +(G′ +r+1, G′ +r+2, · · · , G′m) = (Or+1, Or+2, · · · , Om) | (G′ +1, G′ +2, · · · , G′r) = (O1, O2, · · · , Or) +� +Since ∀j ∈ O1 ∪O2 ∪· · ·∪Or−1, cj = c′ +j, the behavior of running Algorithm 1 on the prefix O1 ∪O2 ∪ +· · ·∪Or−1 of c1, c2, · · · , cT is the same as the behavior of running it on the prefix O1∪O2∪· · ·∪Or−1 +of c′ +1, c′ +2, · · · , c′ +T . Therefore, we have Pr[(G1,G2,··· ,Gr−1)=(O1,O2,··· ,Or−1)] +Pr[(G′ +1,G′ +2,··· ,G′ +r−1)=(O1,O2,··· ,Or−1)] = 1. Similarly, since ∀j ∈ +45 + +Or+1∪Or+2∪· · ·∪Om, cj = c′ +j, when Gr = G′ +r = Or, the behavior of running Algorithm 1 on the suf- +fix Or+1∪Or+2∪· · ·∪Om of c1, c2, · · · , cT is the same as the behabior of running it on the same suffix +of c′ +1, c′ +2, · · · , c′ +T . Therefore, we have Pr[(Gr+1,Gr+2,··· ,Gm)=(Or+1,Or+2,··· ,Om)|(G1,G2,··· ,Gr)=(O1,O2,··· ,Or)] +Pr[(G′ +r+1,G′ +r+2,··· ,G′m)=(Or+1,Or+2,··· ,Om)|(G′ +1,G′ +2,··· ,G′r)=(O1,O2,··· ,Or)]. +Thus, we have: +Pr [(G1, G2, · · · , Gm) = (O1, O2, · · · , Om)] +Pr [(G′ +1, G′ +2, · · · , G′m) = (O1, O2, · · · , Om)] += Pr [Gr = Or | (G1, G2, · · · , Gr−1) = (O1, O2, · · · , Or−1)] +Pr +� +G′r = Or | (G′ +1, G′ +2, · · · , G′ +r−1) = (O1, O2, · · · , Or−1) +�. +Suppose Or = {x + 1, · · · , x + k}. +Consider the first case where r ̸= m. In this case, we have +Pr [Gr = Or | (G1, G2, · · · , Gr−1) = (O1, O2, · · · , Or−1)] += Pr [Gr = Or | Gr−1 = Or−1] += Pr +�� +∀j ∈ [k − 1], νx+j + +j +� +b=1 +cx+b < τr +� � � +νx+k + +k +� +b=1 +cx+b ≥ τr +�� +. +Now, let us fix νx+1, νx+2, · · · , νx+k−1 and let g = maxj∈[k−1] νx+j + �j +b=1 cx+b. Then, +Pr +νx+k,τr [Gr = Or | Gr−1 = Or−1] += +Pr +νx+k,τr +� +τr ∈ (g, νx+k + +k +� +b=1 +cx+b] +� += +� ∞ +−∞ +� ∞ +−∞ +pνx+k(v) · pτr(τ) · 1 +� +τ ∈ (g, v + +k +� +b=1 +cx+b) +� +dvdτ +(3) +where pνx+k(·) and pτr(·) are density functions of νx+k and τr respectively. Let g′ = maxj∈[k−1] νx+j+ +�j +b=1 c′ +x+b. Let v′ = v + g − g′ + �k +b=1 c′ +x+b − �k +b=1 cx+b. Let τ ′ = τ + g − g′. Since |cq − c′ +q| ≤ 1, +it is easy to see that |v′ − v| ≤ 2 and |τ − τ ′| ≤ 1. Note that dv′ = dv and dτ′ = dτ. Therefore, +Equation (3) is equal to the following: +� ∞ +−∞ +� ∞ +−∞ +pνx+k(v′) · pτr(τ ′) · 1 +� +τ + g − g′ ∈ +� +g, v + g − g′ + +k +� +b=1 +c′ +x+b +�� +dvdτ += +� ∞ +−∞ +� ∞ +−∞ +pνx+k(v′) · pτr(τ ′) · 1 +� +τ ∈ +� +g′, v + +k +� +b=1 +c′ +x+b +�� +dvdτ +≤ +� ∞ +−∞ +� ∞ +−∞ +exp(ε0/2) · pνx+k(v) · exp(ε0/2) · pτr(τ) · 1 +� +τ ∈ +� +g′, v + +k +� +b=1 +c′ +x+b +�� +dvdτ += exp(ε0) · +Pr +νx+k,τr +� +τr ∈ (g′, νx+k + +k +� +b=1 +c′ +x+b] +� += exp(ε0) · +Pr +νx+k,τr +� +G′ +r = Or | G′ +r−1 = Or−1 +� +46 + += exp(ε0) · +Pr +νx+k,τr +� +G′ +r = Or | (G′ +1, G′ +2, · · · , G′ +r−1) = (O1, O2, · · · , Or−1) +� +. +Next, consider the second case where r = m. In this case, we have +Pr [Gr = Or | (G1, G2, · · · , Gr−1) = (O1, O2, · · · , Or−1)] += Pr [Gr = Or | Gr−1 = Or−1] += Pr +� +∀j ∈ [k], νx+j + +j +� +b=1 +cx+b < τr +� +. +Now, let us fix νx+1, νx+2, · · · νx+k and let g = maxj∈[k] νx+j + �j +b=1 cx+b. Let g′ = maxj∈[k] νx+j + +�j +b=1 c′ +x+b. Since |cq − c′ +q| ≤ 1, we have |g − g′| ≤ 1. Then, +Pr +τr [τr > g] ≤ exp(ε0) · Pr +τr [τr > g′] . +Thus, we have +Pr [Gr = Or | (G1, G2, · · · , Gr−1) = (O1, O2, · · · , Or−1)] +≤ exp(ε0) · Pr +� +G′ +r = Or | (G′ +1, G′ +2, · · · , G′ +r−1) = (O1, O2, · · · , Or−1) +� +. +Therefore, we can conclude that (G1, G2, · · · , Gm) is always ε0 DP. +Notice that, given any fixed (O1, O2, · · · , Om) and condition on (G1, G2, · · · , Gm) = (O1, O2, · · · , Om), +ˆc1, ˆc2, · · · , ˆcT is ε0-DP by Laplace mechanism. Thus, by composition theorem, the output stream +ˆc1, ˆc2, · · · , ˆcT is ε-DP. +A.2 +Proof of Lemma 3.2 +Let G1, G2, · · · , Gm be the groups produced during Algorithm 1. Let ˜c1, ˜c2, · · · , ˜cm be that ∀i ∈ +[m], ˜ci = ˆcmax(Gi), i.e., ˜ci is the noisy count of the group Gi. +Lemma A.1. With probability at least 1−ξ, the output stream of Algorithm 1 satisfies the following +properties: +1. ∀i ∈ [m − 1], � +j∈Gi\{maxj′∈Gi j′} cj ≤ +7 +ηε0 · ln (3 · T/ξ) + 13 +ε0 · ln(3 · T/ξ). +2. � +j∈Gm cj ≤ +7 +ηε0 · ln (3 · T/ξ) + 13 +ε0 · ln(3 · T/ξ). +3. ∀i ∈ [m − 1], (1 − η) � +j∈Gi cj ≤ ˜ci ≤ (1 + η) � +j∈Gi cj. +Proof. Let E denote the event that +1. ∀i ∈ [m], +���τi − +� +1 +η + 1 +� +· 7 +ε0 · ln (3 · T/ξ) +��� ≤ +2 +ε0 · ln (3 · T/ξ). +2. ∀t ∈ [T ], |νt| ≤ +4 +ε0 · ln (3 · T/ξ). +3. ∀i ∈ [m − 1], |˜ci − � +j∈Gi cj| ≤ +1 +ε0 · ln(3 · T/ξ) +47 + +According to the CDF of Laplace noise, it is easy to show that E happens with probability at least +1 − ξ by a union bound over all i ∈ [m], t ∈ [T ]. In the remaining of the proof we condition on the +event E. +Consider property 1. +Consider any i ∈ [m − 1]. +Let t = maxj∈Gi j − 1. +Then we have +νt + � +j∈Gi,j≤t cj ≤ τi. Since |νt| ≤ +4 +ε0 · ln (3 · T/ξ) and τi ≤ +7 +η·ε0 · ln (3 · T/ξ) + 9 +ε0 · ln(3 · T/ξ), we +have � +j∈Gi,j≤t cj ≤ +7 +ηε0 · ln(3 · T/ξ) + 13 +ε0 · ln(3 · T/ξ). +The proof of property 2 is the same as the proof of property 1. +Consider property 3. +Consider any i ∈ [m − 1]. +Let t = maxj∈Gi j. +Then we have νt + +� +j∈Gi cj ≥ τi. +Since |νt| ≤ +4 +ε0 · ln (3 · T/ξ) and τi ≥ +7 +η·ε0 · ln (3 · T/ξ) + +5 +ε0 · ln(3 · T/ξ), we +have � +j∈Gi cj ≥ +7 +ηε0 · ln(3 · T/ξ). Note that |˜ci − � +j∈Gi cj| ≤ +1 +ε0 · ln (3 · T/ξ). Thus, we know that +� +j∈Gi cj ≥ +7 +ηε0 ·ln (3 · T/ξ). Thus, ∀i ∈ [m−1] we have (1−η) � +j∈Gi cj ≤ ˜ci ≤ (1+η) � +j∈Gi cj. +Now, we are able to prove Lemma 3.2. +Proof of Lemma 3.2. With probability at least 1 − ξ, the properies listed in Lemma A.1 hold. +Let l′ ∈ [m] be the smallest index such that maxj∈Gl′ j ≥ l and r′ ∈ [m] be the largest index +such that maxj∈Gr′ j ≤ r. +We have: +r +� +j=l +ˆcj = +r′ +� +j′=l′ +˜cj′ +≥(1 − η) +r′ +� +j′=l′ +� +j∈Gj′ +cj − +� 7 +ηε0 +· ln (3 · T/ξ) + 13 +ε0 +· ln(3 · T/ξ) +� +≥(1 − η) +maxb∈Gr′ b +� +j=l +cj − +� 7 +ηε0 +· ln (3 · T/ξ) + 13 +ε0 +· ln(3 · T/ξ) +� +≥(1 − η) +r +� +j=l +cj − +7 +ηε0 +· ln (3 · T/ξ) − 26 +ε0 +· ln (3 · T/ξ) , +where the first inequality follows from property 3 and property 2 of Lemma A.1, the second inequal- +ity follows from the choice of l′, and the third inequality follows from property 1 of Lemma A.1. +Similarly, we can show +r +� +j=l +ˆcj = +r′ +� +j′=l′ +˜cj′ +≤(1 + η) +r′ +� +j′=l′ +� +j∈Gj′ +cj + +� 7 +ηε0 +· ln (3 · T/ξ) + 13 +ε0 +· ln(3 · T/ξ) +� +≤(1 + η) +r +� +j=minb∈Gl′ b +cj + +� 7 +ηε0 +· ln (3 · T/ξ) + 13 +ε0 +· ln(3 · T/ξ) +� +≤(1 + η) +r +� +j=l +cj + +7 +ηε0 +· ln (3 · T/ξ) + 26 +ε0 +· ln (3 · T/ξ) , +48 + +where the first inequality follows from property 3 and property 2 of Lemma A.1, the second inequal- +ity follows from the choice of r′, and the third inequality follows from property 1 of Lemma A.1. +49 + diff --git a/W9E5T4oBgHgl3EQfcg_y/content/tmp_files/load_file.txt b/W9E5T4oBgHgl3EQfcg_y/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ede5f045c122bc1777f498fc82e9fa47d4e0881f --- /dev/null +++ b/W9E5T4oBgHgl3EQfcg_y/content/tmp_files/load_file.txt @@ -0,0 +1,1929 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E5T4oBgHgl3EQfcg_y/content/2301.05605v1.pdf,len=1928 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E5T4oBgHgl3EQfcg_y/content/2301.05605v1.pdf'} +page_content='05605v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E5T4oBgHgl3EQfcg_y/content/2301.05605v1.pdf'} +page_content='DS] 13 Jan 2023 Differentially Private Continual Releases of Streaming Frequency Moment Estimations Alessandro Epasto π(i + 1)} ⊂ [n − 1] +and its peak set +Peak(π) = {2 ≤ i ≤ n − 1|π(i − 1) < π(i) > π(i + 1)}. +The peak set of a permutation is said to be peak-lacunar, i.e. it neither contains 1 nor +contains two consecutive integers. +1.2 +Enriched P-partitions and q-deformed generating functions +We recall the main definitions regarding weighted posets, enriched P-partitions and their +q-deformed generating functions. The reader is referred to [1, 3, 4, 7, 8] for more details. +Definition 1 (Labelled weighted poset, [3]). A labelled weighted poset is a triple P = +([n], j, then f(i) < f(j) or f(i) = f(j) ∈ −P. + +Extended peaks +3 +We denote LP±(P) be the set of enriched P-partitions. +Definition 3 (q-Deformed generating function, [4]). Consider the set of indeterminates X = +{x1, x2, x3, . . .}, the ring C [[X]] of formal power series on X where C is the set of complex +numbers, and let q ∈ C be an additional parameter. Given a labelled weighted poset ([n], p − 1 = 1. +Remark 2 (Permutation statistics). Extended peak sets look like an intermediary statistic be- +tween peak and descent sets. Any peak set is a 1-extended peak set and any descent set on +permutations of n elements is a p-extended peak set for p ≥ n. Moreover, given a permutation π +in Sn and an integer p ≥ 1 one may define Peakp(π) as +Peakp(π) = {i ∈ Des(π)|i ≤ p − 1 or ∃ 1 ≤ j ≤ p such that i − j /∈ Des(π)}. +On the one hand one has always Peak(π) = Peak1(π) ⊆ Peakp(π) ⊆ Des(π). On the other +hand, Peakp(π) ⊆p [n − 1]. For instance let π = 54163287. We have Peak1(π) = Peak(π) = +{4, 7} ⊆ Peak2(π) = {1, 4, 5, 7} ⊆ Peak3(π) = {1, 2, 4, 5, 7} = Des(π). +We count the number of extended peak sets. +Proposition 5. Let n, p ∈ P. Denote s(p) +n +be the number of p-extended peak sets on n elements. +Extend the definition with s(p) +0 += 0 for all positive p. One has +s(p) +n += +� +2n−1 if n ≤ p +∑ +p +k=0 s(p) +n−k−1 = ∑ +p+1 +k=1 s(p) +n−k if n > p +(2.1) +Proof. The result is immediate for n ≤ p. For n > p, there is a bijection between p- +extended peak sets on n elements and the union of p-extended peak sets on n − 1 − k +elements for k ∈ [0, p]. Indeed, given a set I ⊆p [n − 1], define k ≤ p as the integer such +that [n − k, n − 1] is the maximum sequence of consecutive integers in I containing n − 1. +If n − 1 /∈ I we define k = 0 and assume [n, n − 1] = ∅. We map I to the unique element +J ⊆p [n − k − 2] such that I = J ∪ [n − k, n − 1]. This mapping is clearly one-to-one and +the result follows. +2.2 +Extended peak quasisymmetric functions +We proceed with the definition the relevant subfamilies of q-fundamentals. + +Extended peaks +7 +Definition 8 (Extended-peak quasisymmetric functions). Let n, p ∈ P and denote ρp the +root of unity ρp = e−iπ(p−1)/(p+1). We have ρ1 = 1, ρ2 = e−iπ/3, ρ4 = e−iπ/2, . . . . Note +that (−ρp) is a primitive p + 1-th root of unity, i.e., (−ρp)p+1 = 1 but (−ρp)j ̸= 1 for +1 ≤ j < p + 1. Given a subset I ⊆ [n − 1] we define the p-extended peak quasisymmetric +function indexed by I +Lp +n,I = L(ρp) +n,I . +(2.2) +Denote P p ⊆ QSym the subalgebra of QSym spanned by (Lp +n,I)n≥0,I⊆[n−1] and P p +n ⊆ QSymn +its subspace composed of quasisymmetric functions of degree n (i.e the vector space spanned by +(Lp +n,I)I⊆[n−1]). We call P p the algebra of p-extended peaks. +Definition 8 gives extended peak functions over all subsets. +However, we know +from Proposition 4 that they do not span QSym. As a result, for all p ∈ P, the fam- +ily (Lp +n,I)n≥0,I⊆[n−1] is not linearly independent and some indices are redundant. We +characterise these set indices. First, for n, p ∈ P, if a set I is not a p-extended peak set, +then Lp +n,I may be expressed in terms of other p-extended peak quasisymmetric functions. +Theorem 1 (Extended peak functions over sets that are not p-extended peaks). Let n, p ∈ +P with n ≥ p + 1, i be an integer such that 0 ≤ i ≤ n − 1 − p and J ⊆ [n − 1] be a subset that +satisfies [i + 1, i + p + 1] ∩ J = ∅ and i ∈ J ∪ {0}. Then, the set [i + 1, i + p] ∪ J ⊈p [n − 1] as +it contains either a sequence of p + 1 consecutive elements or the sequence [1, p]. Notice further +that any set that is not a p-extended peak set may be written as such. We have the following +equality. +∑ +I⊆[i+1,i+p] +(−1)|I|Lp +n,I∪J = 0. +Secondly, we can compute explicitly the dimension of P p +n for n, p ∈ P. +Theorem 2 (Subspaces dimension). Let n, p ∈ P be two positive integers. The dimension of +P p +n is equal to s(p) +n , the number of p-extended peak sets on n elements. +dim P p +n = s(p) +n +We postpone the proofs of Theorems 1 and 2 respectively to Sections 3.1 and 3.2. +Combining them we characterise the subalgebra P p. +Theorem 3 (Basis for the algebra of extended peaks). Let p ∈ P. The family (Lp +n,I)n≥0,I⊆p[n−1] +is a basis of the subalgebra P p of QSym. +Proof. Fix p ∈ P. As p-extended peak quasisymmetric functions are special cases of +q-fundamentals, the stability by multiplication is actually a direct consequence of Equa- +tion (1.2). Then Theorem 1 shows that only p-extended peak quasisymmetric functions +indexed by p-extended peak sets may be linearly independent. Finally, Theorem 2 shows +that for all n ∈ P the dimension of the finite vector space containing homogenous qua- +sisymmetric functions of degree n is exactly the number of p-extended peak sets on n +elements. + +8 +D. Grinberg and E.A. Vassilieva +3 +Proofs of Theorems 1 and 2 +3.1 +Extended peak functions indexed by generic sets +In order to show Theorem 1 compute +∑ +I⊆[i+1,i+p] +(−1)|I|L(q) +I∪J +using Equation (1.3) for integers i, n, p and set J satisfying the conditions of the theorem. +Note that for I ⊆ [i + 1, i + p], i + 1 /∈ Peak(I ∪ J) (either i + 1 = 1 or i ∈ J) and that +Peak(I ∪ J) ∩ J = Peak(J) irrelevant of the choice of I (as i + p + 1 /∈ J). As a result, we +can decompose in Equation (1.3) any subset or peak lacunar subset (i.e. 1-extended peak +subset) of I ∪ J as a (peak lacunar) subset of I and a (peak lacunar) subset of J. Namely, +∑ +I⊆[i+1,i+p] +(−1)|I|L(q) +I∪J += +∑ +U′⊆J +V′⊆Peak(J) +U′∩V′=∅ +(−q)|V′|(q − 1)|U′| ∑ +I⊆[i+1,i+p] +(−1)|I| ∑ +U⊆I +V⊆Peak(I)\{i+1} +U∩V=∅ +(−q)|V|(q − 1)|U|η(q) +U′∪V′−1∪V′∪U∪V−1∪V. +Next, invert summation indices in the last sums. +∑ +I⊆[i+1,i+p] +(−1)|I|L(q) +I∪J += +∑ +U′⊆J +V′⊆Peak(J) +U′∩V′=∅ +(−q)|V′|(q − 1)|U′| +∑ +U⊆[i,i+p] +V⊆1[i+1,i+p] +U∩V=∅ +U∩V−1=∅ +(−q)|V|(q − 1)|U|η(q) +U′∪V′−1∪V′∪U∪V−1∪V ∑ +U∪V⊆I⊆[i+1,i+p]\V−1 +(−1)|I|. +The last sum is obviously 0 except when U ∪ V = [i + 1, i + p] \ V − 1. As a result, +∑ +I⊆[i+1,i+p] +(−1)|I|L(q) +I∪J += +∑ +U′⊆J +V′⊆Peak(J) +U′∩V′=∅ +(−q)|V′|(q − 1)|U′|η(q) +U′∪V′−1∪V′∪[i,i+p] +∑ +V⊆1[i+1,i+p] +(−q)|V|(q − 1)p−2|V|(−1)p−|V|. +The summands in the sum over all 1-extended peak sets V ⊆1 [i + 1, i + p] depend only +on the cardinality of V. It easy to show (left to the reader) that +|{V ⊆1 [i + 1, i + p], |V| = v}| = +�p − v +v +� +. + +Extended peaks +9 +Subsequently, +∑ +I⊆[i+1,i+p] +(−1)|I|L(q) +I∪J += (−1)p∑ +U′⊆J +V′⊆Peak(J) +U′∩V′=∅ +(−q)|V′|(q − 1)|U′|η(q) +U′∪V′−1∪V′∪[i+1,i+p] +p +∑ +v=0 +(−1)v +�p − v +v +� +(−q)v(q − 1)p−2v. +We use the following lemma. +Lemma 1 ([9]). Let n ∈ P and x, y ∈ C, one has +n +∑ +k=0 +(−1)k +�n − k +k +� +(xy)k(x + y)n−2k = +n +∑ +j=0 +xn−jyj. +Denote for n ∈ P, c ∈ C, [n]c = (1 − cn)/(1 − c). As a direct consequence of Lemma 1, +∑ +I⊆[i+1,i+p] +(−1)|I|L(q) +I∪J = +∑ +U′⊆J +V′⊆Peak(J) +U′∩V′=∅ +(−q)|V′|(q − 1)|U′|η(q) +U′∪V′−1∪V′∪[i+1,i+p] +p +∑ +t=0 +(−q)p−t, += [p + 1]−q +∑ +U′⊆J +V′⊆Peak(J) +U′∩V′=∅ +(−q)|V′|(q − 1)|U′|η(q) +U′∪V′−1∪V′∪[i+1,i+p]. +End the proof with +[p + 1]−ρp = 1 − (−ρp)p+1 +1 + ρp += 0. +3.2 +Finite subspaces dimension +For n ∈ P and q ∈ C, denote B(q) +n +the transition matrix between (L(q) +n,I)I⊆[n−1] and +(η(q) +n,J )J⊆[n−1] with coefficients given by Equation (1.3). Columns and rows are indexed +by subsets I of [n − 1] sorted in reverse lexicographic order. A subset I is before subset +J iff the word obtained by writing the elements of I in decreasing order is before the +word obtained from J for the lexicographic order. The column indexed by the subset I +corresponds to L(q) +n,I and the row indexed by J to η(q) +n,J (as a direct consequence B(q) +n +is the +transpose of the similar matrix defined in [4]). For n = 0, assume B(q) +0 +to be the empty +matrix. + +10 +D. Grinberg and E.A. Vassilieva +Example 2. For n = 4, the transition matrix B(q) +4 +between (L(q) +I )I⊆[3] and (η(q) +J )J⊆[3] is given +by +B(q) +4 += +∅ +{1} +{2} +{2, 1} +{3} +{3, 1} +{3, 2} +{3, 2, 1} +∅ +1 +1 +1 +1 +1 +1 +1 +1 +{1} +0 +q − 1 +0 +q − 1 +0 +q − 1 +0 +q − 1 +{2} +0 +0 +q − 1 +q − 1 +0 +0 +q − 1 +q − 1 +{2, 1} +0 +0 +−q +(q − 1)2 +0 +0 +−q +(q − 1)2 +{3} +0 +0 +0 +0 +q − 1 +q − 1 +q − 1 +q − 1 +{3, 1} +0 +0 +0 +0 +0 +(q − 1)2 +0 +(q − 1)2 +{3, 2} +0 +0 +0 +0 +−q +−q +(q − 1)2 +(q − 1)2 +{3, 2, 1} +0 +0 +0 +0 +0 +−q(q − 1) +−q(q − 1) +(q − 1)3 +Our goal is to compute the dimension of the kernel of B(q) +n +to get the dimension of +the vector subspace P p +n as +dim P p +n = rank(B(q) +n ) = 2n−1 − dim ker B(q) +n . +We show the following proposition. +Proposition 6. Let n, p ∈ P be two positive integers. We have +dim ker B(ρp) +n += +� +∑ +p+1 +k=1 dim ker B(ρp) +n−k + [n > p + 1]2n−p−2 for n > p, +0 for n ≤ p. +Proof. The second case is a direct consequence of the fact that the matrix B(ρp) +n +is invertible +for n ≤ p (see [4]). To show the general recurrence, assume that n > p. As in [4], notice +that the matrix B(q) +n +is block upper triangular. For each k ∈ [n], let A(q) +k +denote the transi- +tion matrix from (L(q) +n,I)I⊆[n−1], max(I)=k−1 to (η(q) +n,J )J⊆[n−1], max(J)=k−1 (where max ∅ := 0); +this actually does not depend on n. Note that A(q) +k +is a 2k−2 × 2k−2-matrix if k ≥ 2, +whereas A(q) +1 +is a 1 × 1-matrix. We have +B(q) +n += + + + + + + + + + +A(q) +1 +∗ +∗ +. . . +∗ +0 +A(q) +2 +∗ +. . . +∗ +0 +0 +A(q) +3 +. . . +∗ +0 +0 +0 +... +∗ +0 +0 +0 +0 +A(q) +n + + + + + + + + + +. +We have the following lemma: + +Extended peaks +11 +Lemma 2. The matrices +� +B(q) +n +� +n and +� +A(q) +n +� +n satisfy the following recurrence relations (for +n ≥ 1 and n ≥ 2, respectively): +B(q) +n += +� +B(q) +n−1 +B(q) +n−1 +0 +A(q) +n +� +, +A(q) +n += +� +(q − 1)B(q) +n−2 +(q − 1)B(q) +n−2 +−qB(q) +n−2 +(q − 1)A(q) +n−1 +� +. +Consider for n ≥ 2 and coefficients α, β ∈ C the kernel of the matrix αA(q) +n+1 + βB(q) +n . +Let X be a 2n−1 vector and denote X1 and X2 the two vectors of size 2n−2 such that +X = +�X1 +X2 +� +. We compute +� +αA(q) +n+1 + βB(q) +n +� +X = 0 ⇔ +� +((q − 1)α + β)B(q) +n−1(X1 + X2) = 0 +−qαB(q) +n−1X1 + ((q − 1)α + β)A(q) +n X2 = 0 +(3.1) +Two cases arise from the previous equation. Either ((q − 1)α + β) ̸= 0 and +� +αA(q) +n+1 + βB(q) +n +� +X = 0 ⇔ + + + +B(q) +n−1(X1 + X2) = 0 +� +((q − 1)α + β)A(q) +n ++ qαB(q) +n−1 +� +X2 = 0 +(3.2) +or ((q − 1)α + β) = 0 and +� +αA(q) +n+1 + βB(q) +n +� +X = 0 ⇔ qαB(q) +n−1X1 = 0 +(3.3) +Consider the sequence of coefficients +� +α(q) +0 +β(q) +0 +� += (0 +1) +� +α(q) +n+1 +β(q) +n+1 +� += +� +α(q) +n +β(q) +n +� �q − 1 +q +1 +0 +� +, for n ≥ 0. +Solving the recurrence we have for integer n ≥ 1 +� +α(q) +n +β(q) +n +� += (0 +1) +�q − 1 +q +1 +0 +�n += (−1)n(0 +1) +�[n + 1]−q +−q[n]−q +−[n]−q +q[n − 1]−q +� += (−1)n(−[n]−q +q[n − 1]−q), +where for integer i and complex number c recall that [i]c = (1 − ci)/(1 − c). Finally +notice that for integer n ≥ 1 +α(q) +n (q − 1) + β(q) +n += (−1)n � +[n]−q − q(−q)n−1� += (−1)n[n + 1]−q. + +12 +D. Grinberg and E.A. Vassilieva +Get back to the case q = ρp for some positive integer p ∈ P. As a direct consequence of +the defintion of ρp in Definition 8, we have +[p + 1]−ρp = 0 +[n]−ρp ̸= 0, 1 ≤ n ≤ p +As a result, if q = ρp, one may iterate the recurrence in Equation (3.1) p times with the +case of Equation (3.2) and one more time to go to the case of Equation (3.3). Recall that +β(ρp) +p+1 = ρp[p]−ρp ̸= 0 to conclude the proof. +Noticing that for n > p + 1, 2n−1 = 2n−2 + 2n−3 + · · · + 2n−p−1 + 2 · 2n−p−2 we can +deduce the rank of B(ρp) +n +using Proposition 6. We get +rank +� +B(ρp) +n +� += 2n−1 − dim ker B(ρp) +n += + + + +∑ +p+1 +k=1 rank +� +B(ρp) +n−k +� +for n > p, +2n−1 for 1 ≤ n ≤ p. +(3.4) +We conclude that the sequence of subspace dimensions (P p +n)n follows the same recur- +rence with the same initial conditions as the sequence of the numbers of p-extended +peak sets (sp +n)n. Theorem 2 follows. +References +[1] +I. Gessel. “Multipartite P-partitions and inner products of skew Schur functions”. Contem- +porary Mathematics 34 (1984), pp. 289–317. +[2] +D. Grinberg and V. Reiner. “Hopf Algebras in Combinatorics”. arXiv:1409.8356v7. 2020. +Link. +[3] +D. Grinberg and E. Vassilieva. +“Weighted posets and the enriched monomial basis of +QSym”. Sémin. Loth. de Comb. 85B (FPSAC 2021) (2021). arXiv:2202.04720v1. +[4] +D. Grinberg and E. Vassilieva. “A q-Deformation of Enriched P-Partitions”. Sémin. Loth. de +Comb. 86B (FPSAC 2022) (2022). +[5] +M. E. Hoffman. “Quasi-symmetric functions and mod p multiple harmonic sums”. Kyushu +J. Math. 69 (2015). arXiv:math/0401319v3, pp. 345–366. +[6] +S. K. Hsiao. “Structure of the peak Hopf algebra of quasisymmetric functions”. 2007. +[7] +R. Stanley. Enumerative combinatorics. Vol. 2. Cambridge University Press, 2001. +[8] +J. Stembridge. “Enriched P-partitions.” Trans. Amer. Math. Soc. 349.2 (1997), pp. 763–788. +[9] +B. Sury. “A parent of Binet’s formula”. Mathematics Magazine 77 (2004), pp. 308–310. + diff --git a/_9AyT4oBgHgl3EQfdvew/content/tmp_files/load_file.txt b/_9AyT4oBgHgl3EQfdvew/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6d54fefe4daf405ab9a103a13947043ed7f417d7 --- /dev/null +++ b/_9AyT4oBgHgl3EQfdvew/content/tmp_files/load_file.txt @@ -0,0 +1,425 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf,len=424 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content='00309v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content='CO] 1 Jan 2023 Séminaire Lotharingien de Combinatoire XX (2023) Proceedings of the 35th Conference on Formal Power Article #YY, 12 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' Series and Algebraic Combinatorics (Davis) The algebra of extended peaks Darij Grinberg*1, and Ekaterina A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' Vassilieva†2 Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' Building up on our previous works regarding q-deformed P-partitions, we introduce a new family of subalgebras for the ring of quasisymmetric functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' Each of these subalgebras admits as a basis a q-analogue to Gessel’s fundamental quasisym- metric functions where q is equal to a complex root of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' Interestingly, the basis elements are indexed by sets corresponding to an intermediary statistic between peak and descent sets of permutations that we call extended peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' Résumé.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' En nous appuyant sur nos travaux précédents concernant les P-partitions q-déformées, nous introduisons une nouvelle famille de sous-algèbres pour l’anneau des fonctions quasi-symétriques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' Chacune de ces sous-algèbres admet comme base un q-analogue aux fonctions quasisymétriques fondamentales de Gessel où q est égal à une racine complexe de l’unité.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' Il est notable que les éléments de base sont indexés par des ensembles correspondant à une statistique intermédiaire entre les ensembles de pics et de descentes des permutations que nous appelons pic étendu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' Keywords: Quasisymmetric functions, descent set, peak set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' 1 Introduction In [4], we show that a q-deformation of the generating functions for P-partitions leads to a unified framework between classical and enriched P-partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' In particular, we intro- duce our q-fundamental quasisymmetric functions that interpolate between I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' Gessel’s fundamental ([1], q = 0) and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' Stembridge’s peak ([8], q = 1) quasisymmetric functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' When q is not a root of unity, q-fundamentals are a basis of QSym, the ring of quasisym- metric functions and are indexed by descent sets of permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' If q = 1, they span the subalgebra of QSym named the algebra of peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' The relevant basis elements are those indexed by peak sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' As it turns out, using other complex roots of unity for q, we are able to build new intermediate subalgebras between the algebra of peaks and QSym, the basis of which are q-fundamentals indexed by a new permutation statistic that lies between peak and descent sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' We call this statistic the extended peak set and the cor- responding subalgebras of quasisymmetric functions the algebra of extended peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' We begin with the required definitions and results from [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' Then we introduce and prove the new results regarding the algebra of extended peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' darijgrinberg@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content='com †katya@lix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content='polytechnique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content='fr 2 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' Grinberg and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' Vassilieva 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content='1 Permutation statistics Let P be the set of positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' For m, n ∈ P, write [m, n] = {m, m + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' , n} and simply [n] = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' We denote Sn the symmetric group on [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' Given π ∈ Sn, define its descent set Des(π) = {1 ≤ i ≤ n − 1|π(i) > π(i + 1)} ⊂ [n − 1] and its peak set Peak(π) = {2 ≤ i ≤ n − 1|π(i − 1) < π(i) > π(i + 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' The peak set of a permutation is said to be peak-lacunar, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' it neither contains 1 nor contains two consecutive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content='2 Enriched P-partitions and q-deformed generating functions We recall the main definitions regarding weighted posets, enriched P-partitions and their q-deformed generating functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' The reader is referred to [1, 3, 4, 7, 8] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' Definition 1 (Labelled weighted poset, [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfdvew/content/2301.00309v1.pdf'} +page_content=' A labelled weighted poset is a triple P = ([n], 0, and the electron filling number ne. The summary +of our results and conclusions are presented in Sec. IV. +II. +THEORETICAL FORMALISM +Let us consider a 1D chain of atoms with the s-wave +pairing of electrons in the system and adopt the attrac- +tive Hubbard model within the tight-binding approxima- +tion. The related BdG equations can be written as [14– +18] +Eνuν(i) = +� +i′ +Hii′uν(i′) + ∆ivν(i) +(1) +Eνvν(i) = ∆∗ +i uν(i) − +� +i′ +H∗ +ii′vν(i′), +(2) +where ∆i is the superconducting order parameter (pair +potential) at the lattice site i; Hii′ is the single-particle +Hamiltonian; Eν, uν(i), and vν(i) are the quasiparticle +energy, electron- and hole-like wave functions, respec- +tively. In the absence of external fields, the single-particle +Hamiltonian can be written as [18] +Hii′ = − +� +δ +tδ(δi′,i−δ + δi′,i+δ) − µδii′, +(3) +arXiv:2301.12979v1 [cond-mat.supr-con] 30 Jan 2023 + +2 +where tδ is the hopping parameter, δ enumerates the +neighboring coupled atomic-like orbitals, µ is the chem- +ical potential, and δi,i′ is the Kronecker delta symbol. +The Hartree-Fock mean field interaction is ignored here +as its main effect is reduced to a shift of the chemical +potential, see e.g. Refs. 17 and 19. +The BdG equations are solved in the self-consistent +manner as µ and ∆i are dependent on the electron- and +hole-like wave functions [14, 15, 17, 18]. The chemical +potential is determined via the equation for the averaged +electron filling number (below referred to as the electron +density) +ne = 2 +N +� +ν,i +� +fν|uν(i)|2 + (1 − fν)|vν(i)|2� +, +(4) +where fν = f(Eν) is the Fermi-Dirac distribution. The +site-dependent pair potential ∆i is given by +∆i = g +� +ν +uν(i)v∗ +ν(i)[1 − 2fν], +(5) +where the summation is over the BdG pair states +uν(i)v∗ +ν(i) with the quasiparticle energies 0 < Eν ≤ +ℏωD [20–22], where ωD is the Debye frequency (for the +conventional phonon mediated superconductivity). Here +we notice that the superconductive Hubbard model is of- +ten used without the energy cutoff as the band width is +finite and so, the ultraviolet divergence does not appear. +Obviously, this does not distort results when the band +width is less than the Debye energy ℏωD. However, in +the opposite case one should include the ultraviolet cut- +off to keep the trace of the phonon characteristic energy +and recover the standard BCS results for the parabolic +band approximation. +To solve the BdG equations, we first choose initial val- +ues for ∆i and µ and insert them into Eq. (1). Second, +we derive the quasiparticle energies, electron- and hole- +like wave functions by diagonalizing the corresponding +BdG matrix. Third, the obtained solutions are plugged +in Eqs. (4) and (5) to get new ∆i and µ. +Then, the +procedure is repeated until the convergence is reached. +When solving the formalism, we take into account the +normalization condition +� +i +� +|uν(i)|2 + |vν(i)|2� += 1, +(6) +see e.g. Ref. 4. Notice that ∆i can be chosen real in the +absence of the magnetic field as the Hamiltonian of the +system is time-reversal symmetric. +In the present work we consider the electron densities +ne = 0.8-1.2. In this case the system is close to the half- +filling regime, which steadily guarantees the presence of +the surface enhancement of the critical temperature, as +shown in the previous work [15]. The Debye energy and +the Hubbard coupling strength are taken as free param- +eters. +To avoid unnecessary complications, we restrict +ourselves to the conventional nearest-neighbor approxi- +mation, i.e. +δ = 1 and tδ = t. +Below all the energy +1 +101 +201 +301 +0.0 +0.1 +0.2 +0.3 +0.00 +0.05 +0.10 +0.15 +0.0 +0.1 +0.2 +0.3 +0.4 +1 +101 +201 +301 +0.0 +0.1 +0.2 +0.3 +0.4 +0.00 +0.05 +0.10 +0.15 +0.20 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +1 +101 +201 +301 +0.0 +0.1 +0.2 +0.3 +0.4 +0.0 +0.1 +0.2 +0.3 +0.0 +0.2 +0.4 +0.6 +� i +(a) +� � D=1.5 +T = 1.0 Tcb +g = 2 +ne = 1 +N = 301 +i=1 +� i(T) +(d) +Tcb=0.0925 +Tcs=0.1425 +i=(N+1)/2 +� � D=1.8 +T = 1.0 Tcb +� i +(b) +i=1 +i=(N+1)/2 +� i(T) +(e) +0.13 +0.205 +� � D=2.1 +T = 1.0 Tcb +� i +Site i +(c) +i=(N+1)/2 +i=1 +0.255 +0.205 +� i(T) +T +(f) +FIG. 1. (a,b,c) The pair potential ∆i versus the site number +i, calculated at the bulk critical temperature. (d,e,f) The pair +potentials at the edge (surface) ∆1 and in the center of the +chain (bulk) ∆(N+1)/2 versus the temperature. The calcula- +tions are done at ℏωD = 1.5 (a),(d), 1.8 (b),(e), and 2.1 (c),(f) +for g = 2 and ne = 1, other parameters are discussed in the +text. +related quantities are calculated in units of the hopping +parameter t, i.e. we set t = 1. +Notice that Eq. (3) is written for the case of an infinite +chain. To consider the surface enhancement of supercon- +ductivity, we investigate a finite 1D chain with infinite +potential barriers at the sites i = 0 and i = N + 1. +The number of atoms contributing to the superconduct- +ing condensate is chosen as N = 301, which is sufficiently +large to avoid any quantum-size effects. For such a finite +1D chain one should keep in mind that the first term in +the parenthesis of the right-hand side of Eq. (3) is multi- +plied by 1−δi,0 whereas the second term is multiplied by +1−δi,N+1. In addition, we have the boundary conditions +uν(0) = uν(N + 1) = 0, vν(0) = vν(N + 1) = 0. +(7) +This, taken together with Eq. (5), results in ∆0 = +∆N+1 = 0. +III. +RESULTS AND DISCUSSIONS +A. +Surface superconductivity +Figures 1(a)-(c) show the order parameter ∆i cal- +culated at the bulk critical temperature T = Tcb for +ne = 1, g = 2, and the three values of the Debye en- +ergy ℏωD = 1.5 (a), 1.8 (b) and 2.1 (c). [We recall that + +3 +0% +20% +40% +60% +0 +1 +2 +3 +0.0 +0.1 +0.2 +0.3 +g = 2 +ne = 1 +N = 301 +(Tcs-Tcb)/Tcb + +(a) +(b) +Tcb +Tcb, Tcs +� � D +Tcs +FIG. 2. (a) The difference of Tcs and Tcb in units of Tcb as a +function of ℏωD. (b) Tcs and Tcb versus ℏωD. The microscopic +parameters are the same as in Fig. 1. +all the energy related quantities are given in units of the +hopping parameter t.] In these plots, ∆i vanishes in the +center of the chain (bulk) while it is finite near the edges +(surface). As is seen, the system exhibits the surface en- +hancement of superconductivity. The values of ∆1 = ∆N +are sensitive to the Debye energy. For ℏωD = 1.5 we have +∆1 = 0.29 whereas for ℏωD = 1.8 and ℏωD = 2.1 we ob- +tain ∆1 = 0.39 and ∆1 = 0.35. +For further details, Figs. 1(d)-(f) demonstrate ∆1 and +∆(N+1)/2 (bulk) as functions of the temperature T for +ℏωD = 1.5, 1.8, and 2.1, respectively. +The electron +density and the coupling strength are the same as in +Figs. 1(a)-(c). One sees that ∆1 and ∆(N+1)/2 approach +zero at different temperatures, which is in agreement with +the data shown in Fig. 1(a)-(c). Thus, in addition to the +bulk critical temperature Tcb, associated with the tem- +perature dependence of ∆(N+1)/2, there exists the surface +critical temperature Tcs, associated with the temperature +behavior of the edge order parameter ∆1. +The both critical temperatures Tcb and Tcs increase +with ℏωD: for ℏωD = 1.5, 1.8 and 2.1, we have Tcb = +0.0925, 0.13 and 0.205 and Tcs = 0.1425, 0.205 and 0.255, +respectively. However, Tcb and Tcs are not simply propor- +tional to ℏωD as in the conventional BCS model. This +is clearly seen from Fig. 2, where (Tcs − Tcb)/Tcb and +Tcs, Tcb are shown versus the Debye energy in panels (a) +and (b). +The calculations are done at g = 2 for the +half-filling case, similarly to Fig. 1. If Tcs and Tcb were +proportional to ℏωD, the relative difference between Tcs +and Tcb in Fig. 2(a) would be constant for any value of +the Debye energy. However, (Tcs − Tcb)/Tcb exhibits a +complex nonmonotonic dependence on the Debye energy +when ℏωD < 2 and becomes constant only when ℏωD ex- +ceeds 2. From Fig. 2(b) one can see that Tcb and Tcs are +almost linear in ℏωD only for ℏωD ≲ 0.4. In the region +0.4 < ℏωD ≤ 1.75 the trend becomes different: both Tcs +and Tcb start to rise with ℏωD much faster. Furthermore, +Tcs increases with ℏωD faster than Tcb, which leads to the +notable increase of the relative difference between Tcs and +Tcb, see Fig. 2(a). Then, near ℏωD = 2 both critical tem- +peratures approach their maximal values Tcs,max = 0.25 +and Tcb,max = 0.202. As a result, the relative difference +of the surface and bulk critical temperatures first reaches +its maximum of about 61% at ℏωD = 1.75 and then, +drops to the value (Tcs,max − Tcb,max)/Tcb,max = 23.4% +at ℏωD = 2. For larger values of the Debye energy the +relative difference of Tcs and Tcb remains 23.4%. +To get an insight into the results in Fig. 2, let us con- +sider the system at temperatures T ∼ Tcs. In this case +the order parameter is sufficiently small and the quasipar- +ticle energy approaches the absolute value of the single- +particle energy ξk (absorbing the chemical potential). For +the single-particle Hamiltonian given by Eq. (3) with the +nearest-neighbor hopping, one obtains [18] +ξk = −2cos(ka) − µ, +(8) +with a the distance between the neighboring sites of the +1D chain and k the crystal momentum. +For the half- +filling case µ = 0 and the modulus of the single-particle +energy spans the interval from 0 to 2 and so does the +quasiparticle energy at T ∼ Tcs. According to the selec- +tion rule of Eq. (5), only the BdG pair states correspond- +ing to the quasiparticle energies smaller than ℏωD should +be taken into consideration. Then, for relatively small +Debye energies, the order parameter includes the BdG +pairs states with 0 < Eν < ℏωD < 2. In this case the or- +der parameter and the both critical temperatures should +increase with the Debye energy because a larger num- +ber of the states is incorporated. This increase becomes +more pronounced when the Debye energy approaches 2 +and nearly degenerate BdG pair states associated with +the edges of the Brillouin zone come into play. +How- +ever, when the Debye energy exceeds the band width, +i.e. ℏωD > 2, a further increase of ℏωD does not pro- +duce any effect on the superconducting properties since +all possible pair states are already taken into account. +This is why Tcs and Tcb in Fig. 2 do not change with the +Debye energy for ℏωD > 2. We stress that this conclu- +sion is only related to the half-filling case with µ = 0. For +ne < 1 or ne > 1 the chemical potential deviates from +0, and the maximal energy of the contributing quasipar- +ticles becomes larger than 2, see our results discussed +below. +B. +Interference of the BdG pair states +It is explained in the previous subsection why Tcs and +Tcb increase with the Debye energy for ℏωD ≤ 2 while + +4 +0.0 +0.2 +0.4 +0.6 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.1 +0.2 +0.3 +0.4 +(a) +bulk +1.5 +1.8 +2.0 +D +=2.0 +1.8 + + +D +s,b +(E) +T = 0.5 T +cb +1.5 +surface +D +s,b +(E) +(b) +bulk +D +=1.5 +2.0 + + +E +T = 1.0 T +cb +1.8 +surface +FIG. 3. +The cumulative pair potentials ∆(E) +s +≡ ∆(E) +1 +and +∆(E) +b +≡ ∆(E) +(N+1)/2 calculated for ℏωD = 1.5, 1.8 and 2. Panel +(a) corresponds to T = 0.5 Tcb, and panel (b) is for T = +1.0 Tcb. Other parameters are the same as in Figs. 1 and 2. +remaining the same for ℏωD ≳ 2. However, those argu- +ments cannot explain why we have the surface critical +temperature Tcs > Tcb. From the earlier work [15] we +know that the effect of the surface enhancement of super- +conductivity comes from the constructive interference of +the BdG pair states near the surface (edge) of the system. +Exactly this constructive interference results in the ap- +pearance of the surface critical temperature rather than +any superconducting pair mode localized near the edges +of the chain. This feature has been revealed in Ref. 15 for +ℏωD ≫ 2, when all the solutions of the BdG equations +contribute to the pair potential (5) and the analysis of +their contributions is not complicated by the application +of the selection rule for the quasiparticle energies. How- +ever, it follows from our present results that the surface +enhancement is much more pronounced for the Debye +energies in the interval from 1.5 to 2.0 which was not in- +vestigated in Ref. 15. To fill this gap, below we analyze +the contributions of the BdG pair states uν(i)v∗ +ν(i) to the +order parameter near the edges of the 1D chain and in +its center for ℏωD ≲ 2. +In particular, we follow the paper [15] and investigate +the quantity [for i = 1 and i = (N + 1)/2] +∆(E) +i += g +� +0 0.4 +while for ℏωD = 2.0 the trend changes above E = 0.5. +One can see that there are no pair states that contribute +to ∆(E) +s +but do not make any contribution to ∆(E) +b +at +T = 0.5Tcb. This analysis clearly demonstrates that the +surface amplification of the superconducting critical tem- +perature is a consequence of near-surface constructive in- +terference between the pair states spanning the entire +system volume, and not a correlation between electrons +in localized surface states. +We also cannot find any particular state which makes +a major contribution to ∆(E) +s +at T = Tcb. As is seen from +Fig. 3(b), all solutions of the BdG equations with Eν ≤ E +contribute to ∆(E) +s +and so, ∆1 is controlled by all pair +states with Eν ≤ ℏωD. Thus, we conclude that the con- +structive interference of the BdG pair modes is respon- +sible for a nonzero superconducting condensate near the +chain edges at the bulk critical temperature (and above +Tcb). +This finding is similar to the earlier results [15] +obtained for the Debye energies significantly larger than +the band width ℏωD ≫ 2. +Based on the interference scenario of the surface en- +hancement of superconductivity, the appearance of the +maximum of (Tcs − Tcb)/Tcb as a function of the De- +bye energy can be explained as follows. +At ℏωD = 0 +we have Tcs = Tcb = 0 and so, the relative difference +of the surface and bulk critical temperatures is equal to +zero. As the Debye frequency increases, more and more +pair states appear that contribute to the superconduct- +ing condensate. Obviously, the presence of a significant +number of participating pair states is necessary for a pro- +nounced constructive interference of such states. This is +why the interference effect gets stronger as ℏωD increases. +However, when the number of pair states contributing +to the gap function becomes very large, the interference +may suffer from an almost random summation of a large +number of different terms (similarly to the random phase +approximation). +This suggests that the surface effect +should be maximum at a certain value of ℏωD, which is +in agreement with our results for (Tcs − Tcb)/Tcb shown +in Fig.2(b). + +5 +0 +1 +2 +3 +0% +10% +20% +30% +40% +0 +1 +2 +3 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0 +1 +2 +3 +0% +20% +40% +60% +80% +0 +1 +2 +3 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0 +1 +2 +3 +0% +20% +40% +60% +0 +1 +2 +3 +0.0 +0.2 +0.4 +0.6 +g = 1.5 +ne = 1 +N = 301 +(Tcs-Tcb)/Tcb + +(a) +(d) +Tcb + +Tcb, Tcs +Tcs +g = 2.5 +(Tcs-Tcb)/Tcb + +(b) +(e) +Tcb + +Tcb, Tcs +Tcs +g = 3.5 +(Tcs-Tcb)/Tcb + +� ωD +(c) +(f) +Tcb + +Tcb, Tcs +� ωD +Tcs +FIG. 4. (Tcs − Tcb)/Tcb, Tcs, and Tcb as functions of ℏωD at +g = 1.5 (a,d), 2.5 (b,e) and 3.5 (c,f) for the half-filling case. +C. +Relative difference of Tcs and Tcb as a function +of microscopic parameters +Here we investigate how the surface enhancement of su- +perconductivity is sensitive to the coupling g and electron +density ne. In Fig. 4 one can find (Tcs − Tcb)/Tcb (a,b,c) +and Tcs, Tcb (d,e,f) as functions of the Debye energy cal- +culated for the half-filling case and the couplings g = +1.5 (a,d), g = 2.5 (b,e), and g = 3.5 (c,f). +From Fig. 4(a), we find that for g = 1.5 the maxi- +mal relative difference between Tcs and Tcb is about 38%, +which is by a factor of 1.4 larger than its value 27% for +ℏωD > 2. For small values of the Debye energy from 0 to +≈ 0.5 the quantity (Tcs − Tcb)/Tcb is zero or nearly zero +since Tcs and Tcb approach each other for ℏωD → 0, see +Fig. 4(b). The relative difference between the surface and +bulk critical temperatures starts to sharply increase with +ℏωD only when the Debye energy exceeds 1.5 and then, +the maximum of (Tcs−Tcb)/Tcb is reached at ℏωD = 1.93. +The results change significantly for larger couplings. In +particular, one can see from Fig. 4(b) that for g = 2.5, the +relative difference of the surface and bulk critical temper- +atures can increase up to 67%, which is much larger than +the maximal value of this quantity at g = 1.5 (38%). Fur- +thermore, 67% is about 4 times larger than the value of +(Tcs − Tcb)/Tcb for ℏωD > 2 at the same coupling (16%). +0 +1 +2 +3 +0% +10% +20% +30% +40% +50% +0 +1 +2 +3 +0.00 +0.09 +0.18 +0.27 +0 +1 +2 +3 +0% +20% +40% +60% +0 +1 +2 +3 +0.00 +0.09 +0.18 +0.27 +ne = 0.9 +g = 2 +N = 301 +(Tcs-Tcb)/Tcb + +(a) +(c) +Tcb +Tcb, Tcs +Tcs +� ωD +ne = 0.95 +(Tcs-Tcb)/Tcb +(b) +� ωD +(d) +Tcb +Tcb, Tcs +Tcs +FIG. 5. Beyond the half-filling: (Tcs − Tcb)/Tcb, Tcs, and Tcb +as functions of ℏωD at ne = 0.9 (a,c) and ne = 0.95 (b,d). +The coupling is chosen as g = 2. +In addition, here the relative difference of Tcs and Tcb +begins to rapidly increase with the Debye energy when +ℏωD crosses 0.1, which is much smaller than ℏωD = 1.5, +the onset of such an increase for g = 1.5. One can see +that there are two intervals where (Tcs − Tcb)/Tcb ex- +hibits a significant growth: from 0.1 to 0.3 and from 1.1 +to 1.6. For ℏωD = 0.3-1.1 we have a saturation of this +quantity near 43% whereas its maximal value is reached +at ℏωD = 1.6. +While the data for (Tcs−Tcb)/Tcb at g = 2.5 are signif- +icantly different from those of g = 1.5, the relative differ- +ence between Tcs and Tcb calculated at g = 3.5 and shown +in Fig. 4(c) is close to the result for this quantity given in +Fig. 4(b). The only minor difference is that the values of +(Tcs−Tcb)/Tcb in panel (c) are by about 6% smaller than +those in panel (b) for large ℏωD. However, Tcs and Tcb +shown in Fig. 4(f) are more significantly different from +the critical temperatures given in Fig. 4(e). +Although +the qualitative picture of the Debye-energy dependence +of Tcs and Tcb is the same in both panels, Tcs,max for +g = 3.5 is larger by about 30% than Tcs,max for g = 2.5. +A similar result is obtained for Tcb. +Thus, we find that the maximal value of the rela- +tive difference between Tcs and Tcb at ne = 1 increases +with g at small couplings, then approaches almost 70% +at g ≈ 2.5, and slowly decreases with a further in- +crease in g. Furthermore, one sees in Figs. 3(a)-(c) that +(Tcs − Tcb)/Tcb is above 40% in a wide range of the mi- +croscopic parameters 0.4 < ℏωD < 1.9 and 2.0 ≲ g ≲ 3.5. +Finally, we go beyond the half-filling regime and inves- +tigate how Tcs, Tcb and their relative difference depend +on ℏωD at the densities ne = 0.9 and 0.95. Due to the +symmetry of the Hubbard model, the results for ne = 0.9 +and 0.95 are the same as for ne = 1.1 and 1.05, respec- + +6 +tively. +Figures 5(a,c) and (b,d) demonstrate (Tcs − Tcb)/Tcb, +Tcs, and Tcb as functions of ℏωD calculated for g = 2 at +ne = 0.9 (a,c) and ne = 0.95 (b,d). One sees from Fig. 5 +that for ne = 0.9 and ne = 0.95 the maximal relative +difference between Tcs and Tcb is about 45.0% and 54%, +respectively, which should be compared with the maximal +relative difference (60%) in Fig. 2(a) for the same cou- +pling g = 2. The locus of the maximum of (Tcs−Tcb)/Tcb +is at ℏωD = 1.60 for the both densities. When the density +ne is shifted further to 0.85 and 0.8, the surface enhance- +ment of superconductivity continues to slightly weaken so +that the maximal relative difference between Tcs and Tcb +approaches 38.0% and 29.0%, respectively. These results +are in agreement with the conclusions of Ref. 15 that the +interference surface effect is most pronounced in the half- +filling regime. However, the decrease of (Tcs − Tcb)/Tcb +calculated at ne < 1(> 1) with respect to its value at +ne = 1 is moderate. For example, in the density interval +from 0.8 to 1.2, we obtain for g = 2 that the maximal +relative enhancement of the surface critical temperature +is above 29%. +Notice that this is still larger than the +surface enhancement obtained for the half-filling regime +at ℏωD > 2 in Refs. 14 and 15. +It is seen from Figs. 5(c) and (d) that Tcs and Tcb +for ne = 0.9 are nearly the same as the critical tem- +peratures calculated for ne = 0.95. Qualitatively, their +Debye-energy dependence is similar to that demonstrated +in Fig. 4 for the half-filling case. +However, there is a +new feature to discuss: Tcb and Tcs exhibit the pres- +ence of cusps situated at ℏωD = 1.7 for ne = 0.9 and +at ℏωD = 1.8 for ne = 0.95 (for both Tcb and Tcs). The +reason for the formation of these cusps is the following. +At sufficiently large temperatures we can assume that +the quasiparticle energy approaches the modulus of the +single-particle energy given by Eq. (8). As ne < 1, the +chemical potential µ is not any more in the center of the +band but shifts down, i.e. µ < 0. We can distinguish the +two branches with ξk > 0 and ξk ≤ 0. When the De- +bye energy is smaller than |ξk=0| = 2 + µ and increases, +new contributing BdG states are supplied by the both +branches. +However, when ℏωD exceeds |ξk=0|, the in- +crease of Tcs and Tsc occurs only due to the BdG states +with ξk > 0 and as a result, the cusps in the Debye-energy +dependence of Tcs and Tcb appear. For ne = 0.9 they ap- +pear at ℏωD = 1.7 since |ξk=0| = 1.7 and µ = −0.3. In +turn, for ne = 0.95 one gets µ = −0.2, and the cusps +are situated at ℏωD = 1.8. For the half-filling case their +locus approaches ℏωD = 2. +Obviously, the maximal values of Tcb and Tcs are +reached when the Debye energy exceeds the value +|ξk=π/a| = 2 − µ for µ ≤ 0. +For the half-filling case +µ = 0 and |ξk=π/a| = 2. +Then, the both critical +temperatures approach their maxima at ℏωD = 2, see +Figs. 4(d,e) and (f). For ne = 0.95 we have µ = −0.2 and +|ξk=π/a| = 2.2 while for ne = 0.9 one obtains µ = −0.3 +and |ξk=π/a| = 2.3. +Therefore, the relative difference +of the surface and bulk critical temperatures does not +change when ℏωD becomes larger than 2.2 and 2.3, re- +spectively. +IV. +CONCLUSIONS AND DISCUSSIONS +In summary, we find that tuning the Debye frequency +has a significant impact on the surface enhancement of +superconductivity in the attractive Hubbard model with +the nearest-neighbor hopping (for the phonon-mediated +superconductivity). In particular, our study reveals that +(Tcs − Tcb)/Tcb can increase up to nearly 60-70% for the +Debye energies in the interval 1.6-1.8 (in units of the +hopping parameter). This is significantly larger than 20- +25% reported previously for the same model with ℏωD ≥ +2 [14, 15]. +We demonstrate that a pronounced surface enhance- +ment of superconductivity persists over a wide range of +the microscopic parameters. Indeed, the effect is not very +sensitive to a particular value of the coupling constant in +the interval 2-3.5 where the maximum of (Tcs − Tcb)/Tcb +is about 60-70% for the half-filling case. When the sys- +tem deviates from the half-filling regime, the maximum +of (Tcs − Tcb)/Tcb decreases in agreement with findings +in Ref. 15. However, it remains significant. For exam- +ple, at g = 2 the maximal value of the relative difference +between Tcs and Tcb is still above 29% for the electron +densities from 0.8 to 1.2. +It is important to stress that the obtained results for +the surface superconductivity cannot be explained by an +increase of the local electron density and the normal lo- +cal DOS (LDOS) near the sample boundaries. To go in +a more detail on this point, Fig. 6 demonstrates the site- +dependent electron density and normal LDOS together +with the order parameter near the left edge of the chain. +From Fig. 6(a) one can see that the surface effect in ques- +tion is not related to a rise of the local electron density +near the sample edges since the density is uniform. In- +deed, the Friedel oscillations, present in the local density +near the chain edges beyond the half-filling regime, are +weakened and washed out when ne approaches 1, as is +seen from Fig. 6(a). We find that the local density of +electrons is constant when the surface effect is most pro- +nounced. +The Friedel oscillations in the normal (g = 0) LDOS +near the sample edges are present even in the half-filling +case, as is seen from Fig. 6(b) and (c). However, there +is no any overall enhancement of the normal LDOS near +the sample boundaries. Moreover, one can see that the +Friedel oscillations are significant only in the domain with +i < 21. They are completely washed out for i > 25. How- +ever, the surface-enhanced order parameter is not zero +even for i = 51 while the bulk order parameter is al- +ready zero in this case (we recall that here T = 1.1Tcb). +Thus, one can conclude that the Friedel oscillations of the +LDOS and electron density near the chain edges cannot +explain the surface enhancement of the superconducting +condensate. This confirms our conclusion that the sur- + +7 +0.6 +0.8 +1.0 +1.2 +g = 2 +(a) +electron distribution +0.0 +0.1 +0.2 +0.3 + ∆i +∆i +electron distribution +1 +11 +21 +31 +41 +51 +0.0 +0.5 +1.0 +1.5 +V (t/e) +g = 0 +LDOS +(c) + ne = 1 + N = 301 +�� ωD = 1.5 +T = 1.1 Tcb +Site i +0.02 +0.9 +0.0 +0.2 +0.4 +0.6 +zero-bias LDOS (a.u.) +g = 0 +(b) +FIG. 6. +(a) The site-dependent order parameter and local +electron density at T = 1.1Tcb, ne = 1, ℏωD = 1.5, and g = 2; +(b) and (c) demonstrate the zero-bias and energy dependent +normal LDOS (g = 0) for the same T and ne as in panel (a). +face enhancement of superconductivity found in the at- +tractive Hubbard model for the Debye energies less than +the band width is a result of the constructive interference +of the pair states. This is similar to the results obtained +previously [15] for the same model with ℏωD ≫ 2. +Here the question may arise whether the mean field +results obtained in the present study are reliable since +1D systems suffer from strong superconducting fluctua- +tions [23–27]. The point is that the interference effects +are not sensitive to the system dimensionality and the +surface enhancement of superconductivity occurs also in +2D and 3D systems (see e.g. Ref. 14), where the fluc- +tuations are much less important. This is why we can +expect that our conclusions obtained for the 1D chain +(with a relatively simple formalism) are general and hold +for higher dimensions. For example, the surface super- +conductivity impact in a 3D superconductor occupying +the half-space (say, for x > 0) can be estimated by in- +troducing an additional factor in Eq. (5) that accounts +for the states in the y and z directions (considering that +these states are the plain waves). This changes the total +DOS at the Fermi level but does not alter the construc- +tive interference of the BdG pair states. +In addition, since the interference of pair states can +be influenced by the boundary conditions at the chain +edges, it is necessary to say a few words about their pos- +sible effects in the context of the stability of the surface +enhancement of superconductivity. The study performed +in Ref. 15 has demonstrated that the surface supercon- +ductivity is more sensitive to impurities than the bulk +one. This is the reflection of the fact that the interfer- +ence of the pair states is the origin of the surface en- +hancement. +However, the effect survives at moderate +surface disorder (roughness) unless the surface impurity +potential becomes of the order of the hopping parame- +ter. Further investigations of the boundary effects, in- +cluding more sophisticated variants of the confinement +potential at the sample boundaries, would be a signifi- +cant deviation from the goals of the present study. Our +consideration of the infinite potential walls at the chain +edges (open boundary conditions) is dictated by the fact +that the recent results for the interference-induced sur- +face superconductivity were obtained for infinite confine- +ment barriers [14, 15]. Thus, our choice makes it possible +to avoid any effects of a more elaborated finite potential +when comparing our results with the earlier calculations. +As it follows from the present investigation, controlling +the Debye frequency can be important to increase the su- +perconducting surface temperature effect for the phonon- +mediated superconductors. +The Debye frequency de- +pends on the phonon group velocity. There are several +ways of controlling/tuning the phonon dispersion rela- +tion (phonon engineering [28]) and hence the phonon +group velocity. For example, by properly selecting the +parameters of cladding materials and their thicknesses, +one can control the group velocity of phonons near the +sample surface [29, 30]. In addition, the frequency and +group velocity of acoustic phonons can decrease non- +monotonically with an increasing doping concentration, +revealing pronounced phonon softening effects governed +by the doping level [31]. The phonon hardening can be +reached by isotope substitutions like in H3S, where re- +placement of 32S atoms by the heavier isotopes 33S, 34S, +35S, and 36S produces a significant effect on the lattice +dynamics [32]. +Finally, the strain at the sample sur- +face/interface also affects the phonon structure and dis- +persion relation [7, 8, 33, 34] and so, it can be used to +manipulate the Debye frequency. Notice that the phonon +softening near surfaces can have a dual effect on the sur- +face superconductivity enhancement: firstly, by increas- +ing the electron-phonon coupling, and secondly, by in- +creasing Tcs as compared to Tcb due to changing the ratio +of the Debye energy to the energy band width. Thus, tak- +ing into account the present technological possibilities of +manipulating the Debye frequency in a controllable way, +our research suggests an innovative way of tailoring the +surface superconducting characteristics. +ACKNOWLEDGMENTS +This work was supported by Natural Science Founda- +tion of Zhejiang Province (Grants No. LY18A040002), +Science +Foundation +of +Zhejiang +Sci-Tech +Univer- +sity(ZSTU) (Grant No. 19062463-Y), Open Foundation +of Key Laboratory of Optical Field Manipulation of Zhe- +jiang Province(ZJOFM-2020-007). +The study has also + +8 +been funded within the framework of the HSE Univer- +sity Basic Research Program. +[1] D. Saint-James and P. 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K. +Nayak, Pressure-enabled phonon engineering in metals, +PNAS 111, 8712 (2014). + diff --git a/adFOT4oBgHgl3EQf_zRh/content/tmp_files/load_file.txt b/adFOT4oBgHgl3EQf_zRh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..28ac087a38f9591e513a03b0a892748667b555cd --- /dev/null +++ b/adFOT4oBgHgl3EQf_zRh/content/tmp_files/load_file.txt @@ -0,0 +1,847 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf,len=846 +page_content='Interference-induced surface superconductivity: enhancement by tuning the Debye energy Yunfei Bai,1 Yajiang Chen,1, ∗ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' Croitoru,2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' Shanenko,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='2 Xiaobing Luo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='1 and Yunbo Zhang1 1Key Laboratory of Optical Field Manipulation of Zhejiang Province,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' Zhejiang Sci-Tech University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' 310018 Zhejiang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' China 2HSE University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' 101000 Moscow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' Russia (Dated: January 31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' 2023) In the usual perception,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' surface superconductivity is associated with the surface nucleation of a superconducting condensate above the upper critical field in type-II superconductors or with a rearrangement of phonon properties and the electron-phonon coupling near surfaces/interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' Recently, it has been found that there is another example when the surface superconducting tem- perature is increased up to 20-25% as compared to the bulk one due to constructive interference of superconducting pair states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' In the present work, we demonstrate that in fact, such an interference- induced enhancement can be much more pronounced, up to nearly 70%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' Furthermore, here it is shown that such an interference enhancement persists over a wide range of microscopic parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' INTRODUCTION There are two well-known examples of the surface su- perconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' The first one concerns the surface nu- cleation of a pare condensate in type-II superconductors below the third critical field Hc3, when the applied ex- ternal magnetic field H is in the interval from Hc2 to Hc3, see the pioneering works [1–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' The second vari- ant is related to an enhancement (and also suppression) of superconductivity due to surface modifications of the phonon properties, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' the papers [6–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' However, there also exists the surface superconductiv- ity enhancement at the zero applied field and without any modifications in the phonon degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' For conventional superconductors, the investigations based on both the Ginzburg-Landau (GL) theory [10] and the microscopic Bogoliubov-de Gennes (BdG) equations [11– 13] have shown that the order parameter near the surface can be significantly larger than in bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' This does not necessarily lead to a notable increase of the supercon- ducting transition temperature near the surface Tcs as compared to its bulk value Tcb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' The corresponding rela- tive difference between the surface and bulk critical tem- peratures (Tcs − Tcb)/Tcb was reported to be negligible (≈ 10−3) [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' However, recently it was found within the BdG equations for the Hubbard attractive model with the nearest-neighbor hopping that the relative difference between Tcs and Tcb can increase up to 20-25%, and this increase was attributed to the formation of boundary pair states with elevated critical temperatures [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' Later it was shown [15, 16] that in fact, the enlargement of the surface critical temperature is caused by the constructive interference of the bulk pair states near the sample sur- face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' Such a constructive interference was found to be most pronounced when the conduction band is symmet- ric with respect to the Fermi level (the half-filling case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' ∗ yjchen@zstu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='cn In the present work we demonstrate that the interference-induced surface superconductivity can result in an even more significant increase of (Tcs−Tcb)/Tcb, up to ≈ 70%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' We find that the impact of the interference is notably enhanced by an appropriate tuning of the Debye energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' For illustration, we investigate a one-dimensional (1D) chain of atoms with the s-wave pairing of electrons within the tight-binding treatment of the attractive Hub- bard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' II we out- line the relevant BdG formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' Section III presents our numerical results for Tcs and Tcb in a wide range of microscopic parameters, such as the Debye energy ℏωD, the attractive coupling strength of the Hubbard model g > 0, and the electron filling number ne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' The summary of our results and conclusions are presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' THEORETICAL FORMALISM Let us consider a 1D chain of atoms with the s-wave pairing of electrons in the system and adopt the attrac- tive Hubbard model within the tight-binding approxima- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' The related BdG equations can be written as [14– 18] Eνuν(i) = � i′ Hii′uν(i′) + ∆ivν(i) (1) Eνvν(i) = ∆∗ i uν(i) − � i′ H∗ ii′vν(i′), (2) where ∆i is the superconducting order parameter (pair potential) at the lattice site i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' Hii′ is the single-particle Hamiltonian;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' Eν, uν(i), and vν(i) are the quasiparticle energy, electron- and hole-like wave functions, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' In the absence of external fields, the single-particle Hamiltonian can be written as [18] Hii′ = − � δ tδ(δi′,i−δ + δi′,i+δ) − µδii′, (3) arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='12979v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='supr-con] 30 Jan 2023 2 where tδ is the hopping parameter, δ enumerates the neighboring coupled atomic-like orbitals, µ is the chem- ical potential, and δi,i′ is the Kronecker delta symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' The Hartree-Fock mean field interaction is ignored here as its main effect is reduced to a shift of the chemical potential, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' 17 and 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' The BdG equations are solved in the self-consistent manner as µ and ∆i are dependent on the electron- and hole-like wave functions [14, 15, 17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' The chemical potential is determined via the equation for the averaged electron filling number (below referred to as the electron density) ne = 2 N � ν,i � fν|uν(i)|2 + (1 − fν)|vν(i)|2� , (4) where fν = f(Eν) is the Fermi-Dirac distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' The site-dependent pair potential ∆i is given by ∆i = g � ν uν(i)v∗ ν(i)[1 − 2fν], (5) where the summation is over the BdG pair states uν(i)v∗ ν(i) with the quasiparticle energies 0 < Eν ≤ ℏωD [20–22], where ωD is the Debye frequency (for the conventional phonon mediated superconductivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' Here we notice that the superconductive Hubbard model is of- ten used without the energy cutoff as the band width is finite and so, the ultraviolet divergence does not appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' Obviously, this does not distort results when the band width is less than the Debye energy ℏωD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' However, in the opposite case one should include the ultraviolet cut- off to keep the trace of the phonon characteristic energy and recover the standard BCS results for the parabolic band approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' To solve the BdG equations, we first choose initial val- ues for ∆i and µ and insert them into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' Second, we derive the quasiparticle energies, electron- and hole- like wave functions by diagonalizing the corresponding BdG matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' Third, the obtained solutions are plugged in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' (4) and (5) to get new ∆i and µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' Then, the procedure is repeated until the convergence is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' When solving the formalism, we take into account the normalization condition � i � |uν(i)|2 + |vν(i)|2� = 1, (6) see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' Notice that ∆i can be chosen real in the absence of the magnetic field as the Hamiltonian of the system is time-reversal symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' In the present work we consider the electron densities ne = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='8-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' In this case the system is close to the half- filling regime, which steadily guarantees the presence of the surface enhancement of the critical temperature, as shown in the previous work [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' The Debye energy and the Hubbard coupling strength are taken as free param- eters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' To avoid unnecessary complications, we restrict ourselves to the conventional nearest-neighbor approxi- mation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' δ = 1 and tδ = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' Below all the energy 1 101 201 301 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='4 1 101 201 301 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='5 1 101 201 301 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='6 � i (a) � � D=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='5 T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='0 Tcb g = 2 ne = 1 N = 301 i=1 � i(T) (d) Tcb=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='0925 Tcs=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='1425 i=(N+1)/2 � � D=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='8 T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='0 Tcb � i (b) i=1 i=(N+1)/2 � i(T) (e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='205 � � D=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='1 T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='0 Tcb � i Site i (c) i=(N+1)/2 i=1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='255 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='205 � i(T) T (f) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' (a,b,c) The pair potential ∆i versus the site number i, calculated at the bulk critical temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' (d,e,f) The pair potentials at the edge (surface) ∆1 and in the center of the chain (bulk) ∆(N+1)/2 versus the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' The calcula- tions are done at ℏωD = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='5 (a),(d), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='8 (b),(e), and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='1 (c),(f) for g = 2 and ne = 1, other parameters are discussed in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' related quantities are calculated in units of the hopping parameter t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' we set t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' Notice that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' (3) is written for the case of an infinite chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' To consider the surface enhancement of supercon- ductivity, we investigate a finite 1D chain with infinite potential barriers at the sites i = 0 and i = N + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' The number of atoms contributing to the superconduct- ing condensate is chosen as N = 301, which is sufficiently large to avoid any quantum-size effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' For such a finite 1D chain one should keep in mind that the first term in the parenthesis of the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' (3) is multi- plied by 1−δi,0 whereas the second term is multiplied by 1−δi,N+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' In addition, we have the boundary conditions uν(0) = uν(N + 1) = 0, vν(0) = vν(N + 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' (7) This, taken together with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' (5), results in ∆0 = ∆N+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' RESULTS AND DISCUSSIONS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' Surface superconductivity Figures 1(a)-(c) show the order parameter ∆i cal- culated at the bulk critical temperature T = Tcb for ne = 1, g = 2, and the three values of the Debye en- ergy ℏωD = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='5 (a), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='8 (b) and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='1 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' [We recall that 3 0% 20% 40% 60% 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='3 g = 2 ne = 1 N = 301 (Tcs-Tcb)/Tcb (a) (b) Tcb Tcb, Tcs � � D Tcs FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' (a) The difference of Tcs and Tcb in units of Tcb as a function of ℏωD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' (b) Tcs and Tcb versus ℏωD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' The microscopic parameters are the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' all the energy related quantities are given in units of the hopping parameter t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='] In these plots, ∆i vanishes in the center of the chain (bulk) while it is finite near the edges (surface).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' As is seen, the system exhibits the surface en- hancement of superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' The values of ∆1 = ∆N are sensitive to the Debye energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' For ℏωD = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='5 we have ∆1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='29 whereas for ℏωD = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='8 and ℏωD = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='1 we ob- tain ∆1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='39 and ∆1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' For further details, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' 1(d)-(f) demonstrate ∆1 and ∆(N+1)/2 (bulk) as functions of the temperature T for ℏωD = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='8, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' The electron density and the coupling strength are the same as in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' 1(a)-(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' One sees that ∆1 and ∆(N+1)/2 approach zero at different temperatures, which is in agreement with the data shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' 1(a)-(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' Thus, in addition to the bulk critical temperature Tcb, associated with the tem- perature dependence of ∆(N+1)/2, there exists the surface critical temperature Tcs, associated with the temperature behavior of the edge order parameter ∆1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' The both critical temperatures Tcb and Tcs increase with ℏωD: for ℏωD = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='8 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='1, we have Tcb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='0925, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='13 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='205 and Tcs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='1425, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='205 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='255, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' However, Tcb and Tcs are not simply propor- tional to ℏωD as in the conventional BCS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' This is clearly seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' 2, where (Tcs − Tcb)/Tcb and Tcs, Tcb are shown versus the Debye energy in panels (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' The calculations are done at g = 2 for the half-filling case, similarly to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' If Tcs and Tcb were proportional to ℏωD, the relative difference between Tcs and Tcb in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' 2(a) would be constant for any value of the Debye energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' However, (Tcs − Tcb)/Tcb exhibits a complex nonmonotonic dependence on the Debye energy when ℏωD < 2 and becomes constant only when ℏωD ex- ceeds 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' 2(b) one can see that Tcb and Tcs are almost linear in ℏωD only for ℏωD ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' In the region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='4 < ℏωD ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='75 the trend becomes different: both Tcs and Tcb start to rise with ℏωD much faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' Furthermore, Tcs increases with ℏωD faster than Tcb, which leads to the notable increase of the relative difference between Tcs and Tcb, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' Then, near ℏωD = 2 both critical tem- peratures approach their maximal values Tcs,max = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='25 and Tcb,max = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' As a result, the relative difference of the surface and bulk critical temperatures first reaches its maximum of about 61% at ℏωD = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='75 and then, drops to the value (Tcs,max − Tcb,max)/Tcb,max = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='4% at ℏωD = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' For larger values of the Debye energy the relative difference of Tcs and Tcb remains 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' To get an insight into the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' 2, let us con- sider the system at temperatures T ∼ Tcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' In this case the order parameter is sufficiently small and the quasipar- ticle energy approaches the absolute value of the single- particle energy ξk (absorbing the chemical potential).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' For the single-particle Hamiltonian given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' (3) with the nearest-neighbor hopping, one obtains [18] ξk = −2cos(ka) − µ, (8) with a the distance between the neighboring sites of the 1D chain and k the crystal momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' For the half- filling case µ = 0 and the modulus of the single-particle energy spans the interval from 0 to 2 and so does the quasiparticle energy at T ∼ Tcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' According to the selec- tion rule of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' (5), only the BdG pair states correspond- ing to the quasiparticle energies smaller than ℏωD should be taken into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' Then, for relatively small Debye energies, the order parameter includes the BdG pairs states with 0 < Eν < ℏωD < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' In this case the or- der parameter and the both critical temperatures should increase with the Debye energy because a larger num- ber of the states is incorporated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' This increase becomes more pronounced when the Debye energy approaches 2 and nearly degenerate BdG pair states associated with the edges of the Brillouin zone come into play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' How- ever, when the Debye energy exceeds the band width, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' ℏωD > 2, a further increase of ℏωD does not pro- duce any effect on the superconducting properties since all possible pair states are already taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' This is why Tcs and Tcb in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' 2 do not change with the Debye energy for ℏωD > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' We stress that this conclu- sion is only related to the half-filling case with µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' For ne < 1 or ne > 1 the chemical potential deviates from 0, and the maximal energy of the contributing quasipar- ticles becomes larger than 2, see our results discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' Interference of the BdG pair states It is explained in the previous subsection why Tcs and Tcb increase with the Debye energy for ℏωD ≤ 2 while 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='4 (a) bulk 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='0 D =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='8 D s,b (E) T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='5 T cb 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='5 surface D s,b (E) (b) bulk D =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='0 E T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='0 T cb 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='8 surface FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' The cumulative pair potentials ∆(E) s ≡ ∆(E) 1 and ∆(E) b ≡ ∆(E) (N+1)/2 calculated for ℏωD = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='8 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' Panel (a) corresponds to T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='5 Tcb, and panel (b) is for T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='0 Tcb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' Other parameters are the same as in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' remaining the same for ℏωD ≳ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' However, those argu- ments cannot explain why we have the surface critical temperature Tcs > Tcb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' From the earlier work [15] we know that the effect of the surface enhancement of super- conductivity comes from the constructive interference of the BdG pair states near the surface (edge) of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' Exactly this constructive interference results in the ap- pearance of the surface critical temperature rather than any superconducting pair mode localized near the edges of the chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' This feature has been revealed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' 15 for ℏωD ≫ 2, when all the solutions of the BdG equations contribute to the pair potential (5) and the analysis of their contributions is not complicated by the application of the selection rule for the quasiparticle energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' How- ever, it follows from our present results that the surface enhancement is much more pronounced for the Debye energies in the interval from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='5 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content='0 which was not in- vestigated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' To fill this gap, below we analyze the contributions of the BdG pair states uν(i)v∗ ν(i) to the order parameter near the edges of the 1D chain and in its center for ℏωD ≲ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFOT4oBgHgl3EQf_zRh/content/2301.12979v1.pdf'} +page_content=' In particular, we follow the paper [15] and investigate the quantity [for i = 1 and i = (N + 1)/2] ∆(E) i = g � 0 +-1 +robot +robot +worker1 +-worker1 +.worker2 +-worker2 +2 +2 +3 +4 +5 +4 +5 +(w) x +(w) x +U +(w) +robot +robot +worker1 +-worker1 +worker2 +-worker2 +a +5 +2 +(w)x +X (m)7 +[5] Koide K, Miura J, Menegatti E. Monocular person tracking and identification with on-line deep feature +selection for person following robots. Robotics and Autonomous Systems. 2020 Feb 1;124:103348. +[6] Bochkovskiy A, Wang CY, Liao HY. Yolov4: Optimal speed and accuracy of object detection. arXiv +preprint arXiv:2004.10934. 2020 Apr 23. +[7] Wojke N, Bewley A, Paulus D. Simple online and realtime tracking with a deep association metric. +In2017 IEEE international conference on image processing (ICIP) 2017 Sep 17 (pp. 3645-3649). IEEE. +[8] Lee MC, Park MG. Artificial potential field based path planning for mobile robots using a virtual +obstacle concept. InProceedings 2003 IEEE/ASME international conference on advanced intelligent +mechatronics (AIM 2003) 2003 Jul 20 (Vol. 2, pp. 735-740). IEEE. +[9] Rostami SM, Sangaiah AK, Wang J, Liu X. Obstacle avoidance of mobile robots using modified +artificial potential field algorithm. EURASIP Journal on Wireless Communications and Networking. +2019 Dec;2019(1):1-9. +[10] Bochkovskiy A. Alexeyab/Darknet: Yolov4 / scaled-yolov4 / yolo - neural networks for object detection +(windows and linux version of darknet) [Internet]. GitHub. [cited 2022Oct28]. Available from: +https://github.com/AlexeyAB/darknet + diff --git a/eNE1T4oBgHgl3EQfLgOr/content/tmp_files/load_file.txt b/eNE1T4oBgHgl3EQfLgOr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1799e176939b8f23b6cc9b59443f2a415ce2e5cf --- /dev/null +++ b/eNE1T4oBgHgl3EQfLgOr/content/tmp_files/load_file.txt @@ -0,0 +1,172 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf,len=171 +page_content='1 Human Following Based on Visual Perception in the Context of Warehouse Logistics Yanbaihui Liu1，†, Haibo Wang2，*，† and Dongming Jia3，† 1 Department of Electrical & Computer Engineering, University of Michigan, Ann Arbor, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=" 2 Faculty of Electronic and Information Engineering, Xi'an Jiao Tong University, Xi’an, China 3 School of Mechanical and Aerospace Engineering, Ji Lin University, Changchun, China Corresponding author: intp0whb@stu." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content='xjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content='cn †These authors contributed equally to this work Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' Warehousing and logistics robots, which have benefited from the development of 5G, the internet, artificial intelligence, and robot technology, are commonly used to assist warehouse personnel in picking up or delivering heavy goods at dispersed locations along dynamic routes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' However, traditional programs that can only accept instructions or be preset by the system lack flexibility and existing human auto-following techniques either have difficulty accurately identifying specific targets or rely on a combination of lasers and cameras that are cumbersome and not effective at obstacle avoidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' This paper presents an algorithm that combines DeepSort and a width-based tracking module to track targets and uses artificial potential field local path planning to avoid obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' The algorithm is evaluated in a self-designed flat bounded test field and simulated in ROS, and is found to achieve state-of-the-art results in following and successfully reaching the end-point without hitting obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' Keywords: Computer vision;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' Object tracking;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' Obstacle avoidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' Introduction Currently, in industrial scenarios, most robots refer to the mechanical devices used in the warehousing link and can automatically carry out cargo transfer, handling, and other operations by receiving certain instructions or programs of the system but lack the flexibility and intelligence to meet specific application requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' For example, in a manufacturing factory scenario, warehouse personnel may need to frequently deliver or retrieve goods from various corners of the production line throughout the day, and pre-programmed handling robots cannot meet this demand for flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' Therefore, an automatic following robot that tracks warehouse personnel to deliver goods to a designated location and is also capable of real-time obstacle avoidance is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' To solve the problem of the automatic following of a person, how to track the target consistently is the fundamental issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' Many attempts have been made in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' A large group of them relied on a Laser-based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' Martinez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' used a 2D laser scanner to do object following and obstacle avoidance [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' Arras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' proposed a supervised classifier to detect targets by using 2D lidar data [2] However, due to the limitation of laser, solely relies on it are unable to identify a specific tracking target when multiple similar targets appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' Others either use stereo cameras or combine laser range finders with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' Satake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' utilize SIFT features and distance and then train an SVM-based person verifier to recognize the specific person [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' combined modified Online Ada-Boosting with depth information obtained from a stereo camera [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' To further improve the performance of vision-based methods, Koide et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' proposed a method to determine the target to be focused on using convolutional channel features based on the target predicted by the Unscented Kalman filter given a full body observed [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' Despite the high level of target tracking and tracing achieved by Koide et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' their method still does not take into account obstacle avoidance, which is a practical problem in real-life factories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' 2 To address the above problems, this paper proposes a solely RGB-D camera-based auto-following system, which enables the robot consistently follow the target person in the factory and avoid hitting any present obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' The people-tracking module can deal with multiple people tracking at a close distance, and present switch ID frequently by using DeepSort and a width-based tracking module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' The obstacle avoidance module handles the dynamic local path planning so that the robot can avoid stationary machines, non-targeted walking workers, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=', using the Artificial Potential Field (APF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' The depth information of the target will be obtained directly from the RGB-D camera instead of being approximated by triangulation techniques since the effect of sunlight on the infrared sensors is minimal in the indoor environment of the factory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' Experiments of human following and obstacle avoidance are performed in a flat, bounded test field and simulated in ROS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' The results show that our new system can stable track and follow the target person and reach the pre-specific location without hitting obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' Target detection and path tracking method based on visual perception 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content='1 Overview The warehousing environment typically consists of a large number of workers and goods shelves, which requires a robot system to continuously track a specific worker among multiple worker targets and avoid the goods shelves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' To accomplish this task, the system needs to be able to perform dynamic object detection and tracking, target following, and obstacle avoidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' The procedure is shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' 1 Overview of the system Specifically, we used the depth information of the camera and YOLOv4 to dynamically detect the target [6], then applied the DeepSort algorithm to stably track the moving target [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' The robot then turns its wheels based on the relative place of the bounding box of the whole image frame and does local path planning for obstacle avoidance by APF [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content='2 Object detection In order to improve the real-time performance of target tracking and give consideration to the appropriate tracking accuracy, a tracking algorithm combining YOLOv4 and DeepSort is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' This algorithm can not only track and perceive a single target, but also track and perceive multiple targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' You Only Look Once (YOLOv4) is a one-stage object detection model that is well-known for its exceptional computational speed and accuracy and uses CNN once on the entire image to split it into a grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' For each grid location, the CNN network generates a fixed number of different bounding boxes and computes a class probability for each of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' CNN then uses a threshold for the intersections over unions to eliminate bounding boxes that are likely to refer to the same object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' This structure of YOLO results in faster computation speed and accurate detection rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' This paper adopted a pre-trained YOLOv4 model on COCO to save time [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' After successfully detecting the target, DeepSort is performed to do dynamic tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' The recursive Kalman filtering algorithm is Robot People People Detection Tracking ObstacleAvoidance Robot Controller3 used to predict and track the state of the target candidate box, and then the pedestrian in the continuous multiple frames of the video is tracked and assigned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' Three different matching methods, namely, motion matching, appearance matching and cascade matching, are used to achieve more accurate matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' Finally, the continuous tracking of multiple pedestrian targets is achieved, and the ID value of each pedestrian target is obtained in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content='3 Path tracking To enable the robot to be always oriented toward the target, the target needs to be kept in the center of the camera frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' The center along the horizontal axis (or say orientation in the z-direction) of the bounding box created in the image captured by the camera is used as a reference to move the robot left or right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' A threshold range of 20 pixels is used to tolerate errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' The robot turns left whenever it detects any value lesser than the threshold and turns right when greater than the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' For local path planning with obstacle avoidance, we employ an Artificial Potential Field (APF) approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' The APF method involves constructing an artificial virtual potential field, which consists of two components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' One part is the gravitational field generated by the target point to the mobile robot, with the direction of the robot pointing to the target point, and the other part is the repulsive field generated by obstacles to the mobile robot, with the direction of the obstacle pointing to the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' The total potential field in the running space is the combined action of the repulsive force field and gravitational field, so the mobile robot can be controlled by the combined force of gravity and repulsive force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' Experiment and Analysis 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content='1 Experimental setup In the warehouse, there are many goods shelves, and workers walking around, which makes it difficult for robots to follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' However, the layout of the goods shelves in the warehouse is strictly arranged in a rectangular manner, and the roads in the warehouse are relatively open and flat, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' We test the multi-person tracking effect of our object detection algorithms with video as input in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' And we use a gazebo to simulate this environment, test two different scenarios and map the motion track of the robot and workers to MATLAB in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' 2 Warehouse 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content='2 Performance evaluation of dynamic target detection and tracking As for the actual effect of YOLOv4 and DeepSort tracking algorithms, Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content='3 tests the multi-person tracking effect with video as input, mainly to test whether the algorithm can continuously track each pedestrian target in continuous video frames and whether it can keep the ID consistent after workers block each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' 4 From frames 11 to 60 of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content='3, it can be seen that although the position of each worker target has changed, the ID of each worker target remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' This algorithm can continuously track multiple pedestrian targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' From frames 60 to 115, it can be seen that when workers temporarily occlude each other, the algorithm can accurately track the target, and the IDs of the worker targets do not change before and after occlusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' The algorithms can meet the visual perception requirements of mobile robots for fast and accurate tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' 3 Multi-person tracking effect 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content='3 Analysis of obstacle avoidance performance of right-angle turn Each area of the warehouse is basically arranged in a rectangular way, so there are many right-angle bends like Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' The following robot has certain deviation from the actual walking path of the target when turning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' It is necessary to investigate whether the following robot can move in a small turning radius without collision or target loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' In this experiment, the typical layout gap between goods shelves is used as the running path, and the target is transferred to the channel between two shelves to investigate whether the following robot can successfully follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content='5 is the motion diagram of this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' The dotted line in the figure is the walking route of the human body target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' The target moves at normal walking speed and turns into an aisle with a small turning radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' It can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content='5 that the following robot can successfully follow the target into the equipment aisle even in the case of a small turning radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' Frame 11 Frame40 Frame60 Frame75 Frame 84 Frame 1155 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' 4 Layout Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' 5 Obstacle avoidance performance of right-angle turn 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content='4 Dynamic obstacle avoidance performance analysis In addition to avoiding static good shelves, workers walking in the workshop should also be considered when the robot is following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content='6 simulates that when a robot follows a worker in the corridor, another worker suddenly walks toward the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' It can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content='6 that the following robot can successfully follow the target when another worker suddenly walks towards the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' goodsshelf1 goodsshelf2 robot worker5 4 3 (w) 2 robot worker 0 4 3 2 1 0 2 3 4 (w) x6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' 6 Dynamic obstacle avoidance performance 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' Conclusion We successfully accomplish object detection and path tracking based on visual perception in the context of warehousing logistics The system can continue following even if multiple targets temporarily occlude each other, follow the target as it turns at right-angles, and perform well in dynamic obstacle avoidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' If time permits, we will add human pose prediction, which enables the robot to predict the direction of the target in advance and reduce the reaction time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' References [1] Martinez JL, Pozo-Ruz A, Pedraza S, Fernandez R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfLgOr/content/2301.02978v1.pdf'} +page_content=' Object following and obstacle avoidance using a laser scanner in the outdoor mobile robot Auriga-alpha.' metadata={'source': 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b/edAzT4oBgHgl3EQfaPyX/content/tmp_files/2301.01366v1.pdf.txt @@ -0,0 +1,1233 @@ +arXiv:2301.01366v1 [gr-qc] 3 Jan 2023 +Tunneling analysis of null aether black hole theory in the background of +Newman-Janis algorithm +Riasat Ali,1, ∗ Rimsha Babar,2, † Muhammad Asgher,3, ‡ and G. Mustafa4, § +1Department of Mathematics, GC, University Faisalabad Layyah Campus, Layyah-31200, Pakistan +2Division of Science and Technology, University of Education, Township, Lahore-54590, Pakistan +3Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur-63100, Pakistan +4Department of Mathematics, Shanghai University, +Shanghai-200444, Shanghai, People’s Republic of China +We present a new asymptotically flat black hole solution in null aether theory (NAT) by applying +Newman-Janis process. For this purpose, we study the asymptotically flat NAT black hole solution +in Newman-Janis algorithm and then compute the tunneling radiation for NAT black hole. The +Hawking temperature for NAT black hole depends upon the rotation parameter and charge of +the black hole. +The Hawking temperature describes a black hole with extremal event horizon. +Furthermore, we analyze the graphical interpretation of Hawking temperature w.r.t event horizon +and check the stability of black hole under the influence of different parameters associated with +black hole temperature. +keywords: +Null Aether Black Hole Theory; +Newman-Janis algorithm; +Quantum gravity; +Lagrangian field equation; Hamilton-Jacobi phenomenon; Hawking temperature. +I. +INTRODUCTION +According to general quantum theory of gravity Lorentz symmetry won’t hold precisely in nature. Lately, this +idea has propelled a lot of intentions in Lorentz breaking theories of gravity. Among these frameworks are specific +concepts of vector-tensor speculations with favored direction settled at each point of space-time via vector field with +fixed-norm. Therefore, vector-tensor gravity speculations are of physical significance nowadays since they may reveal +some aspects of the inside framework of quantum theory of gravity. Einstein-Aether theory [1] is one of the main +theory in which the Aether field is supposed to be time-like and thus rests the boost segment of the Lorentz symmetry. +The concept of Aether theory has been explored throughout the years from different aspects [2]. There additionally +showed up some related works [3]-[5] which talk about the plausibility of a space-like Aether field that breaks the +rotational invariance. The dynamics and inner formalism of these theories are yet under consideration, for instance, +the stability issue of the aether field has been studied [6], obviously, to acquire a clear perceptive in this regard, +one additionally desires some specific analytic solutions for the genuinely complex equations of motion which these +speculations holds. Null Aether Theory (NAT) is the new vector-tensor speculations of modified theory of gravity [7]. +According to NAT, the dynamical vector field acts like the Aether and the BH solution via charge is conceivable [8] +as well as the physical properties (ADM mass, thermodynamics, singularity) of the NAT BH have been investigated. +Additionally, NAT charge is capable to decrease the thermodynamics of horizon as that of the Reissner Nordstr¨om- +AdS BH and generalize the circular orbits of massive as well as massless particles around the BH. Xu and Wang +analyzed [9] the quintessence field around the Kerr-Newman-AdS BH solution by using the Newman-Janis method +and complex calculations. +A typical element of different quantum gravity speculations, like loop quantum gravity, string theory and geometry +of non-commutative, is the presence of a minimum observable length [10, 11]. The generalized uncertainty principle +(GUP) is a basic approach to understanding this minimal observable length. The generalized commutation relation +can be defined as [12] +[x, p] = iℏ +� +1 + ρp2� +, +(1) +here ρ = +1 +3M2 +f represents the correction parameter and Mf is the Plank’s mass. Moreover, the generalized uncertainty +∗Electronic address: riasatyasin@gmail.com +†Electronic address: rimsha.babar10@gmail.com +‡Electronic address: m.asgher145@gmail.com +§Electronic address: gmustafa3828@gmail.com + +2 +relation is given as +∆x∆p ≥ ℏ +2 +� +1 + ρp2� +, +(2) +where x and p stands for position and momentum operators, respectively. The GUP effect on the tunneling radiation +of the higher dimensional BHs in the context of the boson phenomenon of the spin-1 particle have been studied in +[13, 14]. The Kerr Newman-NUT-Kiselev solution of BH by applying Newman-Janis approach to the dyonically as well +as electrically charged BH encompassed by quintessence has been examined [15]. The thermodynamical properties of +BH (Temperature, heat capacity, angular momentum and entropy) have also been derived. The Hawking temperature +phenomenon for the different spin particles has been widely analyzed in literature [16]-[29]. Moreover, it has been +studied that the Hawking temperature for different spin of particles remain preserved. +The aim of our paper is to study the NAT BH solution in the context of Newman-Janis algorithm and to investigate +the NAT BH Hawking temperature (TH) under the effects of rotation parameter and to describe a comparison of our +new results with previous literature. Furthermore, to derive the quantum corrected temperature T ′ +H for NAT BH with +rotation parameter accompanying GUP effects and to analyze the stable condition of BH in the presence of quantum +gravity effects. +This article is formatted in the following manner: Section II, contains a brief introduction about the metric of +asymptotically flat NAT BH. Section III investigate the Hawking temperature of BH under the influence of Newman- +Janis algorithm. Both Sec. IV and VI, present the graphical analysis of Hawking temperature w.r.t event horizon. +Section V study the temperature of NAT BH under the influence of quantum gravity and rotation parameter. Section +VII, comprised the summary and discussion of all the results. +II. +ASYMPTOTICALLY FLAT BH IN NULL AETHER THEORY +The spacetime for asymptotically flat BH in NAT can be defined as [30] +ds2 = −E(r)dt2 + +1 +E(r)dr2 + r2dθ2 + r2sin2θdφ2, +(3) +where +E(r) = +� +1 − 2˜a2 +1˜b1 +r1+˜ +q − 2˜a2 +2˜b2 +r1−˜ +q − 2 ¯m +r +( when ˜q ̸= 0), +1 − 2m +r +(when ˜q = 0), +(4) +here E(r) shows the metric function, ˜a1, ˜a2,˜b1 and ˜b2 are (constant null vector denoting the Aether field) integration +constants treated as free parameters, also +˜b1 = 1 +8 [˜c3 + ˜c23˜q − 3˜c2] , +˜b2 = 1 +8 [˜c3 − ˜c23˜q − 3˜c2] , +˜q ≡ +� +9 + 8 ˜c1 +˜c23 +, +(5) +˜q gives the charge, ˜m & m denotes the mass parameter and ˜c1, ˜c2, ˜c3 represents the dimensionless constant parameters. +For the case ˜q = 0, we can observe that the metric converts into the usual asymptotically flat Schwarzschild BH. +Although, for the case ˜q ̸= 0, we get different asymptotically flat boundary conditions by considering the following +cases independently(by def ˜q > 0 [30]): +E(r)|r→∞ = 1 + + + +for 0 < ˜q < 1 +(if ˜a1 ̸= 0 and ˜a2 ̸= 0) or +� +if ˜a1 = 0 or ˜b1 = 0 +� +, +for 0 < ˜q +(if ˜a2 = 0 or ˜b2 = 0 +� +. +(6) +For ˜q = 0, one can obtain the ADM mass as +¯ +MADM = m +¯G , +(7) +where we have defined [32] +¯G = +G +1 − ˜c1˜b2 +1 +. +(8) + +3 +The effective value of Newtonian constant ¯G associated to the constant G can be evaluated through experiments +within the solar system [31]. Also, for the case ˜q ̸= 0, we get +¯ +MADM = 1 +¯G +� +˜m + ˜a2 +1˜b1 +r˜q (1 + ˜q) + ˜a2 +2˜b2 +r ˜ +−q (1 − ˜q) +� ��� +r→∞. +(9) +From the above equations, we can get the ADM mass for the NAT BH in the following form +¯ +MADM = ˜m +¯G +� +for 0 < ˜q < 1 +(if ˜a1 = 0 or ˜b1 = 0 +� +, +for 0 < ˜q +(if ˜a2 = 0 or b2 = 0) . +(10) +If we consider the condition ˜a2 = 0, then, the the Aether field φ(r) and E(r) becomes +E(r) = 1 − 2˜a2 +1˜b1 +r(1+˜q) − 2 ˜m +r , +(11) +φ(r) = +˜a1 +r(1+˜q)/2 . +(12) +We can get the event horizon r0 by considering E (r0) = 0 and the horizon area is A = 4πr2 +0 . By taking ˜a1 = +˜G ˜Qr(˜q−1)/2 +0 +the Eqs. (11) and (12) become +E(r) = 1 − 2 ˜G2 ˜Q2˜b1 +r2 +�r0 +r +�˜q−1 +− 2 ˜m +r , +(13) +φ(r) = +˜G ˜Q +r +�r0 +r +�(˜q−1)/2 +, +(14) +here ˜Q depicts the charge of NAT BH. After putting r = r0 in the above equations, we get +E (r0) = 1 − 2 ¯G2 ˜Q2˜b1 +r2 +0 +− 2 ˜m +r0 += 0, +(15) +φ (r0) = +¯G ˜Q +r0 +. +(16) +It is note worthy to mention here that the horizon condition in Eq. (15) is free from the parameter ˜q. Moreover, φ(r) +looks like the electric potential at r = r0. +After substituting ˜q = 1 in the metric (3), the E(r) and φ(r) get the form +E(r) = 1 − 2˜a2 +1˜b1 +r2 +− 2 ˜m +r , +φ(r) = +˜a1 +r1/2 . +(17) +III. +ASYMPTOTICALLY FLAT NAT BH IN NEWMAN-JANIS ALGORITHM +By applying the Newman-Janis algorithm [33, 35, 36], we generalize the asymptotically flat NAT BH solution. Now +we introduce a coordinate transformation from Boyer Lindquist (BL) coordinates (t, r, θ, φ) to Eddington Finkelstein +(EF) coordinates (u, r, θ, φ) +du = dt − +dr +E(r), +(18) +where u represents the null coordinate. According to new coordinates the Eq. (3) can be rewritten as +ds2 = −E(r)du2 + r2dθ2 − 2dudr + r2 sin2 θdφ2. +(19) +The non-zero components for the inverse metric (19) are defined as +gur = −1, +grr = E(r), +gθθ = 1 +r2 , +gφφ = +1 +r2 sin2 θ. + +4 +Moreover, the inverse metric with complex null tetrad Zx = (lx, nx, mx, ¯mx) can be written as +gxy = −lxny − lynx + mx ¯my + my ¯mx. +(20) +The corresponding components can be defined as +lx = δx +r , +nx = δx +u − 1 +2E(r)δx +r , +mx = +1 +√ +2r δx +θ + +i +√ +2r sin θδx +φ, +¯mx = +1 +√ +2r +δx +θ − +i +√ +2r sin2 θ +δx +φ. +These null tetrad have orthonormal relation and comply with the accompanying characterizing conditions, specifically +all the vectors satisfy the given relations +lxlx = nxnx += mxmx = ¯mx ¯mx = 0, +lxmx = lx ¯mx += nxmx = nx ¯mx = 0, +lxnx = mx ¯mx = 1, +By considering the Newman-Janis method, we enable the coordinates to get complex values, while for real lx and nx +we are able to consider the given transformation [? ], +u′ = u − ia cos θ, +r′ = r + ia cosθ, +(21) +here a represents the spin parameter (due to Newman-Janis algorithm). Furthermore, we consider the transformations +from E(r) → ˜E(r, a, θ) and σ2 = r2 + a2 cos2 θ, whereas the null tetrad transforms as vectors in the form +lx = δx +r , +ny = δx +u − 1 +2 +˜E(r)δx +r , +mx = +1 +√ +2r +� +δx +θ + +i +sin θδx +φ + ia sin θ(δx +u − δx +r ) +� +, +¯mx = +1 +√ +2r +� +δx +θ − +i +sin θδx +φ − ia sin θ(δx +u − δx +r ) +� +. +(22) +By using the Eq. (20) and (22), the gxy components of non-zero in the EF coordinate can be defined as +guu = a2 sin2 θ +σ2 +, +gur = gru = −1 − a2 sin2 θ +σ2 +, +grr = ˜E(r, θ) + a2 sin2 θ +σ2 +, +gθθ = 1 +σ2 , +gφφ = +1 +σ2 sin2 θ, +guφ = gφu = a +σ2 , +grφ = gφr = − a +σ2 . +Furthermore, the lower indices components of matrix in the EF coordinates can be given as +guu = − ˜E(r, θ), +gur = gru = −1, +grr = 0, +gθθ = σ2, +guφ = gφu = a sin2 θ, +gφφ = sin2 θ +� +σ2 + a2( ˜E(r, θ) − 2) sin2 θ +� +, +grφ = gφr = − a +σ2 , +(23) +where +˜E(r, θ) = r2E + a2cos2θ +σ2 +. +(24) +According to transformed tetrad the new line element can be written as +ds2 = − ˜E(r, θ)du2 + σ2dθ2 + 2a sin2 θdrdφ − 2a +� +1 − ˜E(r, θ) +� +sin2 θdudφ − 2dudr ++ sin2 θ +� +σ2 + a2 � +2 − ˜E(r, θ) +� +sin2 θ +� +dφ2. +(25) + +5 +Now we introduce the transformation of EF coordinates to BL coordinates as [34] +du = dt + Y (r)dr, +dφ = dφ + χ(r)dr, +(26) +where the function of Y (r) and χ(r) is to ignore the grφ and gtr components. However, Y (r) and χ(r) appears as +function of r and θ which can be defined as +Y (r) = − +r2 + a2 +(r2E + a2), +χ(r) = − +a +(r2E + a2). +(27) +The dependence of θ from EF to BL coordinates transformation reveals the fact that, we are dealing with modified +theory of gravity and non-vacuum surrounding [35]. Furthermore, we will exclude the dependency on r and θ in +the functions σ2 and ∆r. The asymptotically flat NAT BH with BL coordinates in the context of Newman-Janis +algorithm can be obtained as +ds2 = − +�∆r − a2 sin2 θ +σ2 +� +dt2 + σ2 +∆r +dr2 − 2a +� +1 + a2 sin2 θ − ∆r +σ2 +� +sin2 θdtdφ + σ2dθ2 ++ sin2 θ +� +σ2 + sin2 θ +� +2 − a2 ∆r − a2 sin2 θ +σ2 +�� +dφ2, +(28) +here +∆r = r2 − 2mr + a2 − +2˜a2 +1˜b1 +σ2(˜q−1)/2 − +2˜a2 +2˜b2 +σ2(−1−˜q)/2 . +(29) +Since, BH acts like thermodynamical substance and whose temperature TH can be determined by considering the +surface gravity κ. So, we can compute the Hawking temperature of the metric (28) by using the following formula +[34] +TH = k +2π , +k = +∆′ +r +2(r2 ++ + a2), +(30) +where ∆′ +r = +d +dr(∆r). The corresponding Hawking temperature for NAT BH with Newman-Janis algorithm can be +derived as +TH = +� +r+ − m − ˜a2 +1˜b1r(1 − ˜q)(r2 ++ + a2)(−1−˜q)/2 − ˜a2 +2˜b2r(1 + ˜q)(r2 ++ + a2)(˜q−1)/2 +2π(r2 ++ + a2) +� +. +(31) +The TH for BH depends upon the BH mass m, charge ˜q, rotation parameter a and free parameters ˜a1, ˜a2,˜b1,˜b2. The +above temperature reduces into temperature of Schwarzschild BH for a = 0, ˜q = 0 which implies as [37] +TSBH = (r+ − m) +2πr2 ++ +, +where +r+ = 2m. +(32) +IV. +STABILITY ANALYSIS OF NAT BH +This section is comprised to investigate the graphical interpretation of temperature TH w.r.t event horizon (r+). +We evaluate the physical significance of the plots to analyze the effects of charge ˜q, mass m and rotation parameter +a of BH on temperature to study the BH stability. +Figure 1: depicts the presentation of TH via r+ for the fixed values of mass m = 1, rotation parameter a = 1, +free parameters ˜a1 = 0.1 = ˜b1, ˜a2 = 50,˜b2 = −10 in the range of charge 0.1 ≤ ˜q < 0.3. At first, the TH increases +and attains a maximum height and then it drops down gradually from a height and gets an asymptotically flat sate +by indicating the stability of BH as r+ → ∞. It can be observe that the temperature of BH increases with the +decreasing values of horizon. This physical behavior satisfies the Hawking’s phenomenon and guarantee the stability +of BH. For 0.1 ≤ ˜q ≤ 0.3, we observe an asymptotically flat behavior in temperature that exhibits the stable state of +BH. Figure 2: depicts the behavior of TH via r+ with fixed values of mass m = 1, charge ˜q = 0.1, free parameters +˜a1 = 0.1,˜b1 = 0.2, ˜a2 = 50,˜b2 = −10 and for varying values of rotation parameter a in the range 0 ≤ r+ ≤ 15. +There can be seen that an asymptotically flat behavior of temperature appears after attaining a maximum height for +different values of a . It can be seen that as we rises the value of a the temperature goes on decreasing as well as +for the increasing horizon the temperature decreases. This Hawking’s phenomenon depicts the BH stability in the +domain 0 ≤ r+ ≤ 15. +It has worth to mention here that for TH ≥ 0, the BH shows the physical behavior and it is in complete stable form. + +6 +q˜=0.1 +q˜=0.2 +q˜=0.3 +0 +2 +4 +6 +8 +10 +12 +14 +0 +500 +1000 +1500 +2000 +r+ +TH +m=1, a˜ +1=0.1, b +˜ +1=0.1, a˜ +2=50, b +˜ +2=-10, a=1 +FIG. 1: TH versus r+. +a=1.0 +a=1.2 +a=1.4 +0 +2 +4 +6 +8 +10 +12 +14 +0 +500 +1000 +1500 +r+ +TH +m=1, a˜ +1=0.1, b +˜ +1=0.2, a˜ +2=50, b +˜ +2=-10, q˜=0.1 +FIG. 2: TH versus r+. +V. +TEMPERATURE OF NAT BH UNDER THE INFLUENCE OF QUANTUM GRAVITY +In this chapter, we analyze the TH under the act upon of quantum gravity for boson spin-1 particles. We rewrite +the Eq. (28) in the adopting form +ds2 = −Fdt2 + Gdr2 + Hdθ2 + Kdφ2 + 2Ldtdφ, +(33) +where +F = ∆r − a2 sin2 θ +σ2 +, +G = σ2 +∆r +, +H = σ2, +K = sin2 θ +� +σ2 + +� +2 + a2 sin2 θ + ∆r +σ2 +� +a2 sin2 θ +� +L = −2a +� +1 + a2 sin2 θ + ∆r +σ2 +� +sin2 θ. +In order to evaluate the corrected TH of vector particles from the BHs. The vector particles such as Z and W are +well-known and act as very significance role in Standard Model [17]. We motion the charges bosonic tunneling in +the NAT BH should be more complicated than the Lagrangian field equation as the nontrivial solution interaction +during the charged bosonic field, the electromagnetic field and the Aether field. Firstly, we take the field equation of +charged particles from the Lagrangian field equation given by the GUP and also we use the Hamilton–Jacobi ansatz +phenomenon and WKB approximation to calculate the set of field equation in NAT space-time. By considering the +coefficient matrix determinant equal to zero and the linear equations can be derived for the radial function. Accord- +ingly, we compute the tunneling probability of the vector particles from the NAT BH and discuss the corresponding +temperature. Therefore, we utilize the generalized Lagrangian equation incorporating the GUP influenced by quantum +gravity. The Lagrangian field equation is given [26] by +∂µ(√−gχνµ) + √−g m2 +ℏ2 ϕν + √−g i +ℏAµϕνµ + √−g i +ℏeF νµϕµ + ̺ℏ2∂0∂0∂0(√−gg00ϕ0ν) +−̺ℏ2∂i∂i∂i(√−ggiiϕiν) = 0, +(34) +here determinant of g, ϕνµ and m represent coefficient matrix, anti-symmetric of tensor and particle of mass, since +ϕνµ = (1 − ̺ℏ2∂2 +ν)∂νϕµ − (1 − ̺ℏ2∂2 +µ)∂µϕν + (1 − ̺ℏ2∂2 +ν) i +ℏeAνϕµ − (1 − ̺ℏ2∂2 +ν) i +ℏeAµϕν, +Fνµ = ∇νAµ − ∇µAν, +where ̺, Aµ, ∇µ and e represent the GUP(quantum gravity) parameter, vector potential, covariant derivatives and +the charge of particle, respectively. The elements of non-zero for anti-symmetric tensor can be calculated as +ϕ0 = −Kϕ0 + Lχ3 +FK + L2 +, +ϕ1 = 1 +Gϕ1, +ϕ2 = 1 +H ϕ2, +ϕ3 = Lϕ0 + Fχ3 +FK + L2 , +ϕ12 = +1 +GH ϕ12, ϕ13 = 1 +GFK + L2ϕ13, +ϕ01 = −Kϕ01 + Lϕ13 +G(FK + L2) , +ϕ02 = +−Kϕ02 +H(FK + L2), +ϕ03 = (−FK + F 2)ϕ03 +(FK + L2)2 +, +ϕ23 = Lϕ02 + Fϕ23 +H(FK + L2), + +7 +The WKB approximation can be expressed as +ϕν = cν exp[ i +ℏQ0(t, r, φ, θ) + ΣℏnQn(t, r, φ, θ)]. +(35) +Using variables technique of separation, we can choose +Q0 = − ˜Et + W(r) + ν(φ) + Jθ, +(36) +where ˜E = E −Jω and E, J denote the particle energy and the angular particle momentum corresponding to θ angle. +After substituting Eq. (34) into set of the field equations, we get a matrix of order 4 × 4 +Y (c0, c1, c2, c3)T = 0, +(37) +whose elements are given as follows: +Y00 = +−K +G(FK + L2) +� +W 2 +1 + ̺W 4 +1 +� +− +K +H(FK + L2) +� +ν2 +1 + ̺ν4 +1 +� +, − +FK +(FK + L2)2 +� +J2 + ̺J4� +− +m2K +(FK + L2), +Y01 = +˜ +−K +G(FK + L2) +� +L + ̺ ˜E3 + eA0 + ̺eA0 ˜E2� +W1 + +E +G(FK + L2) + +� +ν1 + ̺ν3 +1 +� +, +Y02 = +−K +H(FK + L2) +� +˜E + ̺ ˜E3 − eA0 − ̺eA0 ˜E2� +J, +Y03 = +− ˜E +B(FK + L2) +� +W 2 +1 + ̺W 4 +1 +� +− +FK +H(FK + L2)2 +� +˜E + ̺ ˜E3 − eA0 − ̺eA0 ˜E2� +J + +m2L +(FK + L2)2 , +Y11 = +˜ +−K +G(FK + L2) +� +˜E2 + ̺ ˜E4 − eA0 ˜E − ̺eA0 ˜EW 2 +1 +� ++ +L +G(FK + L2) − m2 +G ++ +� +J + ̺J3� +˜E − +1 +GH +� +ν2 +1 + ̺ν4 +1 +� +− +1 +G(FK + L2) +� +J + ̺J3� ++ +eA0L +G(FK + L2) +� +J + ̺J3� +− +eA0K +G(FK + L2) +� +˜E + ̺ ˜E3 − eA0 − ̺eA0 ˜E2� +, +Y12 = +1 +GH [W1 + ̺W 3 +1 ]ν1, +Y13 = +˜ +−E +G(FK + L2) +� +W1 + ̺W 3 +1 +� +˜E + +1 +G(FK + L2)2 +� +W1 + ̺W 3 +1 +� +J + +LeA0 +G(FK + L2) +� +W1 + ̺W 3 +1 +� +, +Y22 = +K +H(FK + L2) +� +˜E2 + ̺ ˜E4 − eA0 ˜E − ̺eA0 ˜E +� +− +1 +GH − m2 +H +− +F +H(FK + L2) +� +ν2 +1 + ̺ν4 +1 +� +− +eA0K +H(FK + L2) +� +˜E + ̺ ˜E3 − eA0 − ̺eA0 ˜E2� ++ +L +H(FK + L2) +� +˜E + ̺ ˜E3 − eA0 − ̺eA0 ˜E2� +J, +Y23 = +F +G(FK + L2) +� +ν1 + ̺ν3 +1 +� +J +(38) +Y33 = (FK − ˜ +F 2) +(FK + L2) +� +˜E2 + ̺ ˜E4 − eA0 ˜E − ̺eA0 ˜E3� +− +1 +G(FK + L2) +� +W 2 +1 + ̺W 4 +1 +� +− +F +H(FK + L2) +� +ν2 +1 + ̺ν4 +1 +� +− +m2F +(FK + L2) − eA0(FK − ˜ +F 2) +(FK + L2) +� +˜E + ̺ ˜E3 − eA0 ˜E2� +, +where ν1 = ∂φQ0, W1 = ∂rQ0 and J = ∂θQ0. The determinant of Y is equal to zero for the non-trivial solution and +get +ImW ± = ± +� +� +� +� +�(E − Jω − A0e)2 + X1 +� +1 + ̺ X2 +X1 +� +(FK + L2)/GK +dr, += ±π ( ˜E − A0e) +2k(r+) +� +1 + ̺Ξ +� +, +(39) + +8 +where +X1 = +GL +(FK + L2) +� +˜E − eA0 +� +ν1 + +FG +(FK + L2)J2 − Gm2, +X2 = +GK +(FK + L2) +� +˜E4 − 2eA0 ˜E3 + (eA0)2 ˜E2� +− +FG +(FK + L2)J4 − W 4 +1 ++ +GL +H(FK + L2) +� +˜E3 − eA0 ˜E2� +J. +The bosonic particle tunneling can be expressed as +Γ = Γemission +Γabsorption += exp +� +−2π(E − Jω − A0e) +k(r+) +� � +1 + ̺Ξ +� +. +(40) +where +k = +∆′ +r +2(r2 ++ + a2). +(41) +The modified temperature can be calculated by applying the Boltzmann factor ΓB = exp [(E − Jω − A0e)/T ′ +H] as +T ′ +H = +� +r+ − m − ˜a2 +1˜b1r(1 − ˜q)(r2 ++ + a2)(−1−˜q)/2 − ˜a2 +2˜b2r(1 + ˜q)(r2 ++ + a2)(˜q−1)/2 +2π(r2 ++ + a2) +� � +1 − ̺Ξ +� +. +(42) +The Hawking temperature for BH depends upon the mass m, charge ˜q, quantum gravity parameter ̺, spin parameter +a, arbitrary parameter Ξ and free parameters ˜a1, ˜a2, ˜b1, ˜b2. The expression (42) reduces into BH temperature for +̺ = 0, which leads a temperature in Eq. (31). It has note worthy that the quantum corrections cause a deceleration +in the increment of temperature. +VI. +STABILITY ANALYSIS OF NAT BH WITH QUANTUM CORRECTIONS +This section depicts the graphical presentation of T ′ +H w.r.t event horizon (r+) with fixed value of arbitrary parameter +Ξ = 1. We study the physical existence of the plots and observe the effects of correction parameter ̺ and spin +parameter a of BH on corrected Hawking temperature to study the stable BH condition under the influence of +quantum effects. +Figure 3(i): describes the behavior of T ′ +H via event horizon for the fixed values of mass m = 1, spin parameter +a = 1, free parameters Ξ = 1, ˜a1 = 0.1 = ˜b1, ˜a2 = 50, ˜b2 = −10, charge ˜q = 0.1 and for varying values of correction +parameter ̺ . At a peak value the temperature attains a maximum height and then it drops down gradually and +obtain a condition of asymptotically flat by indicating the stability of BH as r+ → ∞. It can be observed that the +T ′ +H decreases as we increase the correction parameter values. The temperature of BH increases with the decreasing +values of event horizon. This physical presentation reflects the stability state of BH. The maximum temperature at +non-zero horizon left the BH remnant. +Figure 3(ii) represents the behavior of T ′ +H via r+ with fixed values of mass m = 1, correction parameter ̺ = 0.8, +charge ˜q = 0.1, free parameters ˜a1 = 1,˜b1 = 0.1, ˜a2 = 50,˜b2 = −10 and for varying values of a. There can be seen that +for different values of a the corrected temperature gets a height and then it shows an asymptotically flat behavior. It +is notable that when we increase the value of a the corrected temperature decreases as well as for the increasing value +of horizon the corrected temperature also decreases. This Hawking’s phenomenon represents the BH stable condition +in the domain 0 ≤ r+ ≤ 15. From both plots, we can observe that for T ′ +H ≥ 0, the BH gets its stable form while +for T ′ +H < 0 the BH with negative temperature always depicts its unstable form. We can also observe it graphically +that the T ′ +H is less than the original one. So, we can conclude the quantum corrections decelerates the increment in +temperature. + +9 +ρ=0.5 +ρ=0.6 +ρ=0.7 +0 +2 +4 +6 +8 +10 +12 +14 +0 +200 +400 +600 +800 +r+ +TH +(i) m=1, a˜ +1=0.1, b +˜ +1=0.1, a˜ +2=50, b +˜ +2=-10, a=1, q˜=0.1 +a=1.0 +a=1.2 +a=1.4 +0 +2 +4 +6 +8 +10 +12 +14 +0 +50 +100 +150 +200 +250 +300 +350 +r+ +TH +(ii) m=1, a˜ +1=1, b +˜ +1=0.1, a˜ +2=50, b +˜ +2=-10, q˜=0.1, +�=0.8 +Figure 3: T ′ +H versus r+ with Ξ = 1. +VII. +SUMMARY AND DISCUSSION +The theory of null Aether is a vector-tensor gravity theory with null vector and Aether field exist at every point +of the spacetime. In this paper, we have studied a new asymptotically flat BH solution by using Newman-Janis +algorithm. To do so, firstly, we have reviewed the asymptotically flat BH solution in NAT and then by applying +the Newman-Janis algorithm, we have derived a new asymptotically flat NAT BH spacetime influenced by rotation +parameter. By considering the spin parameter (a → 0) in Eq. (28), we get the asymptotically flat BH solution [30] +in general relativity. Furthermore, by taking into account the surface gravity κ, we have computed the temperature +for NAT BH in the presence of rotation parameter. The BH temperature depends upon the charge, mass, spin and +free parameters of the BH. The NAT BH temperature in Eq. (31) recovers the temperature of Schwarzschild BH for +˜q = 0 = a as in Eq. (32). Moreover, we have comprised the graphical representation of Hawking temperature w.r.t +event horizon in order to check the stability of BH. +We have studied the radiation spectrum through bosonic tunneling process of spin-1 particles from NAT BH +involving both spin and quantum gravity parameters. Therefore, we have utilized the generalized Lagrangian equation +incorporating the GUP influenced by quantum gravity. For this investigation, we have applied the Hamilton-Jacobi +ansatz and WKB approximation to the generalized Lagrangian field equation for boson particles. We have obtained +the bosonic corrected tunneling rate of emitted particles and their corresponding corrected temperature T ′ +H. It has +note worthy to analyzed that, when we ignore the quantum gravity effects, i.e., (ρ = 0), then the corrected Hawking +temperature in Eq. +(42) is reduced to the original temperature in Eq. +(31). +The corrected temperature of BH +depends upon spin parameter, quantum gravity parameter and Aether field. +The T ′ +H reduces into Schwarzschild +BH temperature when the spin parameter, quantum gravity parameter and Aether field approaches to zero. It has +been analyzed that the quantum gravity decelerates the increase in T ′ +H in the process of radiation. Moreover, we +have analyzed the physical significance of corrected temperature to check the effects of quantum gravity and rotation +parameter on T ′ +H by seeing the stability of NAT BH over Aether field. +The results from the plots of Hawking +temperature with respect to the horizon in the presence/absence of gravity parameter for the given BH are given as +follows: +• In the absence of gravity parameter the temperature shows the asymptotically flat behavior in the range of +charge 0.1 ≤ ˜q ≤ 0.3 and the TH decreases with the increasing r+. This is physical graphical presentation of +TH w.r.t r+ and depicts the stable condition of BH with positive temperature. +• The TH for varying values of rotation parameter a shows an asymptotically flat behavior and after a maximum +height the temperature goes on decreasing as well as for the increasing horizon. This Hawking’s phenomenon +depicts the BH stability in the domain 0 ≤ r+ ≤ 15. +• In the presence of gravity parameter T ′ +H decreases with the increasing values of correction parameter as well as +horizon. We have observed BH remnant at nonzero horizon with maximum temperature for different values ρ +in the domain 0 ≤ r+ ≤ 15. +• For different values of a the corrected temperature gets a height and then it shows an asymptotically flat +behavior. It is notable that the corrected temperature decreases with the increasing values of a as well as for +the increasing value of horizon. This Hawking’s phenomenon represents the BH stable condition in the domain +0 ≤ r+ ≤ 15. + +10 +• From all the plots, we have observed that for T ′ +H ≥ 0, the BH gets its stable form. We have also observed +it graphically that the T ′ +H is less than the original one. So, we have concluded that the quantum corrections +decelerates the increment in temperature. +VIII. +APPENDIX +After setting the all values in Eq. (34), we get the field equations set as +K +G(FK + L2) +� +c1(∂0Q0)(∂1Q0) + ̺c1(∂0Q0)3(∂1Q0) − c0(∂1Q0)2 − ̺c0(∂1Q0)4 + c1eA0(∂1Q0) ++c1̺eA0(∂0Q0)2(∂1Q0) +� +− +L +G(FK + L2) +� +c3(∂1Q0)2 + ̺c3(∂1Q0)4 − c1(∂1Q0)(∂3Q0) − ̺c1(∂1Q0)(∂3Q0)2� ++ +K +H(FK + L2) +� +c2(∂0Q0)(∂2Q0) + ̺c2(∂0Q0)3(∂2Q0) − c0(∂2Q0)2 − ̺c0(∂2Q0)4 + c2eA0(∂2Q0) + c2eA0̺ +(∂0Q0)2(∂1Q0) +� ++ +FK +(FK + L2)2 +� +c3(∂0Q0)(∂3Q0) + ̺c3(∂0Q0)3(∂3Q0) − c0(∂3Q0)2 − ̺c0(∂3Q0)4 + c3eA0 +(∂3Q0) + c3eA0(∂0Q0)2(∂3Q0) +� +− m2 Kc0 − Lc3 +(FK + L2) = 0, +(43) +−K +G(FK + L2) +� +c1(∂0Q0)2 + ̺c1(∂0Q0)4 − c0(∂0Q0)(∂1Q0) − ̺c0(∂0Q0)(∂1Q0)3 + c1eA0(∂0Q0) ++̺c1eA0(∂0Q0)3� ++ +L +G(FK + L2) +� +c3(∂0Q0)(∂1Q0) + ̺c3(∂0Q0)(∂1Q0)3 − c1(∂0Q0)(∂3Q0) − ̺c1(∂0Q0)(∂3Q0)3� ++ 1 +GH +� +c2(∂1Q0)(∂2Q0) + ̺c2(∂1Q0)(∂2Q0)3 − c1(∂2Q0)2 − ̺c1(∂2Q0)4� ++ +1 +G(FK + L2) +� +c3(∂1Q0)(∂3Q0) + ̺c3 +(∂1Q0)(∂3Q0)3 − c1(∂3Q0)2 − ̺c1(∂3Q0)4� ++ +eA0K +G(FK + L2) +� +c1(∂0Q0) + ̺c1(∂0Q0)3 − c0(∂1Q0) − ̺c0(∂1Q0)3 ++eA0c1 + ̺c1eA0(∂0Q0)2) +� ++ +eA0L +G(FK + L2) +� +c3(∂1Q0) + ̺c3(∂1Q0)3 − c1(∂3Q0) − ̺c1(∂1Q0)3� +− m2c1 +G += 0, (44) +K +H(FK + L2) +� +c2(∂0Q0)2 + ̺c2(∂0Q0)4 − c0(∂0Q0)(∂2Q0) − ̺c0(∂0Q0)(∂2Q0)3 + c2eA0(∂0Q0) + ̺c2eA0(∂0Q0)3� ++ 1 +GH +� +c2(∂1Q0)2 + ̺c2(∂1Q0)4 − c1(∂1Q0)(∂2Q0) − ̺c1(∂1Q0)(∂2Q0)3� +− +L +H(FK + L2) +� +c2(∂0Q0)(∂3Q0) ++̺c2(∂0Q0)3(∂3Q0) − c0(∂0Q0)(∂3Q0) − ̺c0(∂0Q0)3(∂3Q0) + c2eA0(∂3Q0) + ̺c2eA0(∂3Q0)3� ++ +F +H(FK + L2) +� +c3(∂2Q0)(∂3Q0) + ̺c3(∂2Q0)3(∂3Q0) − c2(∂3Q0)2 − ̺c2(∂3Q0)4� +− m2c2 +H ++ +eA0K +H(FK + L2) +� +c2(∂0Q0) + ̺c2(∂0Q0)3 − c0(∂2Q0) − ̺c0(∂2Q0)3 + c2eA0 + c2̺eA0(∂0Q0)2� += 0, +(45) +FK − F 2 +(FK + L2)2 +� +c3(∂0Q0)2 + ̺c3(∂0Q0)4 − c0(∂0Q0)(∂3Q0) − ̺c0(∂0Q0)(∂3Q0)3 + eA0c3(∂0Q0) ++̺c3eA0(∂0Q0)3� +− +K +H(FK + L2) +� +c3(∂1Q0)2 + ̺c3(∂1Q0)4 − c1(∂1Q0)(∂3Q0) − ̺c1(∂1Q0)(∂3Q0)3� +− +L +H(FK + L2) +� +c2(∂0Q0)(∂2Q0) + ̺c2(∂0Q0)3(∂2Q0) − c0(∂2Q0)2 + ̺c0(∂2Q0)4 + eA0c2(∂2Q0) + ̺c2eA0 +(∂0Q0)2(∂2Q0) +� +− +eA0F +H(FK + L2) +� +c3(∂2Q0)2 + ̺c3(∂2Q0)4 − c2(∂2Q0)(∂3Q0) − ̺c2(∂0Q0)(∂3Q0)3� ++eA0FD − F 2 +(FK + L2)2 +� +c3(∂0Q0) + ̺c3(∂0Q0)3 − c0(∂3Q0) − ̺c0(∂3Q0)3 + c3eA0 + ̺eA0(∂0Q0)2� +−m2Lc0 − Fc3 +(FK + L2) += 0. +(46) + +11 +[1] T. 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Lett. 112, 20005(2015). + diff --git a/edAzT4oBgHgl3EQfaPyX/content/tmp_files/load_file.txt b/edAzT4oBgHgl3EQfaPyX/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3131204b2b37c32ad6d2af3b671d570c0d80928c --- /dev/null +++ b/edAzT4oBgHgl3EQfaPyX/content/tmp_files/load_file.txt @@ -0,0 +1,667 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf,len=666 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='01366v1 [gr-qc] 3 Jan 2023 Tunneling analysis of null aether black hole theory in the background of Newman-Janis algorithm Riasat Ali,1, ∗ Rimsha Babar,2, † Muhammad Asgher,3, ‡ and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Mustafa4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' § 1Department of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' GC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' University Faisalabad Layyah Campus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Layyah-31200,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Pakistan 2Division of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' University of Education,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Township,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Lahore-54590,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Pakistan 3Department of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' The Islamia University of Bahawalpur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Bahawalpur-63100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Pakistan 4Department of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Shanghai University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Shanghai-200444,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Shanghai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' People’s Republic of China We present a new asymptotically flat black hole solution in null aether theory (NAT) by applying Newman-Janis process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' For this purpose, we study the asymptotically flat NAT black hole solution in Newman-Janis algorithm and then compute the tunneling radiation for NAT black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' The Hawking temperature for NAT black hole depends upon the rotation parameter and charge of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' The Hawking temperature describes a black hole with extremal event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Furthermore, we analyze the graphical interpretation of Hawking temperature w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='t event horizon and check the stability of black hole under the influence of different parameters associated with black hole temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' keywords: Null Aether Black Hole Theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Newman-Janis algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Quantum gravity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Lagrangian field equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Hamilton-Jacobi phenomenon;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Hawking temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' INTRODUCTION According to general quantum theory of gravity Lorentz symmetry won’t hold precisely in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Lately, this idea has propelled a lot of intentions in Lorentz breaking theories of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Among these frameworks are specific concepts of vector-tensor speculations with favored direction settled at each point of space-time via vector field with fixed-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Therefore, vector-tensor gravity speculations are of physical significance nowadays since they may reveal some aspects of the inside framework of quantum theory of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Einstein-Aether theory [1] is one of the main theory in which the Aether field is supposed to be time-like and thus rests the boost segment of the Lorentz symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' The concept of Aether theory has been explored throughout the years from different aspects [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' There additionally showed up some related works [3]-[5] which talk about the plausibility of a space-like Aether field that breaks the rotational invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' The dynamics and inner formalism of these theories are yet under consideration, for instance, the stability issue of the aether field has been studied [6], obviously, to acquire a clear perceptive in this regard, one additionally desires some specific analytic solutions for the genuinely complex equations of motion which these speculations holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Null Aether Theory (NAT) is the new vector-tensor speculations of modified theory of gravity [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' According to NAT, the dynamical vector field acts like the Aether and the BH solution via charge is conceivable [8] as well as the physical properties (ADM mass, thermodynamics, singularity) of the NAT BH have been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Additionally, NAT charge is capable to decrease the thermodynamics of horizon as that of the Reissner Nordstr¨om- AdS BH and generalize the circular orbits of massive as well as massless particles around the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Xu and Wang analyzed [9] the quintessence field around the Kerr-Newman-AdS BH solution by using the Newman-Janis method and complex calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' A typical element of different quantum gravity speculations, like loop quantum gravity, string theory and geometry of non-commutative, is the presence of a minimum observable length [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' The generalized uncertainty principle (GUP) is a basic approach to understanding this minimal observable length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' The generalized commutation relation can be defined as [12] [x, p] = iℏ � 1 + ρp2� , (1) here ρ = 1 3M2 f represents the correction parameter and Mf is the Plank’s mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Moreover, the generalized uncertainty ∗Electronic address: riasatyasin@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='com †Electronic address: rimsha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='babar10@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='com ‡Electronic address: m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='asgher145@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='com §Electronic address: gmustafa3828@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='com 2 relation is given as ∆x∆p ≥ ℏ 2 � 1 + ρp2� , (2) where x and p stands for position and momentum operators, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' The GUP effect on the tunneling radiation of the higher dimensional BHs in the context of the boson phenomenon of the spin-1 particle have been studied in [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' The Kerr Newman-NUT-Kiselev solution of BH by applying Newman-Janis approach to the dyonically as well as electrically charged BH encompassed by quintessence has been examined [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' The thermodynamical properties of BH (Temperature, heat capacity, angular momentum and entropy) have also been derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' The Hawking temperature phenomenon for the different spin particles has been widely analyzed in literature [16]-[29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Moreover, it has been studied that the Hawking temperature for different spin of particles remain preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' The aim of our paper is to study the NAT BH solution in the context of Newman-Janis algorithm and to investigate the NAT BH Hawking temperature (TH) under the effects of rotation parameter and to describe a comparison of our new results with previous literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Furthermore, to derive the quantum corrected temperature T ′ H for NAT BH with rotation parameter accompanying GUP effects and to analyze the stable condition of BH in the presence of quantum gravity effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' This article is formatted in the following manner: Section II, contains a brief introduction about the metric of asymptotically flat NAT BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Section III investigate the Hawking temperature of BH under the influence of Newman- Janis algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Both Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' IV and VI, present the graphical analysis of Hawking temperature w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='t event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Section V study the temperature of NAT BH under the influence of quantum gravity and rotation parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Section VII, comprised the summary and discussion of all the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' ASYMPTOTICALLY FLAT BH IN NULL AETHER THEORY The spacetime for asymptotically flat BH in NAT can be defined as [30] ds2 = −E(r)dt2 + 1 E(r)dr2 + r2dθ2 + r2sin2θdφ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (3) where E(r) = � 1 − 2˜a2 1˜b1 r1+˜ q − 2˜a2 2˜b2 r1−˜ q − 2 ¯m r ( when ˜q ̸= 0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' 1 − 2m r (when ˜q = 0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (4) here E(r) shows the metric function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' ˜a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' ˜a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='˜b1 and ˜b2 are (constant null vector denoting the Aether field) integration constants treated as free parameters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' also ˜b1 = 1 8 [˜c3 + ˜c23˜q − 3˜c2] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' ˜b2 = 1 8 [˜c3 − ˜c23˜q − 3˜c2] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' ˜q ≡ � 9 + 8 ˜c1 ˜c23 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (5) ˜q gives the charge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' ˜m & m denotes the mass parameter and ˜c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' ˜c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' ˜c3 represents the dimensionless constant parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' For the case ˜q = 0, we can observe that the metric converts into the usual asymptotically flat Schwarzschild BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Although, for the case ˜q ̸= 0, we get different asymptotically flat boundary conditions by considering the following cases independently(by def ˜q > 0 [30]): E(r)|r→∞ = 1 \uf8f1 \uf8f2 \uf8f3 for 0 < ˜q < 1 (if ˜a1 ̸= 0 and ˜a2 ̸= 0) or � if ˜a1 = 0 or ˜b1 = 0 � , for 0 < ˜q (if ˜a2 = 0 or ˜b2 = 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (6) For ˜q = 0, one can obtain the ADM mass as ¯ MADM = m ¯G , (7) where we have defined [32] ¯G = G 1 − ˜c1˜b2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (8) 3 The effective value of Newtonian constant ¯G associated to the constant G can be evaluated through experiments within the solar system [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Also, for the case ˜q ̸= 0, we get ¯ MADM = 1 ¯G � ˜m + ˜a2 1˜b1 r˜q (1 + ˜q) + ˜a2 2˜b2 r ˜ −q (1 − ˜q) � ��� r→∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (9) From the above equations, we can get the ADM mass for the NAT BH in the following form ¯ MADM = ˜m ¯G � for 0 < ˜q < 1 (if ˜a1 = 0 or ˜b1 = 0 � , for 0 < ˜q (if ˜a2 = 0 or b2 = 0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (10) If we consider the condition ˜a2 = 0, then, the the Aether field φ(r) and E(r) becomes E(r) = 1 − 2˜a2 1˜b1 r(1+˜q) − 2 ˜m r , (11) φ(r) = ˜a1 r(1+˜q)/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (12) We can get the event horizon r0 by considering E (r0) = 0 and the horizon area is A = 4πr2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' By taking ˜a1 = ˜G ˜Qr(˜q−1)/2 0 the Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (11) and (12) become E(r) = 1 − 2 ˜G2 ˜Q2˜b1 r2 �r0 r �˜q−1 − 2 ˜m r , (13) φ(r) = ˜G ˜Q r �r0 r �(˜q−1)/2 , (14) here ˜Q depicts the charge of NAT BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' After putting r = r0 in the above equations, we get E (r0) = 1 − 2 ¯G2 ˜Q2˜b1 r2 0 − 2 ˜m r0 = 0, (15) φ (r0) = ¯G ˜Q r0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (16) It is note worthy to mention here that the horizon condition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (15) is free from the parameter ˜q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Moreover, φ(r) looks like the electric potential at r = r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' After substituting ˜q = 1 in the metric (3), the E(r) and φ(r) get the form E(r) = 1 − 2˜a2 1˜b1 r2 − 2 ˜m r , φ(r) = ˜a1 r1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (17) III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' ASYMPTOTICALLY FLAT NAT BH IN NEWMAN-JANIS ALGORITHM By applying the Newman-Janis algorithm [33, 35, 36], we generalize the asymptotically flat NAT BH solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Now we introduce a coordinate transformation from Boyer Lindquist (BL) coordinates (t, r, θ, φ) to Eddington Finkelstein (EF) coordinates (u, r, θ, φ) du = dt − dr E(r), (18) where u represents the null coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' According to new coordinates the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (3) can be rewritten as ds2 = −E(r)du2 + r2dθ2 − 2dudr + r2 sin2 θdφ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (19) The non-zero components for the inverse metric (19) are defined as gur = −1, grr = E(r), gθθ = 1 r2 , gφφ = 1 r2 sin2 θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' 4 Moreover, the inverse metric with complex null tetrad Zx = (lx, nx, mx, ¯mx) can be written as gxy = −lxny − lynx + mx ¯my + my ¯mx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (20) The corresponding components can be defined as lx = δx r , nx = δx u − 1 2E(r)δx r , mx = 1 √ 2r δx θ + i √ 2r sin θδx φ, ¯mx = 1 √ 2r δx θ − i √ 2r sin2 θ δx φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' These null tetrad have orthonormal relation and comply with the accompanying characterizing conditions, specifically all the vectors satisfy the given relations lxlx = nxnx = mxmx = ¯mx ¯mx = 0, lxmx = lx ¯mx = nxmx = nx ¯mx = 0, lxnx = mx ¯mx = 1, By considering the Newman-Janis method, we enable the coordinates to get complex values, while for real lx and nx we are able to consider the given transformation [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' ], u′ = u − ia cos θ, r′ = r + ia cosθ, (21) here a represents the spin parameter (due to Newman-Janis algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Furthermore, we consider the transformations from E(r) → ˜E(r, a, θ) and σ2 = r2 + a2 cos2 θ, whereas the null tetrad transforms as vectors in the form lx = δx r , ny = δx u − 1 2 ˜E(r)δx r , mx = 1 √ 2r � δx θ + i sin θδx φ + ia sin θ(δx u − δx r ) � , ¯mx = 1 √ 2r � δx θ − i sin θδx φ − ia sin θ(δx u − δx r ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (22) By using the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (20) and (22), the gxy components of non-zero in the EF coordinate can be defined as guu = a2 sin2 θ σ2 , gur = gru = −1 − a2 sin2 θ σ2 , grr = ˜E(r, θ) + a2 sin2 θ σ2 , gθθ = 1 σ2 , gφφ = 1 σ2 sin2 θ, guφ = gφu = a σ2 , grφ = gφr = − a σ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Furthermore, the lower indices components of matrix in the EF coordinates can be given as guu = − ˜E(r, θ), gur = gru = −1, grr = 0, gθθ = σ2, guφ = gφu = a sin2 θ, gφφ = sin2 θ � σ2 + a2( ˜E(r, θ) − 2) sin2 θ � , grφ = gφr = − a σ2 , (23) where ˜E(r, θ) = r2E + a2cos2θ σ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (24) According to transformed tetrad the new line element can be written as ds2 = − ˜E(r, θ)du2 + σ2dθ2 + 2a sin2 θdrdφ − 2a � 1 − ˜E(r, θ) � sin2 θdudφ − 2dudr + sin2 θ � σ2 + a2 � 2 − ˜E(r, θ) � sin2 θ � dφ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (25) 5 Now we introduce the transformation of EF coordinates to BL coordinates as [34] du = dt + Y (r)dr, dφ = dφ + χ(r)dr, (26) where the function of Y (r) and χ(r) is to ignore the grφ and gtr components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' However, Y (r) and χ(r) appears as function of r and θ which can be defined as Y (r) = − r2 + a2 (r2E + a2), χ(r) = − a (r2E + a2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (27) The dependence of θ from EF to BL coordinates transformation reveals the fact that, we are dealing with modified theory of gravity and non-vacuum surrounding [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Furthermore, we will exclude the dependency on r and θ in the functions σ2 and ∆r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' The asymptotically flat NAT BH with BL coordinates in the context of Newman-Janis algorithm can be obtained as ds2 = − �∆r − a2 sin2 θ σ2 � dt2 + σ2 ∆r dr2 − 2a � 1 + a2 sin2 θ − ∆r σ2 � sin2 θdtdφ + σ2dθ2 + sin2 θ � σ2 + sin2 θ � 2 − a2 ∆r − a2 sin2 θ σ2 �� dφ2, (28) here ∆r = r2 − 2mr + a2 − 2˜a2 1˜b1 σ2(˜q−1)/2 − 2˜a2 2˜b2 σ2(−1−˜q)/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (29) Since, BH acts like thermodynamical substance and whose temperature TH can be determined by considering the surface gravity κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' So, we can compute the Hawking temperature of the metric (28) by using the following formula [34] TH = k 2π , k = ∆′ r 2(r2 + + a2), (30) where ∆′ r = d dr(∆r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' The corresponding Hawking temperature for NAT BH with Newman-Janis algorithm can be derived as TH = � r+ − m − ˜a2 1˜b1r(1 − ˜q)(r2 + + a2)(−1−˜q)/2 − ˜a2 2˜b2r(1 + ˜q)(r2 + + a2)(˜q−1)/2 2π(r2 + + a2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (31) The TH for BH depends upon the BH mass m, charge ˜q, rotation parameter a and free parameters ˜a1, ˜a2,˜b1,˜b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' The above temperature reduces into temperature of Schwarzschild BH for a = 0, ˜q = 0 which implies as [37] TSBH = (r+ − m) 2πr2 + , where r+ = 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (32) IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' STABILITY ANALYSIS OF NAT BH This section is comprised to investigate the graphical interpretation of temperature TH w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='t event horizon (r+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' We evaluate the physical significance of the plots to analyze the effects of charge ˜q, mass m and rotation parameter a of BH on temperature to study the BH stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Figure 1: depicts the presentation of TH via r+ for the fixed values of mass m = 1, rotation parameter a = 1, free parameters ˜a1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='1 = ˜b1, ˜a2 = 50,˜b2 = −10 in the range of charge 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='1 ≤ ˜q < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' At first, the TH increases and attains a maximum height and then it drops down gradually from a height and gets an asymptotically flat sate by indicating the stability of BH as r+ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' It can be observe that the temperature of BH increases with the decreasing values of horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' This physical behavior satisfies the Hawking’s phenomenon and guarantee the stability of BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' For 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='1 ≤ ˜q ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='3, we observe an asymptotically flat behavior in temperature that exhibits the stable state of BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Figure 2: depicts the behavior of TH via r+ with fixed values of mass m = 1, charge ˜q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='1, free parameters ˜a1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='1,˜b1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='2, ˜a2 = 50,˜b2 = −10 and for varying values of rotation parameter a in the range 0 ≤ r+ ≤ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' There can be seen that an asymptotically flat behavior of temperature appears after attaining a maximum height for different values of a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' It can be seen that as we rises the value of a the temperature goes on decreasing as well as for the increasing horizon the temperature decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' This Hawking’s phenomenon depicts the BH stability in the domain 0 ≤ r+ ≤ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' It has worth to mention here that for TH ≥ 0, the BH shows the physical behavior and it is in complete stable form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' 6 q˜=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='1 q˜=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='2 q˜=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='3 0 2 4 6 8 10 12 14 0 500 1000 1500 2000 r+ TH m=1, a˜ 1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='1, b ˜ 1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='1, a˜ 2=50, b ˜ 2=-10, a=1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' 1: TH versus r+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='0 a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='2 a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='4 0 2 4 6 8 10 12 14 0 500 1000 1500 r+ TH m=1, a˜ 1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='1, b ˜ 1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='2, a˜ 2=50, b ˜ 2=-10, q˜=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' 2: TH versus r+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' TEMPERATURE OF NAT BH UNDER THE INFLUENCE OF QUANTUM GRAVITY In this chapter, we analyze the TH under the act upon of quantum gravity for boson spin-1 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' We rewrite the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (28) in the adopting form ds2 = −Fdt2 + Gdr2 + Hdθ2 + Kdφ2 + 2Ldtdφ, (33) where F = ∆r − a2 sin2 θ σ2 , G = σ2 ∆r , H = σ2, K = sin2 θ � σ2 + � 2 + a2 sin2 θ + ∆r σ2 � a2 sin2 θ � L = −2a � 1 + a2 sin2 θ + ∆r σ2 � sin2 θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' In order to evaluate the corrected TH of vector particles from the BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' The vector particles such as Z and W are well-known and act as very significance role in Standard Model [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' We motion the charges bosonic tunneling in the NAT BH should be more complicated than the Lagrangian field equation as the nontrivial solution interaction during the charged bosonic field, the electromagnetic field and the Aether field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Firstly, we take the field equation of charged particles from the Lagrangian field equation given by the GUP and also we use the Hamilton–Jacobi ansatz phenomenon and WKB approximation to calculate the set of field equation in NAT space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' By considering the coefficient matrix determinant equal to zero and the linear equations can be derived for the radial function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Accord- ingly, we compute the tunneling probability of the vector particles from the NAT BH and discuss the corresponding temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Therefore, we utilize the generalized Lagrangian equation incorporating the GUP influenced by quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' The Lagrangian field equation is given [26] by ∂µ(√−gχνµ) + √−g m2 ℏ2 ϕν + √−g i ℏAµϕνµ + √−g i ℏeF νµϕµ + ̺ℏ2∂0∂0∂0(√−gg00ϕ0ν) −̺ℏ2∂i∂i∂i(√−ggiiϕiν) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (34) here determinant of g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' ϕνµ and m represent coefficient matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' anti-symmetric of tensor and particle of mass,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' since ϕνµ = (1 − ̺ℏ2∂2 ν)∂νϕµ − (1 − ̺ℏ2∂2 µ)∂µϕν + (1 − ̺ℏ2∂2 ν) i ℏeAνϕµ − (1 − ̺ℏ2∂2 ν) i ℏeAµϕν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Fνµ = ∇νAµ − ∇µAν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' where ̺,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Aµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' ∇µ and e represent the GUP(quantum gravity) parameter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' vector potential,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' covariant derivatives and the charge of particle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' The elements of non-zero for anti-symmetric tensor can be calculated as ϕ0 = −Kϕ0 + Lχ3 FK + L2 , ϕ1 = 1 Gϕ1, ϕ2 = 1 H ϕ2, ϕ3 = Lϕ0 + Fχ3 FK + L2 , ϕ12 = 1 GH ϕ12, ϕ13 = 1 GFK + L2ϕ13, ϕ01 = −Kϕ01 + Lϕ13 G(FK + L2) , ϕ02 = −Kϕ02 H(FK + L2), ϕ03 = (−FK + F 2)ϕ03 (FK + L2)2 , ϕ23 = Lϕ02 + Fϕ23 H(FK + L2), 7 The WKB approximation can be expressed as ϕν = cν exp[ i ℏQ0(t, r, φ, θ) + ΣℏnQn(t, r, φ, θ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (35) Using variables technique of separation, we can choose Q0 = − ˜Et + W(r) + ν(φ) + Jθ, (36) where ˜E = E −Jω and E, J denote the particle energy and the angular particle momentum corresponding to θ angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' After substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (34) into set of the field equations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' we get a matrix of order 4 × 4 Y (c0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' c3)T = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (37) whose elements are given as follows: Y00 = −K G(FK + L2) � W 2 1 + ̺W 4 1 � − K H(FK + L2) � ν2 1 + ̺ν4 1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' − FK (FK + L2)2 � J2 + ̺J4� − m2K (FK + L2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Y01 = ˜ −K G(FK + L2) � L + ̺ ˜E3 + eA0 + ̺eA0 ˜E2� W1 + E G(FK + L2) + � ν1 + ̺ν3 1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Y02 = −K H(FK + L2) � ˜E + ̺ ˜E3 − eA0 − ̺eA0 ˜E2� J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Y03 = − ˜E B(FK + L2) � W 2 1 + ̺W 4 1 � − FK H(FK + L2)2 � ˜E + ̺ ˜E3 − eA0 − ̺eA0 ˜E2� J + m2L (FK + L2)2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Y11 = ˜ −K G(FK + L2) � ˜E2 + ̺ ˜E4 − eA0 ˜E − ̺eA0 ˜EW 2 1 � + L G(FK + L2) − m2 G + � J + ̺J3� ˜E − 1 GH � ν2 1 + ̺ν4 1 � − 1 G(FK + L2) � J + ̺J3� + eA0L G(FK + L2) � J + ̺J3� − eA0K G(FK + L2) � ˜E + ̺ ˜E3 − eA0 − ̺eA0 ˜E2� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Y12 = 1 GH [W1 + ̺W 3 1 ]ν1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Y13 = ˜ −E G(FK + L2) � W1 + ̺W 3 1 � ˜E + 1 G(FK + L2)2 � W1 + ̺W 3 1 � J + LeA0 G(FK + L2) � W1 + ̺W 3 1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Y22 = K H(FK + L2) � ˜E2 + ̺ ˜E4 − eA0 ˜E − ̺eA0 ˜E � − 1 GH − m2 H − F H(FK + L2) � ν2 1 + ̺ν4 1 � − eA0K H(FK + L2) � ˜E + ̺ ˜E3 − eA0 − ̺eA0 ˜E2� + L H(FK + L2) � ˜E + ̺ ˜E3 − eA0 − ̺eA0 ˜E2� J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Y23 = F G(FK + L2) � ν1 + ̺ν3 1 � J (38) Y33 = (FK − ˜ F 2) (FK + L2) � ˜E2 + ̺ ˜E4 − eA0 ˜E − ̺eA0 ˜E3� − 1 G(FK + L2) � W 2 1 + ̺W 4 1 � − F H(FK + L2) � ν2 1 + ̺ν4 1 � − m2F (FK + L2) − eA0(FK − ˜ F 2) (FK + L2) � ˜E + ̺ ˜E3 − eA0 ˜E2� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' where ν1 = ∂φQ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' W1 = ∂rQ0 and J = ∂θQ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' The determinant of Y is equal to zero for the non-trivial solution and get ImW ± = ± � � � � �(E − Jω − A0e)2 + X1 � 1 + ̺ X2 X1 � (FK + L2)/GK dr, = ±π ( ˜E − A0e) 2k(r+) � 1 + ̺Ξ � , (39) 8 where X1 = GL (FK + L2) � ˜E − eA0 � ν1 + FG (FK + L2)J2 − Gm2, X2 = GK (FK + L2) � ˜E4 − 2eA0 ˜E3 + (eA0)2 ˜E2� − FG (FK + L2)J4 − W 4 1 + GL H(FK + L2) � ˜E3 − eA0 ˜E2� J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' The bosonic particle tunneling can be expressed as Γ = Γemission Γabsorption = exp � −2π(E − Jω − A0e) k(r+) � � 1 + ̺Ξ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (40) where k = ∆′ r 2(r2 + + a2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (41) The modified temperature can be calculated by applying the Boltzmann factor ΓB = exp [(E − Jω − A0e)/T ′ H] as T ′ H = � r+ − m − ˜a2 1˜b1r(1 − ˜q)(r2 + + a2)(−1−˜q)/2 − ˜a2 2˜b2r(1 + ˜q)(r2 + + a2)(˜q−1)/2 2π(r2 + + a2) � � 1 − ̺Ξ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (42) The Hawking temperature for BH depends upon the mass m, charge ˜q, quantum gravity parameter ̺, spin parameter a, arbitrary parameter Ξ and free parameters ˜a1, ˜a2, ˜b1, ˜b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' The expression (42) reduces into BH temperature for ̺ = 0, which leads a temperature in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' It has note worthy that the quantum corrections cause a deceleration in the increment of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' STABILITY ANALYSIS OF NAT BH WITH QUANTUM CORRECTIONS This section depicts the graphical presentation of T ′ H w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='t event horizon (r+) with fixed value of arbitrary parameter Ξ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' We study the physical existence of the plots and observe the effects of correction parameter ̺ and spin parameter a of BH on corrected Hawking temperature to study the stable BH condition under the influence of quantum effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Figure 3(i): describes the behavior of T ′ H via event horizon for the fixed values of mass m = 1, spin parameter a = 1, free parameters Ξ = 1, ˜a1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='1 = ˜b1, ˜a2 = 50, ˜b2 = −10, charge ˜q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='1 and for varying values of correction parameter ̺ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' At a peak value the temperature attains a maximum height and then it drops down gradually and obtain a condition of asymptotically flat by indicating the stability of BH as r+ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' It can be observed that the T ′ H decreases as we increase the correction parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' The temperature of BH increases with the decreasing values of event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' This physical presentation reflects the stability state of BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' The maximum temperature at non-zero horizon left the BH remnant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Figure 3(ii) represents the behavior of T ′ H via r+ with fixed values of mass m = 1, correction parameter ̺ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='8, charge ˜q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='1, free parameters ˜a1 = 1,˜b1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='1, ˜a2 = 50,˜b2 = −10 and for varying values of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' There can be seen that for different values of a the corrected temperature gets a height and then it shows an asymptotically flat behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' It is notable that when we increase the value of a the corrected temperature decreases as well as for the increasing value of horizon the corrected temperature also decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' This Hawking’s phenomenon represents the BH stable condition in the domain 0 ≤ r+ ≤ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' From both plots, we can observe that for T ′ H ≥ 0, the BH gets its stable form while for T ′ H < 0 the BH with negative temperature always depicts its unstable form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' We can also observe it graphically that the T ′ H is less than the original one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' So, we can conclude the quantum corrections decelerates the increment in temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' 9 ρ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='5 ρ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='6 ρ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='7 0 2 4 6 8 10 12 14 0 200 400 600 800 r+ TH (i) m=1, a˜ 1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='1, b ˜ 1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='1, a˜ 2=50, b ˜ 2=-10, a=1, q˜=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='1 a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='0 a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='2 a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='4 0 2 4 6 8 10 12 14 0 50 100 150 200 250 300 350 r+ TH (ii) m=1, a˜ 1=1, b ˜ 1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='1, a˜ 2=50, b ˜ 2=-10, q˜=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='1, �=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='8 Figure 3: T ′ H versus r+ with Ξ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' SUMMARY AND DISCUSSION The theory of null Aether is a vector-tensor gravity theory with null vector and Aether field exist at every point of the spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' In this paper, we have studied a new asymptotically flat BH solution by using Newman-Janis algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' To do so, firstly, we have reviewed the asymptotically flat BH solution in NAT and then by applying the Newman-Janis algorithm, we have derived a new asymptotically flat NAT BH spacetime influenced by rotation parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' By considering the spin parameter (a → 0) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (28), we get the asymptotically flat BH solution [30] in general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Furthermore, by taking into account the surface gravity κ, we have computed the temperature for NAT BH in the presence of rotation parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' The BH temperature depends upon the charge, mass, spin and free parameters of the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' The NAT BH temperature in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (31) recovers the temperature of Schwarzschild BH for ˜q = 0 = a as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Moreover, we have comprised the graphical representation of Hawking temperature w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='t event horizon in order to check the stability of BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' We have studied the radiation spectrum through bosonic tunneling process of spin-1 particles from NAT BH involving both spin and quantum gravity parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Therefore, we have utilized the generalized Lagrangian equation incorporating the GUP influenced by quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' For this investigation, we have applied the Hamilton-Jacobi ansatz and WKB approximation to the generalized Lagrangian field equation for boson particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' We have obtained the bosonic corrected tunneling rate of emitted particles and their corresponding corrected temperature T ′ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' It has note worthy to analyzed that, when we ignore the quantum gravity effects, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=', (ρ = 0), then the corrected Hawking temperature in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (42) is reduced to the original temperature in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' The corrected temperature of BH depends upon spin parameter, quantum gravity parameter and Aether field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' The T ′ H reduces into Schwarzschild BH temperature when the spin parameter, quantum gravity parameter and Aether field approaches to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' It has been analyzed that the quantum gravity decelerates the increase in T ′ H in the process of radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Moreover, we have analyzed the physical significance of corrected temperature to check the effects of quantum gravity and rotation parameter on T ′ H by seeing the stability of NAT BH over Aether field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' The results from the plots of Hawking temperature with respect to the horizon in the presence/absence of gravity parameter for the given BH are given as follows: In the absence of gravity parameter the temperature shows the asymptotically flat behavior in the range of charge 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='1 ≤ ˜q ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='3 and the TH decreases with the increasing r+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' This is physical graphical presentation of TH w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='t r+ and depicts the stable condition of BH with positive temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' The TH for varying values of rotation parameter a shows an asymptotically flat behavior and after a maximum height the temperature goes on decreasing as well as for the increasing horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' This Hawking’s phenomenon depicts the BH stability in the domain 0 ≤ r+ ≤ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' In the presence of gravity parameter T ′ H decreases with the increasing values of correction parameter as well as horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' We have observed BH remnant at nonzero horizon with maximum temperature for different values ρ in the domain 0 ≤ r+ ≤ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' For different values of a the corrected temperature gets a height and then it shows an asymptotically flat behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' It is notable that the corrected temperature decreases with the increasing values of a as well as for the increasing value of horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' This Hawking’s phenomenon represents the BH stable condition in the domain 0 ≤ r+ ≤ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' 10 From all the plots, we have observed that for T ′ H ≥ 0, the BH gets its stable form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' We have also observed it graphically that the T ′ H is less than the original one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' So, we have concluded that the quantum corrections decelerates the increment in temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' APPENDIX After setting the all values in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (34),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' we get the field equations set as ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='G(FK + L2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='c1(∂0Q0)(∂1Q0) + ̺c1(∂0Q0)3(∂1Q0) − c0(∂1Q0)2 − ̺c0(∂1Q0)4 + c1eA0(∂1Q0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='+c1̺eA0(∂0Q0)2(∂1Q0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='G(FK + L2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='c3(∂1Q0)2 + ̺c3(∂1Q0)4 − c1(∂1Q0)(∂3Q0) − ̺c1(∂1Q0)(∂3Q0)2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='H(FK + L2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='c2(∂0Q0)(∂2Q0) + ̺c2(∂0Q0)3(∂2Q0) − c0(∂2Q0)2 − ̺c0(∂2Q0)4 + c2eA0(∂2Q0) + c2eA0̺ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='(∂0Q0)2(∂1Q0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='FK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='(FK + L2)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='c3(∂0Q0)(∂3Q0) + ̺c3(∂0Q0)3(∂3Q0) − c0(∂3Q0)2 − ̺c0(∂3Q0)4 + c3eA0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='(∂3Q0) + c3eA0(∂0Q0)2(∂3Q0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='− m2 Kc0 − Lc3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='(FK + L2) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='(43) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='−K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='G(FK + L2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='c1(∂0Q0)2 + ̺c1(∂0Q0)4 − c0(∂0Q0)(∂1Q0) − ̺c0(∂0Q0)(∂1Q0)3 + c1eA0(∂0Q0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='+̺c1eA0(∂0Q0)3� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='G(FK + L2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='c3(∂0Q0)(∂1Q0) + ̺c3(∂0Q0)(∂1Q0)3 − c1(∂0Q0)(∂3Q0) − ̺c1(∂0Q0)(∂3Q0)3� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='GH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='c2(∂1Q0)(∂2Q0) + ̺c2(∂1Q0)(∂2Q0)3 − c1(∂2Q0)2 − ̺c1(∂2Q0)4� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='G(FK + L2) 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='c1(∂0Q0) + ̺c1(∂0Q0)3 − c0(∂1Q0) − ̺c0(∂1Q0)3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='+eA0c1 + ̺c1eA0(∂0Q0)2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='eA0L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='G(FK + L2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='c3(∂1Q0) + ̺c3(∂1Q0)3 − c1(∂3Q0) − ̺c1(∂1Q0)3� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='− m2c1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='G ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='= 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (44) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='H(FK + L2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='c2(∂0Q0)2 + ̺c2(∂0Q0)4 − c0(∂0Q0)(∂2Q0) − ̺c0(∂0Q0)(∂2Q0)3 + c2eA0(∂0Q0) + ̺c2eA0(∂0Q0)3� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='GH ' metadata={'source': 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+page_content='c2(∂0Q0)(∂3Q0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='+̺c2(∂0Q0)3(∂3Q0) − c0(∂0Q0)(∂3Q0) − ̺c0(∂0Q0)3(∂3Q0) + c2eA0(∂3Q0) + ̺c2eA0(∂3Q0)3� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='H(FK + L2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='c3(∂2Q0)(∂3Q0) + ̺c3(∂2Q0)3(∂3Q0) − c2(∂3Q0)2 − ̺c2(∂3Q0)4� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='− m2c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='eA0K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='H(FK + L2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='c2(∂0Q0) + ̺c2(∂0Q0)3 − c0(∂2Q0) − ̺c0(∂2Q0)3 + c2eA0 + c2̺eA0(∂0Q0)2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='= 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='(45) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='FK − F 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='(FK + L2)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='c3(∂0Q0)2 + ̺c3(∂0Q0)4 − c0(∂0Q0)(∂3Q0) − ̺c0(∂0Q0)(∂3Q0)3 + eA0c3(∂0Q0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='+̺c3eA0(∂0Q0)3� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='H(FK + L2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='c3(∂1Q0)2 + ̺c3(∂1Q0)4 − c1(∂1Q0)(∂3Q0) − ̺c1(∂1Q0)(∂3Q0)3� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='H(FK + L2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='c2(∂0Q0)(∂2Q0) + ̺c2(∂0Q0)3(∂2Q0) − c0(∂2Q0)2 + ̺c0(∂2Q0)4 + eA0c2(∂2Q0) + ̺c2eA0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='(∂0Q0)2(∂2Q0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='eA0F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='H(FK + L2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='c3(∂2Q0)2 + ̺c3(∂2Q0)4 − c2(∂2Q0)(∂3Q0) − ̺c2(∂0Q0)(∂3Q0)3� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='+eA0FD − F 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='(FK + L2)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='c3(∂0Q0) + ̺c3(∂0Q0)3 − c0(∂3Q0) − ̺c0(∂3Q0)3 + c3eA0 + ̺eA0(∂0Q0)2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='−m2Lc0 − Fc3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='(FK + L2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content='= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' (46) 11 [1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Jacobson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Mattingly, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' D 64, 024028(2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' [2] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Eling and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Jacobson, Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Quantum Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' 23, 5625(2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' [3] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Rizzo, JHEP 09, 036(2005).' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Carroll, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Dulaney, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Gresham, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Tam, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' D 79, 065011(2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' [7] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Carroll, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Lim, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' D 70, 123525(2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' [8] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Zlosnik, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Ferreira, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Starkman, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' D 75, 044017(2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' [9] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Xu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Wang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' D 95, 064015(2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' [10] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Konishi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Paffuti, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Methods Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' 19, 2250017(2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' [14] R.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Javed, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Ali, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' High Energy Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' 2018, 11(2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' [20] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Ali, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Javed, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Babar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' ¨Ovg¨un, Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' A 34, 1950057(2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Malik, Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' A 35, 2050225(2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' [26] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Javed, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Ali, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Babar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' ¨Ovg¨un, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} +page_content=' Plus 134, 511(2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfaPyX/content/2301.01366v1.pdf'} 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and Localization of Information in +Quantum Gravity +Eyoab Bahirua,b,c Alexandre Belind,e Kyriakos Papadodimasf Gabor Sarosif Niloofar +Vardiana,b +aSISSA, International School for Advanced Studies, via Bonomea 265, 34136 Trieste, Italy +bINFN, Sezione di Trieste, via Valerio 2, 34127 Trieste, Italy +cInternational Centre for Theoretical Physics, Strada Costiera 11, Trieste 34151 Italy +dDipartimento di Fisica, Universit`a di Milano - Bicocca, I-20126 Milano, Italy +eInstitute of Physics, Ecole Polytechnique F´ed´erale de Lausanne, CH-1015 Lausanne, Switzerland +fTheoretical Physics Department, CERN, CH-1211 Geneva 23, Switzerland +E-mail: ebahiru@sissa.it, alexandre.belin@unimib.it, +kyriakos.papadodimas@cern.ch, gabor.sarosi@cern.ch, nvardian@sissa.it +Abstract: Within the AdS/CFT correspondence, we identify a class of CFT operators +which represent diff-invariant and approximately local observables in the gravitational dual. +Provided that the bulk state breaks all asymptotic symmetries, we show that these operators +commute to all orders in 1/N with asymptotic charges, thus resolving an apparent tension +between locality in perturbative quantum gravity and the gravitational Gauss law. +The +interpretation of these observables is that they are not gravitationally dressed with respect +to the boundary, but instead to features of the state. We also provide evidence that there are +bulk observables whose commutator vanishes to all orders in 1/N with the entire algebra of +single-trace operators defined in a space-like separated time-band. This implies that in a large +N holographic CFT, the algebra generated by single-trace operators in a short-enough time- +band has a non-trivial commutant when acting on states which break the symmetries. It also +implies that information deep in the interior of the bulk is invisible to single-trace correlators +in the time-band and hence that it is possible to localize information in perturbative quantum +gravity. +arXiv:2301.08753v1 [hep-th] 20 Jan 2023 + +Contents +1 +Introduction +1 +2 +Aspects of locality in field theory and gravity +7 +2.1 +Classical field theories +8 +2.2 +Localization of information in QFT +9 +2.2.1 +Comments on the split property +9 +2.2.2 +Subtleties with gauge invariance +11 +2.3 +Classical and Quantum Gravity +13 +2.3.1 +On the initial value problem of general relativity +13 +2.3.2 +Diff-invariant observables in classical GR +15 +2.3.3 +State-dressed observables +16 +2.3.4 +A time-band in AdS +18 +3 +Holographic setup +19 +3.1 +Gravitional states in AdS, large diffeomorphisms and asymptotic symmetries +20 +3.2 +Locality in AdS +23 +3.3 +The CFT description and the time band algebra +23 +3.4 +Formulating the main goal +25 +3.5 +Time-shifted states and return probability +26 +3.6 +The return probability +27 +3.7 +Other asymptotic charges +30 +4 +State-dressed operators +31 +4.1 +Vanishing commutator with H to all orders in 1/N +32 +4.2 +Similar action as HKLL operators +33 +4.3 +Interpretation and comments +34 +4.4 +A similarity transformation +36 +4.5 +Other asymptotic charges +37 +5 +A more general argument for the commutant +37 +5.1 +On the consistency of the defining equations +39 +5.2 +Proof that operators have the desired properties +40 +6 +Examples +41 +6.1 +Coherent states +41 +6.2 +Thermofield double state +43 +6.3 +Weakly coupled, large N gauge theories +45 +6.4 +Perturbative states around empty AdS +45 +6.5 +LLM geometries +47 +– i – + +6.5.1 +Computation of the return probability +48 +6.6 +Kourkoulou-Maldacena states in SYK model +51 +6.6.1 +Analytical computation of the return probability at large N +52 +6.6.2 +Some numerical checks +54 +6.7 +Holographic boundary states +56 +6.7.1 +Computation of the return probability and correlators +57 +7 +Black Hole microstates +59 +7.1 +States with macroscopic time-dependence +59 +7.2 +Typical states +60 +7.3 +Two entangled CFTs +61 +7.4 +Island discussion +62 +8 +Discussion +63 +8.1 +The variance of the energy from semi-classical gravity +64 +8.2 +Gravitational proof for the decay of the return probability +64 +8.3 +Microscopically time-dependent states +65 +8.4 +Microcanonical states and small energy variance +66 +8.5 +The AdS vacuum and low-energy states +66 +A Changing the variance of H +67 +B Boosts in global AdS +70 +C Early time decay of the return probability +71 +C.1 Overlap of coherent states and large N factorization +71 +C.2 The return probability +72 +D LLM solutions in the bulk +73 +E Notes on boundary states +74 +E.1 +Boundary states in 2D CFT +75 +E.2 +Boundary states in higher dimensions +77 +E.3 +Correlation functions in BCFTs +77 +1 +Introduction +It is generally believed that in quantum gravity, space-time locality is an emergent notion +which becomes accurate and useful in certain limits of the underlying theory. This perspective +is realized in the AdS/CFT correspondence [1]: bulk locality becomes precise in the large N, +– 1 – + +strong coupling limit and when probing the theory with simple enough operators. Moreover, +a large number of proposals aiming to resolve the black hole information paradox rely on +a certain amount of non-locality [2–13]. A natural question is to understand whether non- +local features of quantum gravity are visible only in the non-perturbative regime, or whether +remnants of non-locality are also visible at the perturbative level. +Even in classical general relativity it is not entirely straightforward to formulate the +concept of locality, as it is non-trivial to define local observables. Physical observables need +to be diff-invariant and, in order for them to also be local, they have to be associated to +points in space-time which have to be specified in a diff-invariant way. If the space-time has +a boundary, a standard approach is to define points relationally with respect to the boundary +or by completely fixing the gauge. We say that these observables are gravitationally dressed +with respect to the boundary. However, the resulting observables, while diff-invariant, are +not strictly localized and have non-vanishing Poisson brackets at space-like separation. A +particular aspect of this difficulty is related to the gravitational Gauss law: in gravitational +theories defined with asymptotically flat or AdS boundary conditions, the Hamiltonian, and +other asymptotic symmetry charges, are boundary terms. +Acting with a candidate local, +diff-invariant observable in the interior of space will generally change the energy of the state, +which is immediately measurable at space-like separation due to Gauss’s law. +Despite these difficulties, at the classical level, there are ways of defining local and diff- +invariant observables in the neighborhood of a state, provided that the state is sufficiently +complicated. A class of such observables introduced a long time ago [14–16] will be reviewed +in sub-section 2.3.3, see also [17–19] for more recent discussions. These observables respect the +causal structure of the underlying space-time, in the sense that their Poisson brackets at space- +like separation vanish. In particular, provided that the state we are considering is complicated +enough, the action of these observables is not visible by the boundary Hamiltonian, as these +observables only rearrange energy in the interior of space. The price we have to pay is that +these observables are not defined globally on the phase space of solutions. They have desired +properties only for certain states. +A natural question is to what extent can such local diff-invariant observables be defined +at the quantum level. +As mentioned above, we do not expect to be able to find exactly +local diff-invariant observables at the non-perturbative level, however it may be possible to +do so in perturbation theory. This question is important in order to be able to quantify +departures from locality in quantum gravity and to understand if there is a way to generalize +the structure of algebras of observables of quantum field theory to situations where gravity +is included perturbatively. +It is useful to formulate these questions in the context of the AdS/CFT correspondence. +We consider a CFT state |Ψ0⟩ that is dual to a semi-classical asymptotically AdSd+1 geometry +in global coordinates and a short time-band near the boundary as shown in Fig. 1. We +consider the algebra A of observables in semi-classical gravity which are localized in this time +band. This algebra includes the Hamiltonian and other asymptotic charges. From the point +of view of the dual CFT, it is natural to identify the algebra A with the algebra generated by +– 2 – + +Figure 1: The single-trace operators localized in the time band t ∈ (−ϵ, ϵ) × Sd−1 (dark blue +region on boundary) form an algebra A which is conjectured to be dual to the causal wedge +of the region (light blue). If the state |Ψ0⟩ of the system breaks all symmetries, then the +causal diamond in the middle (light red), which is spacelike separated from the time-band, +corresponds to the commutant A′ of the algebra A when acting on the code subspace of the +state |Ψ0⟩. +single-trace operators localized in this time-band, we will call it the ”single-trace algebra”. +The expectation is that the single-trace algebra A corresponds to the causal wedge of the +time-band [20]1. Notice that here we have causal-wedge reconstruction and not entanglement +wedge reconstruction, as we are looking only at the single-trace subalgebra. +In the CFT +the notion of a time-band algebra only makes sense at large N, since large N generates a +natural hierarchy between operators that are small combinations of single-trace operators and +arbitrarily complicated operators. For finite N there is no such hierarchy and the time-slice +axiom would imply that A is the full CFT algebra2. Algebras of single-trace operators in +holographic CFTs have been discussed in [6,25,26] and more recently in [27–31]. +If the time-band is short enough, then there is a region in the bulk which is space-like +with respect to the time-band. We will refer to this region as the ”diamond”3. If we were +able to define diff-invariant observables localized in the diamond, they should commute with +the algebra A. As already mentioned, the question is non-trivial as these observables must be +gravitationally dressed and if we use the boundary to dress them, then they will not commute +with A. For example, it appears that since the Hamiltonian H is an element of A it would +be able to detect any excitation added in the interior of the diamond using the gravitational +Gauss law. To summarize, the question we want to examine: +1A different approach for studying time-bands based on gravitational entropy and minimal surfaces was +initiated in [21–24]. It would be interesting to understand possible connections between those ideas and the +results presented in this paper. +2We do not include in the single-trace algebra elements like eiHtO(t = 0, x)e−iHt with t = O(N 0) and large +enough to exit the time-band. Such ”precursor” operators are complicated from the point of view of operators +in the time-band and go beyond the semi-classical description. +3For now we assume that the state has simple topology and there are no black hole horizons in the interior. +– 3 – + +Does the algebra A, when acting on the state |Ψ0⟩ and small perturbations around +it, have a non-trivial commutant in the 1/N expansion? +As we will discuss later, we need to refine the question by demanding that the commutant acts +non-trivially within the code-subspace of the state, in order to avoid obvious but uninteresting +constructions4. We emphasize that we do not expect the algebra to have a commutant at +finite N [20]. +A closely related question is that of localization of information. According to AdS/CFT +the quantum state of the CFT at any moment in time contains the full information of the +bulk. In particular, if we had considered the full algebra of all operators in the time-band, +as opposed to the algebra generated by few (relative to N) single-trace operators, then we +would be able to reconstruct the interior of the diamond. Suppose however, that we only have +access to the algebra A of single-trace operators in the time band. Can we then reconstruct +the information of whatever is hidden inside the diamond? This can also be rephrased as +follows: +Given a state |Ψ0⟩, can we find another state |Ψ0⟩′ such that the correlators of +the single-trace algebra A in the time-band, evaluated on these two states agree to +all orders in 1/N, but correlators of single-trace operators differ at O(N0) outside +the time-band? +The intuition here is that we want to find a state |Ψ0⟩′ which contains an additional excitation +relative to |Ψ0⟩ in the interior of the diamond which becomes visible by single-trace operators +only after a light-ray has reached the boundary i.e. in the future or past of the time-band. +If the algebra A had a commutant then we could take |Ψ0⟩′ = U(A′)|Ψ0⟩ for some unitary U +built out of operators A′ in the commutant. +We will provide evidence that the answer to the two aforementioned questions is positive, +provided that the state |Ψ0⟩ is complicated enough. The reasoning was first outlined in [32]. +In this paper we extend the construction in a few ways and provide additional arguments and +examples. +Standard approaches to bulk reconstruction lead to observables which are relationally +defined with respect to the boundary. This is the case for the HKLL reconstruction [33–39], +as well as approaches based on the Petz map [40, 41] or modular reconstruction [42, 43], as +they all require some sort of boundary dressing. For concreteness we start with a standard +HKLL operator given by +Φ(t, r, Ω) = +� +bdry +dt′ dΩ′ +d−1K(t, r, Ω; t′, Ω′)O(t′, Ω′) . +(1.1) +4 For example, for a complicated state with energy of O(N 2), a unitary which rotates the phase of a single +energy eigenstate will have commutators of O(e−N2) with all elements of A. However, this would not be an +interesting example, as this operator is generally ”invisible” from the bulk point of view and does not create +excitations inside the diamond. +– 4 – + +Here K is a particular Green’s function which depends on the background metric. Implicit in +this expression is a gauge-fixing scheme in a particular coordinate system, which is uniquely +determined by making use of the boundary. If we pick the point (t, r, Ω) to be in the diamond, +the operator (1.1) commutes with all single-trace operators in the time band at large N. +At subleading orders multi-trace corrections need to be added to (1.1) to ensure vanishing +commutators. However the commutator with the Hamiltonian and other asymptotic charges, +which is nonzero at order 1/N, cannot generally be corrected by multi-trace corrections. +The physical reason is that the operator (1.1) is gravitationally dressed with respect to the +boundary. The non-vanishing commutator with H appears to be an obstacle in identifying +(1.1) as an element of the commutant of A [44,45]. +In this paper we present a way to find operators which commute with the asymptotic +charges to all orders in 1/N, while at the same time create excitations in the interior of the +diamond similar to those of the HKLL operator. These operators can be defined provided +the state |Ψ0⟩ that we are considering breaks all asymptotic symmetries. These operators +correspond to observables gravitationally dressed with respect to features of the state. +A crucial starting observation is that if a state |Ψ0⟩ is dual to a bulk geometry which +breaks the asymptotic symmetries, then the overlap +⟨Ψ0|U(g)|Ψ0⟩ +g ∈ SO(2, d) , +(1.2) +is generally exponentially small, of order O(e−aN2) with Re(a) > 0 provided that the element +g is sufficiently far from the identity5. Here SO(2, d) represents the asymptotic symmetry +group of AdSd+1. We will quantify this statement more precisely in the later sections. In +fact, we will provide evidence that if we introduce the code subspace around the state |Ψ0⟩, +defined as +H0 = span{|Ψ0⟩, O(t, Ω)|Ψ0⟩, ..., O1(t1, Ω1)...On(tn, Ωn)|Ψ0⟩} , +(1.3) +and similarly Hg for the state U(g)|Ψ0⟩ then any inner product between unit normalized +states of H0, Hg will also be of order O(e−aN2). +Starting with a standard HKLL operator Φ we consider the operator +�Φ = c +� +B +dµ(g)U(g)P0ΦP0U(g)−1 , +(1.4) +where P0 denotes the projector on (1.3) and dµ(g) is the Haar measure on SO(2, d) and B is +a reasonably sized neighborhood of SO(2, d) around the identity. The overall normalization +constant c will be specified later. The main claim, which will be discussed in section 4, is that +operators (1.4) have the desired properties: their commutators with the asymptotic symmetry +charges Q of SO(2, d) are exponentially small +[Q, ˆΦ] = O(e−N2) , +(1.5) +5But not too far. The state may return to itself in compact directions of the conformal group or approxi- +mately back to itself due to Poincare recurrences. +– 5 – + +when acting on the code subspace, while at the same time, the leading large-N action of ˆΦ +on the code subspace (1.3) is the same as that of the corresponding HKLL operator Φ, that +is +⟨Ψ1|ˆΦ|Ψ2⟩ = ⟨Ψ1|Φ|Ψ2⟩ + O(1/N) +∀ |Ψ1⟩, |Ψ2⟩ ∈ H0 . +(1.6) +The interpretation is that by performing the integral (1.4) we have removed the gravitational +dressing of the operators from the boundary and moved it over to the state. This is only +possible on states where (1.2) decays sufficiently fast. +The operators (1.4) have vanishing commutators with the asymptotic charges to all orders +in 1/N. This demonstrates that the apparent obstacle to identifying a commutant due to +Gauss’s law can be overcome. In order to find a true commutant we need to ensure vanishing +commutators to all orders in 1/N with all single-trace operators in the time-band algebra. It +would be interesting to explore whether a formula achieving this goal and similar to (1.4) can +be derived, possibly by integrating over the unitary orbits generated by A. +We provide an alternative formal argument supporting the idea that the algebra A has +a nontrivial commutant when acting on the code subspace Hcode of a complicated state |Ψ0⟩. +To see that we consider an operator ˆΦ defined by +ˆΦA|Ψ0⟩ = AΦ|Ψ0⟩ +∀A ∈ A , +(1.7) +where again Φ is a standard HKLL operator. This represents a set of linear equations, one for +each A ∈ A, which define the action of ˆΦ on H0. A sufficient condition for the consistency of +these equations is that for all non-vanishing operators A ∈ A we have A|Ψ0⟩ ̸= 0. In section +5 we provide evidence that this is true in the 1/N expansion. Given that these equations are +consistent, we will show in section 5 that the operators ˆΦ defined by (1.7) obey the following +properties: i) by construction they commute with operators in A and ii) to leading order at +large N act like HKLL operators. This provides evidence that the algebra A has a commutant +in the 1/N expansion. As mentioned earlier, a commutant is not expected at finite N. Indeed, +at finite N it is possible to find complicated operators in the time-band which annihilate the +state |Ψ0⟩ and equations (1.7) do not have a consistent solution. +If we take the state |Ψ0⟩ to be the vacuum, i.e. empty AdS, then the previous construction +fails: since the vacuum is invariant under the asymptotic symmetries we no longer have the +decay of (1.2) and (1.6) fails. Also (1.7) fails because there are operators in the time-band, in +particular H, which annihilate the state. We emphasize that this failure is not a limitation +of our particular construction. Instead the interpretation of this failure is that since empty +AdS has no bulk features, the only way to specify a point in the bulk is by dressing it to +the boundary. Hence any bulk diff-invariant operators acting around the vacuum will not +commute with the asymptotic charges [44,45]. This can also be seen from the fact that even +classically, the local diff-invariant observables cannot be defined properly in the vacuum. +We emphasize that the results of this paper do not contradict the claim of [46] that +specifically for perturbative states around empty AdS, it is possible to reconstruct the state +from correlators in the time-band. However we notice that interesting states, that is, states +– 6 – + +which have bulk observers capable of performing physical experiments, are expected to be of +the form where the symmetries are broken and the construction presented in this paper can +be applied. +If the state |Ψ0⟩ corresponds to a black hole state, and if the variance of the asymptotic +charges scales like N2 6 we find that using the operators (1.4) we can create excitations +behind the horizon which cannot be detected by correlators of single-trace operators in the +1/N expansion. Understanding how to diagnose these excitations from a CFT calculation +remains an outstanding open problem. We emphasize that this does not contradict the fact +that, generally, excitations created by unitaries on top of typical states with small energy +spread can be detected by single-trace correlators [25,47,48]. Such states with small energy +spread are those for which our construction cannot be applied. +The operators we identify provide evidence supporting the idea that locality is respected +in perturbative quantum gravity and that information can be localized in subregions at the +level of perturbation theory, provided that the underlying state is sufficiently complicated. +It also suggests that it should be possible to associate algebras of observables to subregions. +However these observables have certain features of state-dependence, since both (1.4) and +(1.7) give operators which are defined only on the code-subspace of the original state |Ψ0⟩. It +is certainly possible to extend the domain of definition of our operators by combining together +code subspaces of sufficiently different states, each one of which must break the asymptotic +symmetries, thus partly eliminating the state-dependence of the operators. +However the +number of these states must not be too large, otherwise the small overlaps between the code +subspaces start to accumulate and modify the correlators. This becomes particularly relevant +for black hole states, where we do not expect to have operators with the desired properties +defined globally for most microstates and some genuine state-dependence is expected. +The plan of the paper is as follows: in section 2 we review background material about +various aspects of locality in field theory and gravity. In section 3 we describe the setup in +AdS/CFT and study the decay of the inner product (1.2). In section 4 we introduce the +operators (1.4) and discuss their basic properties. +In section 5 we provide an alternative +argument for the existence of a commutant based on equations (1.7). In section 6 we consider +various examples. In section 7 we consider aspects of our operators in the presence of black +holes. Finally we close with a discussion of open problems in 8. +2 +Aspects of locality in field theory and gravity +In this section, mostly addressed to non-experts, we review some background necessary to +explore the question of localizing information in different regions of space. A closely related +question is the association of algebras of observables to subregions and the factorization of +the Hilbert space. +We start with non-gravitational field theories, where a non-dynamical +background space-time can be used in order to define sub-regions and their causal relations, +and then we consider the additional complications when gravity is taken into account. +6For example, this is true for black hole states with energy spread similar to the canonical ensemble. +– 7 – + +In relativistic theories we expect that signals and information cannot travel faster than +light. We then want to address the following question: consider an initial space-like slice +Σ and divide it into a compact subregion D and its complement D′. We denote by J(D′) +the domain of dependence of D′. The question is the following: is it possible to modify the +state7 in region D without affecting the state in J(D′). If the answer is positive then an +observer initially in D′, and confined to move in J(D′), cannot reconstruct information about +the interior of D. Then we say that information can be localized. +2.1 +Classical field theories +At the classical level this question can be addressed by studying the initial value problem: +we specify initial data C on a spacelike slice Σ and then look for a solution in the entire +space-time, or at least a neighborhood of the slice Σ, compatible with the initial data. The +initial data will typically include the values and time-derivatives of various fields of the theory. +The theories we will be considering have gauge invariance. One of the implications is that +the existence of a solution is guaranteed only if the initial data satisfy certain constraints. +In relativistic field theories theories the dynamical equations are hyperbolic, which ensures +that signals propagate forward from Σ at most at the speed of light. On the other hand the +constraint equations for initial data are of elliptic nature. This makes the question of being +able to specify the initial data independently in region D and its complement D′ non-trivial. +It is thus convenient to divide the question formulated above in two steps: +A. Localized preparation of states: for given initial data C1 on Σ satisfying the +constraints, to what extent can we deform to other initial data C2, also satisfying the +constraints, such that C1, C2 agree on D′, possibly up to a gauge transformation, but +differ essentially8 on D? +B. No super-luminal propagation: suppose we are given two initial data C1, C2 +which satisfy the constraints, which agree on D′ and differ on D. We then want to +show that the two corresponding solutions agree on J(D′), possibly up to a gauge +transformation. +We will return to the classical problem in theories with gauge invariance in the following +subsections. For now we briefly consider the simplest example of a free Klein-Gordon field in +flat space obeying □φ = m2φ. We consider initial data on the slice Σ corresponding to t = 0. +The initial data on this slice are parametrized by C = {φ(t = 0, x), ∂0φ(t = 0, x)}. In this +case condition A mentioned earlier is clearly satisfied: the initial data do not need to obey +any constraint, so we can simply select the functions φ, ∂0φ to have any smooth profile with +7Either classical state, or quantum density matrix. +8i.e. cannot be matched by a gauge transformation on D. +– 8 – + +features strictly localized in D. Notice that this requires the use of non-analytic initial data. +Condition B is also satisfied, see [49] for a basic review.9 +2.2 +Localization of information in QFT +In non-gravitational QFT we can associate algebras of observables to space-time regions +[50–52]. +Locality is exact, and is expressed by the condition that algebras corresponding +to space-like separated regions commute. An analogue of the initial value problem in QFT +is expressed by the condition of primitive causality or relatedly the time-slice axiom which +postulates that the only operators commuting with the algebra generated by operators in +a time-band are proportional to the identity. Moreover a local version of these statements +postulates that the algebra of operators in a subregion coincides with the algebra of operators +in the causal domain of dependence of the subregion [53]. +An intuitive way to see that that information can be localized in QFT is as follows: +suppose |Ψ0⟩ is a state in the Hilbert space of the QFT. Consider a unitary operator UD +constructed out of observables localized in D and the new state |Ψ⟩ = UD|Ψ0⟩. The unitary +UD modifies the state by creating an excitation in region D which encodes the desired infor- +mation in that region. For any observation OD′ in region D′, and more generally in J(D′), +we have +⟨Ψ|OD′|Ψ⟩ = ⟨Ψ0|U † +DOD′UD|Ψ0⟩ = ⟨Ψ0|OD′|Ψ0⟩ , +(2.1) +where we used [UD, OD′] = 0. Hence states |Ψ⟩, |Ψ0⟩ are indistinguishable by measurements +in J(D′) and the excitation created by UD in D is invisible in J(D′). +2.2.1 +Comments on the split property +More generally we would like to know whether it is possible to independently specify the +quantum state in space-like separated regions. +The question is non-trivial since in most +quantum states these regions will be entangled. It is believed that, as long as the regions in +question are separated by a finite buffer region, then the answer should be positive. This is +related to the split property of quantum field theory [52,54–56]. +The split property can be defined as follows: consider the causal diamond whose base is +a ball D1 and the corresponding operator algebra AD1. Consider a slightly larger ball D2, +containing D1, with corresponding operator algebra AD2 in its causal diamond. The split +property is satisfied if we can find a type I von Neumann algebra of operators N such that +AD1 ⊂ N ⊂ AD2. It has been shown that quantum field theories with a reasonable thermody- +namic behavior, as expressed in terms of nuclearity conditions (see [52] for an introduction), +satisfy the split property. Using the algebra N we can have strict localization of quantum +information which is completely inaccessible from J(D′ +2). +9In the case of non-relativistic theories, for example the heat equation, which is first order in time and +hence not hyperbolic, we are able to specify the initial data in subregions independently but the speed of +propagation is unbounded. Hence the heat equation obeys condition A but not B. +– 9 – + +Equivalently, the split property can be defined by the existence of state |φ⟩ which is cyclic +and separating for the algebra AD1∪D′ +2 and such that +⟨φ|a b|φ⟩ = ⟨0|a|0⟩⟨0|b|0⟩ +∀ a ∈ AD1, b ∈ AD′ +2 , +(2.2) +where |0⟩ is the Minkowski vacuum and D′ +2 denotes the complement of D2. In the state |φ⟩ +the mutual information between regions D1 and D′ +2 is vanishing. Such a state is not uniquely +defined, since for any unitary U ∈ A(D′ +1∩D2) a state of the form U|φ⟩ will also satisfy (2.2). +Starting with a split state |φ⟩ we can construct more general states by exciting the two +regions D1 and D′ +2 acting with localized operators in the corresponding algebras. Since there +is no entanglement between D1 and D′ +2 in the split state |φ⟩ the two algebras act independently +and we can arbitrarily approximate an excited state in D1 and another state in D′ +2. +An interesting question is to estimate the energy of a split state10. We do not expect a +split state to be an energy eigenstate, so in general it will have non-vanishing energy variance. +Here we provide some very heuristic arguments about the expectation value of the energy. +As a starting point, let us consider a CFT on R1,d−1 with coordinates x0, x1, ..., xd−1. We +define two regions to be the causal domains of two slightly displaced Rindler wedges with +bases x0 = 0, x1 < −ϵ and x0 = 0, x1 > ϵ respectively. The two wedges are separated by +the buffer region −ϵ < x1 < ϵ. In this case the total energy of the split state will be infinite +due to the infinite planar extension of the regions in the transverse directions. However, we +expect to have a finite energy per unit area E. Since we are dealing with a CFT then the +only scale in the problem is the size ϵ of the buffer region. Hence by dimensional analysis the +energy per unit area will scale like E = +s +ϵd−1 where s is a constant depending on the CFT. If +we now consider a more general compact region D1 of typical size R, which is separated by +a small buffer region of typical size ϵ from D′ +2 then we would expect that a split state with +respect to D1, D′ +2 will have energy which in the ϵ → 0 limit will scale like +E = s A(∂D1) +ϵd−1 ++ O( ϵ +R) , +(2.3) +where A(∂D1) is the area of the boundary of D1. This is a heuristic estimate and it would be +interesting to investigate it more carefully. +As mentioned above, this is the expectation value of the energy and it would be interesting +to understand the spectral decomposition of a split state in the energy basis. Notice that a +split state does not respect the Reeh-Schlieder property with respect to the algebra AD1 +11. +This implies in particular that the split state should have non-compact support in energy, +since otherwise the Reeh-Schlieder property would have to hold for D1, see for example [57]. +10Since the split state is not unique, a reasonable question might be finding the lowest possible expectation +value for the energy of a split state. +11Since there is no entanglement between D1 and D′ +2 we cannot create excitations in region D′ +2 by acting +with operators in D1. +– 10 – + +2.2.2 +Subtleties with gauge invariance +Consider U(1) gauge theory minimally coupled to a charged scalar with Lagrangian L = +− 1 +4FµνF µν − (Dµφ)∗Dµφ , Dµφ = ∂µφ − igAµφ. +The system has U(1) gauge invariance +Aµ → Aµ + ∂µΛ, φ → eigΛφ. The dynamical equations are +∂νFµν = ig(φ∂µφ∗ − φ∗∂µφ) − 2g2Aµφ∗φ +□φ = ig(∂µAµ)φ + 2igAµ∂µφ + g2AµAµφ . +(2.4) +In this case the initial data are C = {Aµ(t = 0, x), ∂0Aµ(t = 0, x), φ(t = 0, x), ∂0φ(t = 0, x)}. +Here we encounter the subtleties mentioned for gauge systems: initial data related by a +gauge transformation are physically equivalent and initial data are admissible (i.e. lead to a +solution) only if the obey a constraint, the Gauss law, which is the µ = 0 component of the +first equation in (2.4) +∂i(∂0Ai − ∂iA0) = ig(φ∂0φ∗ − φ∗∂0φ) − 2g2A0φ∗φ . +(2.5) +We now revisit the two properties mentioned in subsection 2.1. The fact that the dynamical +part of (2.4) obey condition B follows from general properties of hyperbolic equations of this +type. Let us now examine question A in this theory. From (2.5) we see that if we try to +deform the initial data in region D, then we may be forced to change them in D′ too. For +example if we turn on a profile for the scalar in region D with total non-zero charge, then +the gauge field has to be turned on in region D′. The Gauss law constraint (2.5) is of the +familiar form ∇ · ⃗E = ρ. This imposes the constraint that +� +∂D ⃗E · d⃗S = QD. +However it is clear that once we make sure that the initial data in D′ are compatible +with the Gauss constraint from the total charge QD enclosed in D, there are many ways of +rearranging the initial data in region D keeping those in D′ fixed. In other words there are +deformations of the constraint equation (2.5), which are not gauge-equivalent, and which have +compact support localized in D. This means that theory under consideration obeys condition +A. +Moving on to the quantum theory, we can consider U(1) gauge theory weakly coupled to +matter. As in the classical theory the total charge Q enclosed in a region can be measured on +its boundary and the total charge of the entire state can be measured at space-like infinity. +At the quantum level we can get information not only about the expectation value of the +charge but all the higher moments +⟨Ψ|Qn|Ψ⟩ +, +n = 1, 2, .... +(2.6) +To proceed it is useful to consider observables in this theory. Physical observables must be +gauge invariant. In a U(1) gauge theory there are several examples of such observables which +are also local, for example local operators constructed out of Fµν(x) or φ∗(x)φ(x). Other inter- +esting gauge invariant operators which are not completely local, but can be contained in com- +pact regions are closed Wilson loops eig +� +C Aµdxµ or bilocals of the form φ∗(x)eig +� y +C,x Aµdxµφ(y). +– 11 – + +All these operators are neutral and do not change the electric charge of the region D, if they +are entirely contained in D. We can use such operators localized in region D to construct +unitaries UD which can be used to modify the state inside D leaving all correlators outside +invariant, as in (2.1). So information can be localized in this theory if we work with neutral +operators. +But what if we want to create an excitation in region D which has non-zero charge? +We already know from the classical problem that it will not be possible to add a charge in +D without affecting the exterior due to Gauss law (2.5). The same is true at the quantum +level. A charged operator φ in D is not gauge invariant. It can be made gauge invariant +by dressing it with a Wilson line extending all the way to infinity. +We can think of the +Wilson line as a localized tube of electric flux ensuring that Gauss law is satisfied. It may +be energetically better to smear the Wilson line in a spherically symmetric configuration. +The important point is that the dressed operator Φ(x) = eig +� x +∞ Aµdxµφ(x) is no longer a local +operator, though it is gauge invariant. If we act with a unitary made out of this operator, +we will modify correlators outside D and (2.1) will fail. This means that the addition of the +charge in D can be detected immediately outside. This is not surprising, as the same thing +is already visible at the classical level. +However, looking a bit more carefully, we run into certain somewhat surprising features +of the quantum theory. Suppose we have several charged fields φi, labeled by a flavor index +i, with the same electric charge. We construct the corresponding dressed operators Φi(x) = +eig +� x +∞ Aµdxµφi(x), using some particular prescription for the Wilson line. These obey +[Q, Φi(x)] = g Φi(x) , +(2.7) +where Q = +� +S2∞ ∗F is the charge operator which can be measured at space-like infinity. +Suppose the point x = 0 is inside D. We create a charged excitation of type i in region D +by acting on |0⟩ with a unitary Ui = eiϵΦi(0). Then we study correlators in region D′ in the +state Ui|0⟩ in perturbation theory. Consider a correlator of Q and Φj(x) in region D′. +⟨0|U † +i Φj(x)QUi|0⟩ = ⟨0|Φj(x)|0⟩ + iϵ⟨0|[Φj(x)Q, Φi(0)]|0⟩ + O(ϵ2) , +(2.8) +where to leading order in the perturbative expansion the second term is +⟨0|[Φj(x)Q, Φi(0)]|0⟩ = g⟨0|Φj(x)Φi(0)|0⟩ ∝ δij . +(2.9) +Hence by measuring correlators of all φj(x) and Q in D it seems that we can detect not only +the presence of a charge in D, which is expected by Gauss’s law, but we can even identify +the flavor of the charged particle, i.e. the value of the index i in the interior of D. A similar +argument in the gravitational case was discussed in [25, 48] for black hole states and in [58] +around empty AdS. +The reason we were able to get information beyond the total charge in D is that in +the vacuum the fields have non-trivial entanglement, on which the non-vanishing 2-point +function (2.9) depends. When we act with the unitary containing the Wilson line, the Wilson +– 12 – + +line disturbs the pattern of entanglement in such a way that it breaks the symmetry between +the fields φi and we can detect from D′ the flavor of the excitation in D +This suggests a way to avoid the issue and succeed in hiding the flavor of charge in D. +We start with the analogue of a split state in the U(1) gauge theory, see the discussion in [59], +and then create the charged excitation in D by acting with the same unitary. In that case +there is no entanglement bewtween D and D′ and hence (2.9) will vanish making it impossible +to tell from measurements in D′ what is the type of charged excitation in D.12 This requires +creating the charged excitation on top of the split state, with typical energy scaling like (2.3), +rather than the ground state. +2.3 +Classical and Quantum Gravity +First we notice that in non-perturbative quantum gravity we do not expect to be able to +localize information in space: holography and AdS/CFT suggest that the fundamental degrees +of freedom in quantum gravity are not local, but rather lie at the boundary. Moreover there +is strong evidence that an ingredient towards the resolution of the black hole information +paradox is that the naive factorization of the Hilbert space in space-like separated subregions +may not be true in the underlying theory of quantum gravity. +On the other hand at the classical level in General Relativity we do have an exact notion of +locality and information can be localized, as we will discuss below. An interesting question, +which is the main focus of this paper, is to understand the fate of locality at the level of +perturbative quantum gravity. +2.3.1 +On the initial value problem of general relativity +In General Relativity the initial value problem is formulated by starting with a spacelike +slice Σ and specifying the data C = (hab, Kab) where hab is the intrinsic metric and Kab the +extrinsic curvature of Σ. If we have matter then the values of the fields and their normal +derivatives need to be specified. Initial data related by spatial diffeomorphisms on the slice +Σ are gauge-equivalent and have to be physically identified. In general relativity there is +one more subtlety: even if we have two initial data on the slice Σ which are not related by a +spatial diffeomorphism, they may still correspond to the same physical solution in space-time. +This is related to the freedom of choosing the initial slice Σ in space-time and diffeomorphism +invariance in full space-time. +Admissible initial data, which can be extended into a solution of the Einstein equations +must obey the following constraints +R + (Ka +a)2 − KabKab = 16πGρ +(2.10) +∇aKab − ∇bKc +c = −8πGJb , +(2.11) +12A more mundane way to hide the charge is to add ”screening charges” in the buffer region, but here we +want to discuss how information can be localized even though a Wilson line extends all the way to infinity. +– 13 – + +where R is the Ricci scalar of hab on Σ, the covariant derivatives are with respect to hab on +Σ, na is the unit normal to Σ and ρ = Tabnanb and Jb = −hc +bTcana. +We now want to address the question of localization of information in classical general +relativity, as formulated in subsection 2.1. A theorem, see for example [49,60], settles question +B for pure general relativity: if we have two admissible initial data which agree, up to spatial +diffeomorphism, on a part D′ of Σ, then the corresponding solutions will agree, up to a space- +time diff, on the development of D′. This continues to be true in the presence of matter +provided certain reasonable conditions are satisfied. This shows that in general relativity +signals propagate at most at the speed of light: if we modify the initial data only in the +region D, then the signals will propagate in the causal future of D. +Then we come to question A, that of localizing information on compact regions on Σ: to +what extent is it possible to find two initial data satisfying the constraints (2.10), (2.11), which +agree on D′ but differ on D?13 The equations (2.10) and (2.11) are non-linear and of elliptic +nature, though underdetermined. +Understanding the space of solutions of the constraint +equations is an interesting problem which has been studied extensively in the literature. Here +we summarize some relevant points: +1. Gravitational Gauss law: in asymptotically flat or AdS space-times, the energy and +other conserved charges are defined at space-like infinity. The constraints of general rel- +ativity relate these asymptotic charges to contributions from excitations in the interior +of space-time. For example, in the Newtonian limit the constraint equations reduce to +the gravitational analogue of Gauss’s law +□φ = 4πGρ . +As in electromagnetism this implies that the initial data in region D′ know about the +total mass enclosed in D. +2. Existence of localized deformations: it is possible to find many solutions of the +constraint equations which look the same in the domain D′ but differ on D. For example, +if we restrict our attention to spherically symmetric solutions, Birkhoff’s theorem implies +that there is a large number of solutions of (2.10) and (2.11) which all look like the +Schwarzschild metric of mass M in D′ but differ in D. Examples include static, interior, +star-like geometries supported by matter or more generally spherically symmetric, time- +dependent collapsing geometries of mass M. More generally, it has been shown [61] that +under reasonable conditions a compact patch D of a solution of the constraints (2.10) +and (2.11) can be glued to a boosted, Kerr solution in D′ of appropriate mass, angular +momentum, momentum and center of mass position. The existence of a large number +of solutions, which all look exactly the same in D′ demonstrates that it is possible to +localize information in classical general relativity. +13Here we need to keep in mind that even if the initial data differ on D they may correspond to the same +solution in space-time, as they may correspond to two different choices of the slice Σ in the same space-time +solution. +– 14 – + +3. Comments on the vacuum: For asymptotically AdS geometries, if a solution looks +like empty AdS in D′14 then it is guaranteed to be empty AdS in D as well. In other +words, starting with the vacuum it is not possible to modify the initial data in D into +a new solution, without at the same time modifying the solution in D′. +2.3.2 +Diff-invariant observables in classical GR +We now consider the question of defining local diff-invariant observables in gravity. This is a +long-standing problem which is subtle even at the classical level. Let us consider general rel- +ativity, possibly coupled to other fields, defined with certain asymptotic boundary conditions +at infinity (for example asymptotically flat or AdS) or on a closed manifold of fixed topology. +We denote by X the space of solutions of the equations of motion, in any possible coordinate +system. On this space we have the action of the group Diff of diffeomorphisms15. Solutions +related by a diffeomorphism are physically identified and we introduce +X = X/Diff . +(2.12) +We can think of a diff-invariant observable as a function which has definite values on points of +X. However, we do not demand an observable to be necessarily defined on the entire space of +solutions X. Instead we will allow observables to possibly have a limited domain of definition. +Hence a diff-invariant observable is a map +A : U ⊂ X → R , +(2.13) +where U is an open subset of X. Such observables can also be expressed as functions on X +which must obey A(s) = A(f∗s), where s denotes a solution in some coordinate system and +f∗ the action of a diffeomorphism. +In order for a diff-invariant observable to be local we need to impose additional conditions. +To formulate these conditions it is useful to introduce the Peierls bracket {A, B} between two +diff-invariant observables [62], which is a covariant generalization of the Poisson bracket. To +compute the value of {A, B} we consider a modification of the action as S → S + ϵA and +compute the difference of the first order change of observable B on the perturbed solutions +with advanced (+) and retarded (−) boundary conditions. The Peierls bracket is defined as16 +{A, B} = δ− +AB − δ+ +AB . +(2.14) +It can be shown that the Peierls bracket has similar properties as the Poisson bracket, for +example linearity, antisymmetry and the Jacobi identity, and in fact coincides with the Poisson +bracket if a Hamiltonian formalism is introduced. One of the advantages of the Peierls bracket +14Here we assume that D is compact so D′ includes the region near space-like infinity. +15If the space-time is non-compact along space we only consider small diffeomorphism, i.e. those which +become trivial fast enough at infinity. +16The first order solutions are not unique due to diffeomorphism invariance, however the ambiguity drops +out when computing the change of the diff-invariant observable B. +– 15 – + +is that we do not need to pass to the Hamiltonian formalism which is somewhat complicated +due to the constraints. Notice that to define the Peierls bracket of two observables A, B they +must have a common domain of definition on X and the bracket will be generally a non-trivial +function on this overlap. +We would like to define diff-invariant observables which can be associated to points in +space-time with the property that if two such observables are associated to space-like sep- +arated points the corresponding Peierls bracket must vanish. The difficulty in doing this is +that in order to define an observable we need to define it at least in an open neighborhood +around a state as in (2.13), so we need some prescription for following ”the same point”, on +which the candidate diff-invariant observable will be localized, as we move on the space of +solutions X. General covariance implies that there is no canonical way to keep track of the +point as we change the state. +If the space-time has a well-defined boundary we can find prescriptions which select a +point in space-time for each solution in X relationally with respect to the boundary. For +example in AdS we can define a diff-invariant observable which seems to be localized at a +point by considering a radial geodesic at right angle from a specific point on the boundary, +moving a fixed regularized distance along it and measuring the value of a scalar quantity, for +example a scalar field or a scalar combination of the curvature, at the resulting point. This +gives a map from the space of solutions X to R, so it is a diff-invariant observable which +could potentially be local. Notice however that the location of the resulting point depends +on the entire geometry along the geodesic, all the way from the boundary. Changing the +metric anywhere along this geodesic will move the resulting point. Hence the value of the +observable will not strictly depend on local data near the point. Similarly, if we act with one +of the asymptotic symmetries the boundary starting point will move and also the resulting +bulk point will move. This implies that the Peierls brackets of this candidate observable with +the boundary charges, or other observables along the geodesic will be non-zero, even though +these regions are space-like separated. Hence this relational observable is not really local. +Another way to to define candidate local diff-invariant observables is to consider a com- +plete gauge fixing scheme. +Then observables in the particular gauge labeled by a space- +time coordinates are automatically diff-invariant. However they will generally have non-local +Peierls brackets, since the assignment of a coordinate value to a point in space-time in the +particular gauge, will generally depend on the solution everywhere. +Additional difficulties arise in space-times without boundaries, for example in de Sitter +space. A boundary is an (asymptotic) part of the spacetime where gravity is not dynamical +anymore. +This is why we can for example anchor geodesics to the boundary, and define +relational diff-invariant observables. Without a boundary, there is no part of the space-time +where gravity is turned off, and consequently no place to anchor geodesics. +2.3.3 +State-dressed observables +If we consider a solution that is sufficiently complicated it is possible to specify points, and +hence define local diff-invariant observables, by using features of the state. We emphasize that +– 16 – + +these observables will not have all the desired properties over the entire space of solutions +X, so these observables have certain aspects of state-dependence as discussed around (2.13). +One approach based on this idea was studied by DeWitt [16], building on [14, 15]. For a +D-dimensional space-time we start by identifying D scalar quantities Za, a = 1, ..., D. These +can be combinations of curvature invariants and other scalars formed by the fields of the +theory. We could try to fix a coordinate system by using these D-scalars as coordinates. We +can use this intuition to introduce candidate local diff-invariant observables of the form +φ(Za +0) = +� +dDx φ(x) δD(Za − Za +0) det∂Z +∂x . +(2.15) +Here Za are the D scalar quantities introduced above and φ is any other scalar combination of +the fundamental fields of the theory. Similar constructions can be done for fields with tensor +indices. +Some comments are in order: +1. For a general space-time which is in-homogenous, and for certain choices of the values +Za +0, the delta function in (2.15) will click on a finite number of points, so the quantity +above is well-defined and finite. In symmetric space-times it will either not click at all, +hence the observable will be zero, or an infinite number of times so the observable will +be ill-defined. This shows that (2.15) is a quantity which is defined only on part of the +phase space. This is in accordance with our expectation that state-dressed observables +have to be state-dependent (2.13). +2. Suppose that the observable (2.15) is well defined on a state s and a neighborhood U +of the space of solutions X around it. It is clear that, at least at the classical level, this +observable is diff-invariant, i.e. a well defined map φ(Za +0) : U ⊂ X → R and hence a +good observable according to the definition (2.13). +3. One can show that under certain conditions, observables (2.15) are also local. If we +have a state s on which two such observables φ(Za +A), φ(Zb +B) are well defined, with the +property that the delta functions click at single points A, B and that these points are +space-like separated with respect to the metric of s, then the corresponding observables +have vanishing Peierls brackets {φ(Za +A), φ(Zb +B)} = 0, see [63] for a review. This follows +from the causality properties of linearized Green’s functions appearing in (2.14) around +the solution s. Notice that if two points A, B are spacelike separated on a solution +s, then there is a small enough neighborhood of s in which they remain space-like +separated. Hence their Peierls bracket will vanish in this entire neighborhood. +4. This shows that, as long as we accept that observables may be defined only locally on +the phase space of solutions, it is possible to find local, diff-invariant observables in +classical general relativity around states which are complicated enough. These are also +the interesting states, i.e. those containing bulk observers who want to study physics +in their environment. +– 17 – + +5. Similar ideas are useful in cosmology, where the value of a scalar field can be used as +as clock [64–66]. +The next question is whether it is possible to define similar observables at the quantum +level. Aspects of this question were discussed in [17] and [18], where it was argued that there +is a quantum version of these observables which retain their locality properties to all orders +in the ℏ expansion, even though they are not expected to be local at the non-perturbative +level. Various difficulties are encountered at the quantum level including the question of the +renormalization of the composite operators (2.15), establishing diffeomorphism invariance at +the quantum level and the role of Poincare recurrences which will generally introduce infinite +copies where the delta function will have support [18]. In this paper we provide support +in favor of this conjecture by finding observables with certain similarities in spirit to (2.15) +directly in CFT language. This has the advantage that any object built directly in the CFT +is by construction diff-invariant. +2.3.4 +A time-band in AdS +We now specialize to a setup that will allow us to make contact with AdS/CFT. We consider +geometries that are asymptotically AdSd+1 and we consider a short time-band T−ϵ,ϵ on the +boundary in global coordinates, defined as the set of points (−ϵ, +ϵ)×Sd−1 , ϵ > 0, where the +first interval refers to the time coordinate t. Near the boundary we can select a Fefferman- +Graham coordinate system where the fields, for example the metric and a scalar of mass m2, +have the behavior +ds2 = dr2 +r2 +r2(−dt2 +dΩ2 +d−1)+r2−dgµν(r, x) dxµdxν +gµν(r, x) = g(0) +µν (x)+g(2) +µν (x)r−2 +... +φ = r−∆(φ(0)(x) + φ(2)(x)r−2 + ...) , +(2.16) +where x = (t, Ωd−1) and ∆ = d +2 + +� +d2 +4 + m2. Here we consider normalizable states so the +growing modes, which would be dual to sources in the CFT, are set to zero17. The Fefferman- +Graham coefficients g(0) +µν (x), φ(0)(x) are diff-invariant observables and are labelled by boundary +coordinates18. This set of observables includes the asymptotic charges, for example the ADM +Hamiltonian can be computed as +H = +1 +const +� +Sd−1 dΩd−1g(0) +00 (x) . +(2.17) +We focus on these Fefferman-Graham observables restricted in the time band T−ϵ,ϵ. This set +of observables is closed under Peierls brackets and form a Poisson algebra A. Notice that in +this algebra we do not include observables which would be finite distance under Poisson flow, +17We only assume that the sources are zero in the time band T , they could be turned on in the far past in +order to prepare a state. +18The subleading coefficients are fixed by the equations of motion in terms of the leading ones. +– 18 – + +otherwise flowing by finite distance with H would take us out of the time-band, see also the +discussion in [67]. +Starting with the classical theory, we ask whether we can find observables localized deep +in the interior of AdS which are space-like with respect to the time-band and which have +vanishing Peierls brackets with observables in the time-band algebra A. These candidate +observables are to be defined as in (2.13), in particular they need to be defined on a neigh- +borhood U ⊂ X of a solution s ∈ U and not necessarily on the entire space of solutions +X. +It is clear that observables defined relationally with respect to the boundary, or with +a gauge fixing condition which makes use of the boundary, do not satisfy these conditions. +Due to their gravitational Wilson lines they will have non-vanishing Peierls brackets with the +Hamiltonian and other charges on the boundary [44,45]. Such observables generally change +the energy of the state, which due to the gravitational Gauss law can be measured in the +time band T−ϵ,ϵ by (2.17). Another point of view is that such observables identify a point +in the bulk, and in particular a moment in time, relationally with respect to the boundary. +Thus an infinitesimal motion in time of the starting point on the boundary is translated via +the relational prescription into an infinitesimal time motion of the corresponding bulk point. +Then the Peierls bracket of the candidate bulk observable with H generates time-derivatives +of the point in the bulk and is non-vanishing. +The discussion of the previous subsection implies that if we start with an asymptotically +AdSd+1 solution s of the bulk equations which is complicated enough, then we can define +diff-invariant observables of the form (2.15) in a neighborhood of s so that they have vanish- +ing Peierls bracket with all elements of the time-band algebra A including charges like the +Hamiltonian (2.17). Such observables do not change the total energy of the state but instead +they rearrange the energy, ”absorbing” from the background solution the amount of energy +they themselves create. These observables select a point in the bulk, and a moment in time, +by using features of the state. +In what follows we will provide evidence that the same conclusions are true in perturbative +quantum gravity. We will proceed by translating the question in CFT language and using +the AdS/CFT correspondence. +3 +Holographic setup +In this paper, we will study the question of locality in quantum gravity in the context of +the AdS/CFT correspondence. A question we would like to understand is how certain bulk +subregions are encoded in the boundary CFT. There are cases where this is well understood. +For example, the bulk dual of a boundary subregion is known as the entanglement wedge, +which is the bulk region extending between the boundary subregions and the relevant Ryu- +Takayanagi surface extending in the bulk [68]. This correspondence between parts of the +boundary and bulk is known as subregion-subregion duality [42,69,70], and it is worthwhile +to mention that in general, the entanglement wedge of a boundary subregion is much larger +– 19 – + +than its causal wedge (the part of the bulk contained by lightrays shot from the causal +developments of the boundary subregion). +Subregion-subregion duality and entanglement wedge reconstruction utilizes the organi- +zation and entanglement of CFT degrees of freedom organized spatially. We will be interested +in rather different bulk subregions, which lie deep down in the bulk and never extend to the +boundary CFT. What is the CFT dual of a causal diamond located deep near the center of +AdS? The answer to this question remains elusive, and in particular it is understood that +in general, these bulk regions do not correspond to the entanglement wedge of any bound- +ary subregion. There have been previous attempts to understand the CFT mapping of such +regions, see for example [21–24] which attempt to assign a meaning to the entropy of a gen- +eral closed codimension-2 spatial curve in AdS. Here we will follow a different approach by +focusing on the algebra of single-trace operators [20]. +We will start by reviewing some basic but relevant features of AdS/CFT, before turning +to a discussion of the class of states that we will be considering throughout this paper and +their salient properties. +3.1 +Gravitional states in AdS, large diffeomorphisms and asymptotic symme- +tries +We will be interested in gravitational solutions which are asymptotically AdSd+1. We have in +mind an embedding in a top-down setup with a holographic dual CFT, like N = 4 SYM at +strong coupling, on S3 × R and the N-scaling we indicate in most of the paper refers to this +theory. However for most of the discussion the details of the embedding in string theory, the +extra fields, as well as the presence of a compact internal manifold are not important unless +explicitly stated. +Solutions to the bulk equations of motion can be thought of as states in the dual CFT. If +we think of a bulk geometry described by a Penrose diagram, the diagram really represents +the entire time-history of the state. We can take the state to live at t = 0 on a boundary +Cauchy slice, and the portion of the geometry relevant to describing the state is an initial +data surface given by a bulk Cauchy slice (or the Wheeler-de Witt patch associated to the +boundary Cauchy slice). To view these geometries as states of the dual CFT, it is important +that the bulk fields have a fall-off corresponding to normalizable modes with vanishing CFT +sources.19 +We want to consider semi-classical solutions with non-trivial bulk geometries, i.e. where +backreaction is strong. The corresponding CFT states |Ψ0⟩, which we take to be pure, have +large energies which scale as +⟨Ψ0| H|Ψ0⟩ ∼ O(N2) , +(3.1) +and as we will see, they will generally also have an energy variance of the same order. We +will also consider perturbative excitations of the quantum fields on top of the background +19If these states are prepared by a Euclidean path-integral [71–74], sources can be turned on in the Euclidean +past which prepares the state, but it is important that they vanish as tE → 0 for the geometries to be interpreted +as states in the undeformed CFT. +– 20 – + +geometry. These excitations add/subtract quantum particles which change the energy by an +O(N0) amount, and whose backreaction on the geometry is thus generally small. +Geometries of this type will often be macroscopically time-dependent, such that the +initial data on a bulk Cauchy slice changes as we perform time-evolution of the state. This +has consequences for the variance of the energy, as we will now see. Any state |Ψ0⟩ can be +expanded in the basis of CFT energy eigenstates as +|Ψ0⟩ = +� +i +ci|Ei⟩ . +(3.2) +The time-dependence of the bulk geometry implies that such states will have energy variance +(∆H)2 ≡ ⟨Ψ0|H2|Ψ0⟩ − ⟨Ψ0|H|Ψ0⟩2 ∼ O(N2) . +(3.3) +To see this, consider the inequality +1 +2|⟨[H, A]⟩| = 1 +2 ⟨∂tA⟩ ≤ ∆H · ∆A , +(3.4) +where in the first equality we assumed that the operator A is not explicitly time-dependent. +Then we have +∆H ≥ 1 +2 +⟨∂tA⟩ +∆A ∼ O(N) , +(3.5) +where we have used large N factorization for the operator A. +This shows that provided +there is macroscopic time-dependence (the classical vev of A changes at leading order), the +variance of the energy scales at least as N2.20 Some bulk geometries we will consider are +macroscopically time-dependent, but only inside the horizon. In this case, we cannot use the +argument above, but we still expect the variance to be of order N2. It is interesting to ask +whether the variance is a quantity that can be extracted from the semi-classical geometry +alone. In general, we expect that the quantum state of the fields in the bulk is important as +well. We discuss this further in Appendix A. +There are various types of explicit constructions of states of this kind. There are states +prepared by Euclidean path integral with sources for single-trace operators [71–74]. These +states should be interpreted as coherent states of the quantum gravitational dual, which +are labelled by phase-space points corresponding to initial data21. +There are also states +prepared by a boundary state of the CFT, further evolved by some amount of Euclidean +time [76–79]. The bulk interpretation of these states is that they correspond to black hole +geometries with End-of-the-World branes sitting behind the horizon. +This is an example +where the bulk geometry is macroscopically time-dependent, but only behind the horizon. +Similarly, for two-dimensional CFTs, we can construct pure states by performing the path +20Note that if the variance is parametrically larger than O(N 2), the state may no longer have a good +semi-classical interpretation. An example would be a superposition of black holes of different masses. +21It appears that one may not construct arbitrary initial data this way, see [75]. This will not affect our +construction and for states prepared by a Euclidean path integral, we should simply keep in mind that we +have access to a restricted class of initial data. +– 21 – + +integral over a surface of higher topology, for example half a genus-2 surface, see [80]. These +geometries are also macroscopically time-dependent behind the horizon, but instead of having +a brane behind the horizon, they have topology. Finally, it is worth noting that there are +semi-classical geometries that also preserve supersymmetry, the most famous of which are the +LLM geometries [81]. In these cases, one can obtain a better understanding of the dual CFT +states. We will come back to these geometries in section 6. +As usual in gravity, we should identify solutions which are related by small diffeomor- +phisms, i.e. diffeomorphisms that vanish near the AdS boundary. There is also a class of +large diffeomorphisms, which are compatible with the boundary conditions imposed in the +definition of our theory of AdS gravity. This set of diffeomorphisms forms what is called +the asymptotic symmetry group. In the case of AdSd+1, d ≥ 3 this is the conformal group +SO(2, d), while for d = 2 it gets enhanced to the Virasoro group [82]. When acting on a given +bulk solution these large diffeomorphisms will generally transform the geometry into a new +state, which is physically distinguished from the previous one, unless of course the original +state happens to be invariant under the symmetry. We will later also discuss solutions with +two asymptotic boundaries, such as the eternal black hole in AdS, in which case the asymp- +totic symmetry group is larger. Let us now discuss the various elements of the asymptotic +group/conformal group: +• Time translations: One particular class of states we will discuss are those with semi- +classical time-dependence in the bulk, for example a state corresponding to the gravita- +tional collapse of a star. In this case large diffeomorphisms corresponding to asymptotic +time translations transform the state as |Ψ0⟩ → e−iHt|Ψ0⟩. The initial data correspond- +ing to |Ψ0⟩ is not the same as that of e−iHt|Ψ0⟩. Our end goal will be to provide local +operators whose gravitational dressing is done towards a feature of the state. If the +state is time-dependent then we can select a moment in time by using the features of +the state, as opposed to the boundary time coordinate. On the other hand if the state +is static, then the only way to identify a moment in time is by dressing to the boundary. +This is why it will be important for us to consider time-dependent states. +• SO(d) rotations: If the state breaks SO(d), then asymptotic rotations transform it +to a new state. In this case we can use the features of the state to identify the angular +location of a point. On the other hand, if the state is SO(d) invariant it will generally +not be possible and at best we can obtain an operator smeared over the bulk angular +coordinates, or alternatively we can fix the angular location by dressing to the boundary. +• AdS boosts: The Lorentzian conformal group acting on Sd−1 × R has another 2d +generators which correspond to boosts in various directions. +These can be realized +as d non-independent copies of an SL(2, R) algebra, see for example [83]. Any state +with finite energy cannot be annihilated by Hermitian combinations of these generators, +which we show in Appendix B. The only state which is annihilated by these generators +is the global vacuum and any other state will necessarily transform under the action of +– 22 – + +these boosts22. Therefore, in any non-trivial state, we can fix the radial position of an +operator without referring to the boundary. +In a top-down setup, the gravity dual may have an internal manifold, like the S5 in the +context of N = 4 SYM. In such cases, we would need to break the R-symmetry to localize +a bulk operator in the internal space. In this paper, we will mostly restrict to a bottom up +construction without an internal manifold but it would be an interesting generalization. +3.2 +Locality in AdS +We are now ready to discuss locality in quantum gravity with asymptotically AdS boundary +conditions. We would like to understand whether one can define local observables and whether +we can localize information deep in the center of the AdS. +The presence of the AdS boundary allows us to define one natural class of diff-invariant +observables: The fields in AdS can be expanded in a Fefferman-Graham expansion. +The +coefficients of this expansion are themselves diff-invariant observables, which are dressed to +the boundary since the Fefferman-Graham gauge is chosen with respect to the boundary. +Let us call these observables FG-observables. +For example, the AdM Hamiltonian is one +particular observable in this class. In perturbative quantum gravity, we can also consider +the expectation values of these observables as well as their higher-point correlation functions. +As we will discuss below, if we want to stay within the regime which can be described by +semi-classical gravity we may need to restrict the complexity of the correlators (for example +the number of operator insertions in the correlation function). We emphasize again that all +these observables are dressed with respect to the boundary. In particular, they will generally +not commute with the Hamiltonian or the other charges described in the previous section. +The question we would like to address is the following. If we start with a state with a +semi-classical geometric description, is there a way to modify the state in the interior of AdS, +without modifying any of the correlators of FG-observables localized in a short time-band of +the boundary? If the answer is yes, this means we can localize information since an observer +living near the boundary will have no way to know whether or not we modified the state. +Rather than trying to come up with bulk objects that achieve this goal, we will address this +question directly in the dual CFT. This has the following advantage: any object built out +of CFT degrees of freedom is necessarily diff-invariant and non-perturbatively well defined. +Provided the object acts in the right away, we can be assured that the construction is fully +consistent. +3.3 +The CFT description and the time band algebra +Consider a large N holographic CFT which is dual to semi-classical general relativity coupled +to matter fields. In the large N limit, we can define the algebra A generated by single-trace +operators in a time-band Dt1,t2, where we allow products of single-trace operators where the +22States with infinite energy like the AdS-Rindler vacuum could also potentially be annihilated by some +boost generators. +– 23 – + +number of factors is arbitrary but scales like O(N0).23 This was originally discussed in [20], +inspired by the earlier work [6, 25, 26]. In [20] it was proposed that the algebra A can be +thought of as being dual to the causal wedge of the region Dt1,t2 in the bulk (see Fig. 1). +This picture also suggests that the algebra A has a commutant which can be idenfitied with +a spacelike-separated causal diamond in the interior. Algebras of this type have received +attention recently [27–31]. +The work [20] studied this setup for states which are small perturbations around the +AdS vacuum. The geometry of AdS is homogeneous and featureless since it is a maximally +symmetric space. As already discussed in the previous section, this makes the definition of +local diff-invariant observables challenging. We would like to revisit the time-band algebra, +this time in cases where the bulk state has features, which in particular are time-dependent. +This means the state must be highly excited as can be seen for example from its energy (3.1). +At infinite N the problem can be understood in terms of QFT on a curved and in general +time-dependent background. In particular, gravitational backreaction of the quantum fields +can be ignored and one does not need to talk about gravitational dressing, which is a form +of backreaction. In this case, the existence of the commutant is obvious because we are in a +QFT situation. Note that if the Hamiltonian (which is always an element of the time band +algebra) is normalized appropriately24, its commutator with the other single-trace operators +is suppressed by 1/N and thus vanishes when N is infinite. +At the level of 1/N corrections, the existence of the commutant is less obvious. Backreac- +tion must now be taken into account and the gravitational Gauss law can spoil the commutator +between H and the other operators of the time-band algebra. For example, the standard way +to write bulk fields in terms of CFT operators is the HKLL construction [33–39] +Φ(t, r, Ω) = +� +bdry +dt′ dΩ′ +d−1K(t, r, Ω; t′, Ω′)O(t′, Ω′) , +(3.6) +where K is related to a Green’s function of the Klein-Gordon operator on the appropriate bulk +geometry. This operator is defined purely within the CFT so it is manifestly diff-invariant. +To leading order at large N, it acts as a bulk field and commutes with other bulk fields at +spacelike separation. Notice however that in order to define the kernel K we have to choose +a coordinate system in the bulk, which often is taken using Fefferman-Graham gauge. As we +already mentioned, this gauge choice is defined by making use of the asymptotic boundary, +and an HKLL operator is thus dressed to the boundary. Because of this, the commutator +between an HKLL operator and the Hamiltonian will not vanish at subleading orders in the +1/N expansion. +The physical origin of this effect is the gravitational Gauss law: acting with (3.6) will +generally create or destroy a particle in the bulk, thus changing the energy of the state, which +23Notice that at finite N the algebra in a time-band would be the same as the full algebra. In the large +N limit, a natural hierarchy emerges between ”small products” of single-trace operators and the rest of the +algebra, which allows us to consider the notion of a time-band algebra. +24A useful normalization is h = +1 +N (H − ⟨Ψ0|H|Ψ0⟩), which ensures that ⟨Ψ0|h2|Ψ0⟩ ∼ O(N 0). +– 24 – + +can be immediately measured at spacelike infinity by H. One can try to correct the HKLL +operators at higher orders in 1/N by mixing it with other single- and multi-trace operators, +see [39, 84, 85], but the commutator with the Hamiltonian is universal and generally cannot +be removed in this way. It is also possible to think about the dressing in terms of (smeared) +gravitational Wilson lines connecting the bulk operator to the boundary, which make it diff- +invariant at the price of making it non-local [86–89]. The commutator with H is nonzero +because H picks up the contribution of the Wilson line. +This raises the question of whether the algebra A still has a commutant at subleading +orders in 1/N. The main goal of this paper is to provide evidence for the existence of such a +commutant. We will do so by identifying a class of operators that are gravitationally dressed +with respect to features of the state, rather than dressed to the boundary. In particular, these +operators will have vanishing commutators with the Hamiltonian, to all orders in 1/N. In this +paper, we will focus mostly on ensuring that bulk operators have a vanishing commutator +with the Hamiltonian (and the other charges), but it would be important to extend our +construction to all single-trace operators in Dt1,t2. We given an alternative argument for the +existence of a commutatant to all orders in 1/N in section 5. +The existence of a commutant for A in 1/N perturbation theory would imply that in- +formation can be localized in regions of the bulk and is not visible from the boundary at the +level of perturbative quantum gravity25. We are now ready to formulate the concrete goal +that we will achieve in this paper. +3.4 +Formulating the main goal +Our goal is to improve the locality properties of (3.6) by moving the gravitational dressing +from the boundary to the state. From a technical point of view, we will find CFT operators +�Φ which obey two properties: +1. [Qi, �Φ] = 0 to all orders in 1/N, for all asymptotic charges Qi ∈ SO(2, d). +2. The correlators of �Φ agree with those of ΦHKLL to leading order in the large N expansion, +on the code subspace of |Ψ0⟩. +In taking the large N limit it is important to track how various effects scale with N. As we +will see, our new operators �Φ have vanishing commutator with Qi to all orders in the 1/N +expansion, but have a non-vanishing commutator at the level of e−N2 corrections. +In what follows we will first focus on ensuring a vanishing commutator of ˆΦ with the +Hamiltonian H to all orders in 1/N and then discuss the generalization to the other charges +in SO(2, d). +As we will see, our construction will not work for |Ψ0⟩ = |0⟩. Technically, this is because +the vacuum does not comply with the properties (3.1) and (3.3). Physically, it is because +25See [46,58,59,67,90–94] for other discussions of localization of information in perturbative quantum gravity, +with varying conclusions. +– 25 – + +the AdS vacuum has no feature that we can use to attach the dressing of our local operator. +Note that this is in line with the results of [46], where a protocol to reconstruct the bulk state +from correlators in the time-band was discussed. +3.5 +Time-shifted states and return probability +We will now present the main technical tool that will enable us to define state-dressed opera- +tors: the return probability. Let us start with a state |Ψ0⟩ satisfying the properties (3.1) and +(3.3). We define the following one-parameter family of states +|ΨT ⟩ = e−iTH|Ψ0⟩ +T ∈ R . +(3.7) +In the bulk, the states |ΨT ⟩ are related to |Ψ0⟩ by a large diffeomorphism, i.e. one that does +not vanish near the boundary and induces a boundary time-translation. It is important to +emphasize that they are different quantum states, even though they are related by a symmetry. +If we think about the phase space of gravity in AdS, the family of states correspond to different +phase space points, just like a particle moves on phase space as a function of time in classical +mechanics. From the bulk perspective, if |Ψ0⟩ was a coherent state, we can also think of |ΨT ⟩ +as coherent states. +We would now like to consider the overlap of such states. In particular, we would like to +study the overlap +⟨Ψ0|ΨT ⟩ . +(3.8) +Thinking of these states as coherent states is useful to gain intuition about such overlaps. For +the simple harmonic oscillator, the overlap of two coherent states is +⟨α|β⟩ = e− 1 +ℏ f(α,β) , +(3.9) +for a very simple quadratic function f. For states on the gravitational phase space, recalling +that ℏ ∼ GN ∼ 1/N 2, we thus expect +⟨Ψ0|ΨT ⟩ = e−N2f0(T) , +(3.10) +for a function f0 whose real part is positive. In the gravitational setting, it is not straight- +forward to directly compute f0(T) from the phase space information, see [47] for a discussion +on nearby states. There is a general way to compute f0(T) based on a Euclidean preparation +of the states [74], but it requires some effort (in particular solving the non-linear Einstein +equations). The computation of f0(T) directly from the information on an initial data slice, +which specifies the point on phase-space, is an interesting problem.26 +It is also instructive to think about the overlap from a microscopic point of view. In the +CFT, the overlap is given by +⟨Ψ0|ΨT ⟩ = +� +i +|ci|2e−iTEi . +(3.11) +26Similarly, we do not know of a gravitational argument that guarantees that the real part of f0(T) is +positive, which must be the case if the geometries have a state interpretation in the dual CFT. We comment +on this further in the discussion. +– 26 – + +Note that there are eS(E) terms here, each of size e−S(E). +The suppression (3.10) must +therefore come from the summation over a large number of phases. +If the bulk state has no periodicities in time, we expect the real part of f0(T) to increase +as we increase T. However, this increase will not continue forever. We will shortly give an +estimate of the time-average of (3.11), and argue that the decay will saturate at some point. +Physically, the non-trivial overlaps (3.11) imply that it is not correct to think that all the states +|ΨT ⟩ are independent, see also [47, 95, 96] for related discussions. In particular, even if the +bulk state is not macroscopically periodic, there will still be a microscopic periodicity of the +state due to Poincare recurrences, that will happen at very large T ∼ O(eeN2 +). Throughout +this paper, we will be interested in much earlier time scales so it will be sufficient for us to +treat the states |ΨT ⟩ as quasi-orthogonal since all overlaps will be exponentially small. +We will also need to define the notion of code subspace. Starting with the state |Ψ0⟩ we +define the code subspace as +H0 = span{|Ψ0⟩, O(t, Ω)|Ψ0⟩, ..., O1(t1, Ω1)...On(tn, Ωn)|Ψ0⟩} , +(3.12) +generated by acting on |Ψ0⟩ with a small number (n ≪ N) of single-trace operators27. It will +also be useful to define the projector P0 on this subspace. Similarly, a code subspace can be +defined for each of the time-shifted states +HT = span{|Ψ⟩T , O(t, Ω)|ΨT ⟩, ..., O1(t1, Ω1)...On(tn, Ωn)|ΨT ⟩} , +(3.13) +with the corresponding projector PT . The projectors P0 and PT are simply related by time- +evolution, i.e. we have +PT = e−iTHP0eiTH , +(3.14) +and in particular, we emphasize again that PT ̸= P0. In what follows, it will be convenient to +work with real quantities rather than the overlap (3.8), and we are now ready to define the +return probability. +3.6 +The return probability +We now ready to examine the T-dependence of the overlap (3.11) in more detail. As explained +above, it is more convenient to work with a real quantity so let us define the return probability +R(T) := |⟨Ψ0|e−iTH|Ψ0⟩|2 . +(3.15) +It is similar to the spectral form factor (the two coincide when |Ψ0⟩ = |TFD⟩ and H = +HL + HR). Recently, the spectral form factor has been extensively discussed in connection to +the black hole information paradox and quantum chaos, see for example [97]. The time-scales +of interest in that context are again late times such as t ∼ eN2 (note this is much shorter than +27To be precise, we should also give a small smearing to the single-trace operators in order to avoid UV +divergences of operator insertions at coincident points. We will leave it as implicit in what follows. +– 27 – + +the Poincare recurrence time which is doubly exponential). Here again, we will be interested +in much earlier time-scales. +In general, it is difficult to compute (3.15). As we mentioned above, the overlaps can be +computed from time-shifted coherent states in gravity but the best known technology to do +so uses the Euclidean path integral and involves solving the non-linear Einstein’s equations. +Nevertheless, we can compute the very early time dependence using large N factorization. +We present this calculation in Appendix C. At early times, we have +R(T) = e−(∆H)2T 2 , +(3.16) +which is generally valid for times up to T ∼ O(N−1). For the purposes of this paper, we want +to understand how the return probability behaves at time-scales T ∼ O(1). Here, the decay +does not follow from large N factorization and it is in general not an easy task to compute it. +In Appendix C, we review that for the TFD state, the return probability (which is the +spectral form factor) decays as +RTFD(T) = e−N2fTFD(T), +(3.17) +where fTFD(T) is O(N0) and for early times T ∼ O(N0) ≪ β behaves like fTFD(T) ≈ αT 2, +where α is an O(N0) constant which depends on the temperature. This is an extremely fast +decay, much faster than thermalization where the prefactor in the exponent is of order N0, +and shows that thermofield double states at different times orthogonalize exponentially fast. +We expect similar behaviour for many other semi-classically time-dependent states, that +is for timescales of T ∼ O(1), we expect +R(T) ∼ e−N2 ˜f0(T) , +(3.18) +for a positive and O(N0) function ˜f0(T) which depends on the state |Ψ0⟩. We expect that +for small T the function ˜f0(T) starts quadratically, as in (3.16). Note that this fast decay +is not even a consequence of quantum chaos, as it can occur at weak coupling or even in +free theories, provided they have a large number of degrees of freedom (see [98] for a study +of this question in weakly coupled N = 4 SYM). The difference between a free theory and +a holographic one will manifest itself in the time-scale during which the exponentially small +overlap remains valid. For free N = 4 SYM, the spectrum is integer spaced and so the return +probability will be periodic with period 2π, while in a chaotic theory it will take doubly +exponentially long for the signal to return to unity. +The average late-time value of the signal is also highly dependent on whether the theory +is chaotic or not. For a system with no degeneracies,28 +R = lim +t∗→∞ +1 +2t∗ +� t∗ +−t∗ +dT R(T) = +� +i +|ci|4 . +(3.19) +28Systems like N = 4 SYM will have degeneracies due to superconformal symmetry. For example, for every +primary, there are towers of descendants with degenerate energy levels. Nevertheless, the number of degenerate +states is exponentially smaller than the number of all states, at least in the high-energy sector of the theory, +so the degeneracy only contributes a subleading effect. +– 28 – + +For the type of states we are considering, i.e. those with a large energy variance, this is +exponentially small, and scales as e−α′N2, where α′ is an O(1) constant which depends on the +particular |Ψ0⟩ we have picked. This value is often referred to as the plateau, especially in +the context of the spectral form factor. +Between the initial decay (3.17) and the plateau (3.19), there can be other regimes, +which are particularly interesting in connection to quantum chaos [99,100]. For example, in +the spectral form factor, the plateau is preceded by a ramp where the signal grows linearly. +These effects will not be important for the present work, as we will only consider O(1) +timescales. +The crucial point we will exploit throughout the paper is that the signal is +already exponentailly small in N2 at those timescales. +The overlap (3.8) obeys the property +⟨Ψt0|Ψt0+T ⟩ = ⟨Ψ0|ΨT ⟩ . +(3.20) +This may appear trivial, but it means that even if the bulk geometry appears to be static at +the semi-classical level, the return probability may still decay following (3.17) if the state had +a period of manifest bulk time-dependence in the far past. Said differently, the variance in +energy which determines the decay is unchanged under time-evolution, so even if the 1-point +functions have stabilized, the variance remains large. This observation is particularly relevant +in the case of a black hole formed by gravitational collapse. +The exponential decay (3.17) can be extended to more general correlators of the form +⟨Ψ0|O(t1) . . . O(tn)|ΨT ⟩, where O are single-trace operators. We expect +⟨Ψ0|O(t1) . . . O(tn)|ΨT ⟩ = F(T)⟨Ψ0|ΨT ⟩ , +(3.21) +where F(T) is finite in the large N limit and satisfies +F(0) = ⟨Ψ0|O(t1) . . . O(tn)|Ψ0⟩ +, +dkF(T) +dT k +|T=0 = O(N0) . +(3.22) +To see the exponential decay we write (3.21) as +⟨Ψ0|O(t1) . . . O(tn)|ΨT ⟩ = ⟨Ψ0|O(t1) . . . O(tn)|ΨT ⟩ +⟨Ψ0|ΨT ⟩ +⟨Ψ0|ΨT ⟩ . +(3.23) +The second term in this product is really responsible for the decay of the correlator. The +first term is hard to evaluate from first principles, but in holography its meaning is clearer. +In the bulk theory, it is computed by computing a correlation function on a background +dictated by the Euclidean path integral with different sources on the northern and soutern +hemisphere (corresponding to |Ψ0⟩ and |ΨT ⟩, respectively). This correlator is O(1) and a +smooth function of the background, which will generally change slowly with T, so we expect +its time derivatives not to scale with N as indicated in (3.23). We check this statement in a +few examples in section 6. +– 29 – + +To sum up, any state in the code subspace (3.12) has an exponentially small overlap with +any state in the code subspace (3.13). This can be summarized by the relation +Rcode(T) = +1 +dcode +Tr[PT P0] = O(e−N2 ˜f(T)) +(3.24) +where dcode is the dimensionality of the code subspace, and for the time-scales we have dis- +cussed. The decay (3.24) can be used in combination with other useful inequalities. For +example, for a Hermitian operator O with eigenvalues λi, and if [P0, O] = 0, we have +|⟨Ψ0|O|ΨT ⟩|2 ≤ +� +Tr[O4] +� +Tr[PT P0] and |⟨Ψ0|O|ΨT ⟩|2 ≤ max(λ2 +i ) Tr[PT P0]. +3.7 +Other asymptotic charges +More generally we can consider the change of the state by large diffeomorphisms corresponding +to the other asymptotic symmetries of the theory, in the case of AdSd+1 the conformal group +SO(2, d) with the generators we discussed in section 3.1. This leads us to define a natural +generalization of the return probability +R(g) = |⟨Ψ0|U(g)|Ψ0⟩|2 +, +g ∈ SO(2, d) , +(3.25) +where U(g) is the unitary realizing the conformal transformation of the CFT on Sd−1 × time. +What can we expect for these overlaps? To start, let us suppose the state |Ψ0⟩ breaks +rotational SO(d) symmetry at the classical level. By this, we mean that bulk dual geometry +breaks the symmetry, which would be the case for some spherically asymmetric lump of +matter. Take J to be the angular momentum generator, then we expect that the variance of +J will be of O(N2) for such a state. Hence we expect that for small values of a rotation angle +φ dual to J we will have +R(φ) = e−(∆J)2φ2 = e−κN2φ2 , +(3.26) +for κ ∼ O(1). For more general angles, we expect +R(φ) = e−N2frot(φ) . +(3.27) +However, because angular momentum is quantized, we have +R(φ + 2π) = R(φ) , +(3.28) +hence the function frot(φ) has period 2π. In this direction of the conformal group the return +probability has a very short Poincare recurrence equal to 2π. +All in all we find that as we increase φ away from 0 the return probability R(φ) very +quickly dips down to exponentially small values and stays there until the Poincare recurrence +at φ = 2π. As we see from (3.27), for any fixed φ which is in the range (0, 2π), we have R(φ) +being exponentially small in the large N limit. +Of course if the state respects spherical symmetry then the return probability will not +decay in the corresponding SO(d) directions. +It is worthwhile to discuss several distinct +– 30 – + +scenarios. In the simplest case, the state preserves the symmetry and is thus annihilated by +the generators of rotations. The second simplest situation is the case where the symmetry is +manifestly broken at the classical level (for example an asymmetric lump of matter). In this +case, the breaking of the symmetry is manifest, and would be visible in the 1-point function +of single-trace operators. There are also more subtle situations where the state breaks the +symmetry classically in the bulk, but this may be invisible in the 1-point functions. +An +example of this are states by prepared by the path integral on higher genus surfaces in d = 2, +and have topology behind the horizon [80].29 +Finally as discussed in section 3.1, we expect that semi-classical states also break the other +conformal symmetries. We can get some intuition by considering a state dual to a conformal +primary of dimension ∆. In this case the return probability along one of the conformal boost +directions is determined by a group theoretic computation +R(s) = |⟨∆|e−isK|∆⟩|2 = +� +1 +cosh2 s +�2∆ +. +(3.29) +For primary states with ∆ ∼ O(N2), we get exponential decay of the form e−N2f(s) for any +non-zero s. Notice that for the conformal boosts we do not expect any Poincare recurrence +for large s, which in the case of primaries is obvious from the formula above, since such a +transformation monotonically increases the energy of the state. +In the case of AdS3 the asymptotic symmetry group is enhanced to Virasoro and similar +statements hold for the flow of the state under more general large diffeomorphisms generated +by Ln, Ln. +To summarize, if we start with a state |Ψ0⟩ which breaks all conformal symmetries at +the level of the semi-classical geometry we expect that R(g) defined in (3.25) will decay +exponentially fast in all directions away from the identity element on the conformal group +manifold. +4 +State-dressed operators +We are now in a position to introduce operators ˆΦ which satisfy the two properties described +in section 3.4, namely their commutator with the Hamiltonian and other asymptotic charges +is zero to all orders in the 1/N expansion and they act like HKLL operators to leading order at +large N on the code subspaces {HT , T ∈ (−t⋆, t⋆)}. Here t∗ is an order one (i.e. N0)) time of +our choice. We define the HKLL operator Φ, (3.6), in the N → ∞ limit. In this limit the bulk +is described by a quantum field theory on a curved spacetime and code subspaces for different +T will be strictly orthogonal to one another. In addition, Φ is a local bulk operator which +commutes with all the boundary single-trace operators in the time band algebra, including +29The thermofield double also has this property. It breaks rotational symmetry of each CFT individually, +but the breaking is invisible in 1-point functions. It would be interesting to understand if this type of breaking +always requires a horizon. +– 31 – + +the appropriately normalized Hamiltonian [36,84]. But it will no longer be commuting once +1/N corrections are included. In particular, we will have +[Φ, H − ⟨H⟩ +N +] = O(1/N) ̸= 0 . +(4.1) +Again, the physical reason behind this is that (3.6) is a diff-invariant operator that is dressed to +the boundary. Note that for the naive HKLL operator (3.6), the commutator with other single- +trace operators will also be non-zero at order O(1/N). For almost all single-trace operators, +this can be removed order by order in 1/N by adding the appropriate corrections to Φ [84]. +However, these modifications will not be able to remove the non-vanishing commutator with +the Hamiltonian (4.1). Thus, to remove the gravitational dressing to the boundary CFT, a +more sophisticated procedure is required. +We start by focusing on setting the commutator with the Hamiltonian to zero and discuss +the extension to other asymptotic charges later. +To this end, we introduce the following +operator30 +�Φ = c +� t∗ +−t∗ +dT e−iTHP0ΦP0eiTH , +(4.2) +where t∗ is an O(N0) timescale of our choice, and c is an overall normalization constant +c−1 = +� t∗ +−t∗ +dT⟨Ψ0|PT |Ψ0⟩ . +(4.3) +As we will see, the projector P0 will be key and will make �Φ act appropriately on the code +subspaces. The range (−t⋆, t⋆) determines the set of code subspaces on which �Φ acts in the +desired fashion, and ultimately cannot be taken to be bigger than the time range where the +exponential decay of the return probability (3.17) is valid. To make the operator (4.2) have +the desired properties on as many states as possible, we can take this range to be the time +range where the return probability decays exponentially, though this is not strictly necessary +and a t∗ of O(N0) is sufficient. We also provide an alternative presentation of the operators +in subsection 4.4. In the following subsections, we will study the action of these operators in +the relevant code subspaces, and will be particularly interested in their commutator with the +Hamiltonian. +4.1 +Vanishing commutator with H to all orders in 1/N +We now show that the operator (4.2) has vanishing commutator with H to all orders in 1/N. +We start by rewriting the commutator as +[H, �Φ] = −i d +ds +� +eisH �Φe−isH���� +s=0, +(4.4) +30Recall that P0 is the projector on the code subspace of |Ψ0⟩, and thus [Φ, P0] = 0 in that code subspace. +Therefore, we could have defined operators with the same action on the code subspace as (4.2), using a single +projector on the left (or right) of Φ. Even though the resulting operators would act in the same way on the +relevant code subspace, the operators would not be exactly identical: they would have additional non-zero +matrix elements associated to subspaces orthogonal to H0. +– 32 – + +and performing a change of variables, we find +[H, �Φ] = −i d +ds +� +c +� t∗−s +−t∗−s +dT e−iTHP0ΦP0eiTH���� +s=0 += ic(Pt∗Φt∗Pt∗ − P−t∗Φ−t∗P−t∗) , +(4.5) +where we defined Φt∗ = e−iHt∗ΦeiHt∗. Using the decay of the return probability through +(3.24), we see that the commutator inserted inside a correlator of a small number of single- +trace operators and evaluated on the state |ΨT ⟩ will give an exponentially small answer, since +each of the two terms in (4.5) give exponentially small numbers. This is valid for any T as +long as |T| < t⋆ and |T| − t⋆ ∼ O(N0). Thus, +[H, �Φ] = O(e−γN2) , +(4.6) +where γ is positive and O(N0), proving property 1, defined in subsection 3.4, for these opera- +tors. Note (4.6) is true for our set of code subspaces with T constrained as above, but not for +all states. For example, the commutator is not exponentially suppressed in the state |Ψt∗⟩. +4.2 +Similar action as HKLL operators +A vanishing commutator with the Hamiltonian is necessary but not sufficient. There are many +CFT operators that commute with the Hamiltonian up to exponentially small corrections in +N2, but they will not have the same effect as acting with a local bulk operator, see for +example footnote 4. Therefore, we also need to show that the operator ˆΦ behaves in the same +way as the HKLL operator (3.6) to leading order at large N inside correlation functions of +single-trace operators. For that we consider +⟨Ψ0|O...�Φ...O|Ψ0⟩ = +=c +� t∗ +−t∗ +dT ⟨Ψ0|O...e−iTHP0ΦP0eiTH...O|Ψ0⟩ +=c +� t∗ +−t∗ +dT ⟨Ψ0|O...P0PT (e−iTHΦeiTH)PT P0...O|Ψ0⟩. +(4.7) +In the last line, we have inserted two projectors P0, which we are free to do since the correlators +is evaluated in the state |Ψ0⟩. +The integrand above corresponds to TrPT P0, up to some +operator insertions that do not affect its general structure. +From (3.24) we see that the +integrand will be exponentially suppressed as |T| increases (and is not O(1/N)) because of +the exponentially small overlap of the code subspaces. We can thus evaluate the integral by a +saddle-point method controlled by the large N limit. The dominant contribution comes from +T = 031. Using (3.21) and (4.3) we have +⟨Ψ0|O...�Φ...O|Ψ0⟩ = ⟨Ψ0|O...Φ...O|Ψ0⟩ + O(1/N), +(4.8) +31One might worry about the possibility of rapidly oscillating phases, such as the one in ⟨Ψ0|ΨT ⟩ displacing +the location of the saddle point. Notice however that from (3.21),(3.22) it follows that such rapidly oscillating +phases cancel between the bra and ket contribution. +– 33 – + +as desired. The 1/N corrections can be thought of coming from corrections to the leading +saddle-point, and would be sensitive to the more detailed form of F(T) in (3.21). +Notice that if we apply the operator �Φ to one of the time-shifted states, then as long as +|T| < t∗, we find +⟨ΨT |O...�Φ...O|ΨT ⟩ = ⟨ΨT |O...(e−iTHΦeiTH)...O|ΨT ⟩ + O(1/N) +(4.9) +Thus in the code subspace HT , ˆΦ acts as e−iTHΦeiTH to leading order at large N. To make +this more manifest, we can also write (4.2) as +�Φ = c +� t∗ +−t∗ +dT PT (e−iTHΦeiTH)PT . +(4.10) +Since we have shown that, to leading order at large N, ˆΦ and Φ have the same matrix elements +on the entire code subspace it follows that higher point functions of ˆΦ will also agree at large +N with those of Φ. Consider for instance, +ˆΦi = c +� t∗ +−t∗ +dT e−iTHP0ΦiP0eiTH +(4.11) +where Φi ≡ Φ(xi) is an HKLL operator located at a certain spacetime point xi, then in the +large N limit +⟨Ψ0|O...�Φ1�Φ2...�Φn...O|Ψ0⟩ =cn +� t∗ +−t∗ +dT1...dTn ⟨Ψ0|O...PT1(e−iT1HΦ1eiT1H)PT1PT2 +(e−iT2HΦ2eiT2H)PT2...PTn(e−iTnHΦneiTnH)PTn...O|Ψ0⟩ +≈ ⟨Ψ0|O...Φ1Φ2...Φn...O|Ψ0⟩ . +(4.12) +In addition, this implies that the commutator of ˆΦi’s is the same as that of HKLL operators +in the large N limit. Two operators, ˆΦ(xi) and ˆΦ(xj), will have zero commutator at spacelike +separated points whereas they have O(1) commutator if they are timelike-separated. This +is true even though these operators do not translate under commutation with the boundary +Hamiltonian, up to exponentially small corrections in N. Nevertheless, they still have bulk +space-time labels and preserve the causal properties of HKLL operators in the large N limit. +4.3 +Interpretation and comments +We have just seen that to leading order in the large N limit, the operator (4.2) acts like the +HKLL operator (3.6) in the appropriate code subspace. However, it commutes with H to all +orders in 1/N. The existence of these operators provides strong evidence that the algebra of +single-trace operators in a short time band can have a non-trivial commutant when acting on +time-dependent states of high energy. +The vanishing of the commutator with H should be interpreted as (4.2) being gravita- +tionally dressed not with respect to the boundary, but instead with respect to features of +– 34 – + +the bulk state, in particular its time-dependence. This can be seen by the fact that ˆΦ acts +differently on different states. On the time-shifted states |ΨT ⟩ and their code subspaces, it +acts as e−iTHΦeiTH. For example, imagine that in the state |Ψ0⟩ we have a supernova explo- +sion taking place at t = 0 and we chose the operator (3.6) so that it acts right next to the +explosion. In the state |ΨT ⟩ the explosion obviously takes place at t = −T. From equation +(4.9), we can see that the operator �Φ will act again right next to the supernova explosion, +even though the supernova is now at t = −T. Therefore, one and the same operator �Φ knows +how to always act at the correct moment (right next to the explosion) for the entire family of +states |ΨT ⟩, as long as |T| < t∗. The finiteness of t∗ indicates that there is still some residual +boundary dressing, which however is not visible in pertubation theory32. +The property of being dressed with respect to features of the state is also present in +the local observables one defines in general relativity, discussed in section 2.3.3. These state +dressed observables are defined at points where a set of D scalars, like the Ricci scalar or +RµνρσRµνρσ where Rµνρσ is the Riemann tensor, ’click’ with a certain set of numbers. The +observables are labeled by these values and they are evaluated precisely where the scalars +take those values in each state. Locality of these observables requires them to be defined only +in some neighbourhood of a classical solution. In the same spirit, the operators discussed in +this section are also local for a certain family of code subspaces, see section 4.1. +As mentioned earlier, if the spacetime is so symmetric that the scalars take the same +values throughout the spacetime, then these classical observables are not well defined. Since +every point in the spacetime is physically equivalent, it is reasonable that local observables are +ill defined for these solutions. For this reason, the observables are state dependent. Similarly, +it is not possible to apply the same logic discussed in the previous subsections to empty AdS, +or other static states, as there are no time-dependent features in the bulk that can be used +as a ’clock’ to define a moment in time where the operator acts. Technically, the return +probability for such states does not exhibit the rapid decay (3.10). We thus see a nice parallel +between the classical and quantum situations. +The definition of our operator gives a bulk operator which is dressed with respect to +features of the state, but in an implicit manner. Our construction does not permit us to +extract the details of the dressing. Going back to our example of a supernova explosion, +one might guess that the dressing is with respect to the supernova and that one could in +principle define a gravitational Wilson line between the operator and the supernova. But +what if the state described instead two supernovas exploding at the same or different times? +To which explosion would our operator be dressed to? +The construction does not give a +definite answer, and the way to address this question would be to enlarge the set of code +subspaces on which our operator correctly acts. For example, if our operator did not move +under the time-translation of one of two supernovae, we would say that it is dressed to the +other one. We hope to return to this question in the future, but see subsection 7.3 for some +related remarks. +32Similar remarks were made in [18] for the DeWitt observables in AdS. +– 35 – + +4.4 +A similarity transformation +We briefly mention a variant of operators with properties similar to those of (4.2). We first +define the shifted Hamiltonian33 +ˆH = H − ⟨Ψ0|H|Ψ0⟩I . +(4.13) +Then we introduce +V = +c +√ +2 +� t∗ +−t∗ +dTe−i ˆHT P0 , +(4.14) +with c given in (4.3). We have +V V † = c2 +2 +� t∗ +−t∗ +dT +� t∗ +−t∗ +dT ′e−i ˆHT P0ei ˆHT ′ , +(4.15) +where we used P 2 +0 = P0. Following arguments similar to those of the previous subsection, +we find that to leading order at large N, and when computing the matrix elements of (4.15) +within the code subspace, the two integrals in (4.15) can be computed by a saddle point +method, where the dominant saddle is T = T ′ = 0. We then find that in this class of states +and at large N +V V † ≃ I, +and +V †V ≃ I . +(4.16) +in the sense that, within the code subspace V behaves like a unitary, up to 1/N corrections. +Then we start with a boundary-dressed operator Φ and define +ˆΦ = V ΦV † . +(4.17) +Following similar arguments as before we can show that the operator (4.17) satisfies properties +1 and 2 of subsection 3.4. To check the commutator of ˆΦ with H. We write +[H, ˆΦ] = −i d +ds +� +ei ˆHsV ΦV †e−i ˆHs� +|s=0 += −i d +ds +c2 +2 ( +� t∗−s +−t∗−s +dTe−i ˆHT )P0ΦP0( +� t∗−s +−t∗−s +dT ′ei ˆHT ′)|s=0 , +(4.18) +which again localizes on boundary terms and is thus exponentially suppressed. +Second, to show that the leading large N correlators of ˆΦ are the same as those of Φ we +follow exactly the same reasoning as in the previous subsection, but now we will have two +time-integrals. Each one of these time integrals will lead to a sharply suppressed Gaussian +around T = T ′ = 0 and can be evaluated by saddle-point at large N, reproducing the desired +result. +33This shift is useful in order to avoid rapidly oscillating phases in the discussion below. +– 36 – + +4.5 +Other asymptotic charges +More generally we need to make (3.6) commute with all boundary symmetry generators +corresponding to asymptotic symmetries. For asymptotically AdSd+1 space-times this is the +conformal group SO(2, d) and we consider a generalization of the form +�Φ = c +� +B +dµ(g)U(g)P0ΦP0U(g)−1, +(4.19) +where now +c−1 = +� +B +dµ(g)⟨Ψ0|U(g)P0U(g)−1|Ψ0⟩ . +(4.20) +Above, dµ(g) is the Haar measure on SO(2, d) and B is a reasonably sized connected sub- +manifold of SO(2, d) containing the identity. The commutator with conformal generators will +then be given by operators in the code subspace of states U(g∗)|Ψ0⟩, where g∗ lies on the +boundary ∂B. For the construction to work in this generalization we must make sure that +the overlaps +R(g) = |⟨Ψ0|U(g)|Ψ0⟩|2, +(4.21) +decay exponentially in the geodesic distance of g from the identity. As discussed in subsection +3.7 we expect this to be true for states which break all symmetries at the semiclassical level34. +The quantity R(g) is an interesting generalization of the return probability (3.15) that would +be interesting to study further. +5 +A more general argument for the commutant +The operators (4.19) constructed in the previous section commute with the asymptotic charges +to all orders in 1/N, however they commute with the other single-trace operators in the time- +band generally only to leading order in 1/N. To identify a commutant for the time-band +algebra A, the operators (4.19) have to be improved. +In this short section we outline a +somewhat different argument suggesting that it is indeed possible to find a commutant to +all orders in 1/N. We caution the reader that the argument that follows is based on certain +assumptions which seem physically plausible, but for which a rigorous proof is still lacking. A +more careful treatment for the existence of a commutant (as well as a mathematically precise +definition of the time-band algebra in the first place) would be desirable. +Let us start with a standard HKLL operator Φ. We also introduce the notation qi = +Qi−⟨Qi⟩ +N +for where Qi denotes any of the asymptotic SO(2, d) charges and Oj a general single- +trace operator in the time-band. Our goal is to find an operator ˆΦ which has the following +properties: +1. [ˆΦ, qi] = 0 and [ˆΦ, Oj] = 0 for all qi ∈ SO(2, d) and Oj ∈ A, to all orders in 1/N. +34For compact symmetries, such as rotations, R(g) will have recurrences every 2π. Hence along the compact +directions we take g∗ ∼ O(1) < 2π. +– 37 – + +2. To leading order at large N the correlators of ˆΦ with qj, Oi must be the same as those +of Φ. In particular this means that for single-trace operators Oi outside the time-band +we generally expect [Oi, ˆΦ] = O(N0). +The first condition is obvious. The second condition is necessary in order to ensure that the +operator ˆΦ acts in the expected way, at least to leading order at large N, and creates particles +that can be detected with an O(1) effect by operators outside the time-band when light rays +from the diamond hit the boundary. +Here we remark that in order for the two conditions to be mutually consistent, it is +important that we impose the second condition only to leading order at large N. The point is +that [qi, Φ] = O(1/N) hence when looking at leading order correlators it is indeed consistent +to demand simultaneously that i) ˆΦ commutes with qi and that ii) ˆΦ acts like Φ. However, +when moving on to subleading corrections we have a non-vanishing commutator [qi, Φ] hence +we cannot impose both conditions at the same time. We choose to impose that our operators +ˆΦ continue to commute with qi to all orders in 1/N, but we allow their correlators to depart +from those of Φi at subleading orders in 1/N. +We now define the desired operators ˆΦ by specifying how they act on the code subspace +H0. Earlier we defined the code subspace as the space generated by acting on |Ψ0⟩ with +single-trace operators, which are not necessarily restricted in the time-band. However, by +an analogue of the Reeh-Schlieder theorem35 we expect that for reasonable bulk states |Ψ0⟩ +the code subspace H0 can also be generated by acting on |Ψ0⟩ with only elements of the +time-band algebra A +H0 = span{A|Ψ0⟩} . +(5.1) +We now define the action of the the operator ˆΦ on the code subspace by the following condi- +tions +ˆΦA|Ψ0⟩ = AΦ|Ψ0⟩ , +∀A ∈ A . +(5.2) +This set of linear equations, one for every element of the small algebra A, defines the action +of ˆΦ on the code subspace, in a way which satisfies the desired properties as we will see below. +Notice that these equations can also be represented as follows: we first select a basis +of linearly independent elements Ai of the algebra A. then we define the matrix of 2-point +functions +gij = ⟨Ψ0|A† +iAj|Ψ0⟩ . +(5.3) +From (5.1), it follows that the set of states |i⟩ = Ai|Ψ0⟩ form a (possibly over-complete) basis +of the code subspace. Since ˆΦ is an operator on the code subspace it can be written as +ˆΦ = Kij|i⟩⟨j| = KijAi|Ψ0⟩⟨Ψ0|A† +j . +(5.4) +35This was discussed in [20] for the case of empty AdS and at large N. We believe that a similar result +should hold for more general heavy states and even when taking 1/N corrections into account, but it would +be interesting to develop a more careful proof. +– 38 – + +for an appropriate choice of Kij. To find the matrix K, we start with the desired relation +(5.2) written as +ˆΦAl|Ψ0⟩ = AlΦ|Ψ0⟩ , +(5.5) +then we replace ˆΦ with (5.4) and multiply from the left with ⟨Ψ0|A† +k to get +gjl gki Kij = ⟨Ψ0|A† +kAlΦ|Ψ0⟩ . +(5.6) +If the set of states |i⟩ = Ai|Ψ0⟩ are linearly independent then the matrix gij is positive definite +and invertible. In that case we can solve for K as +Kij = gikgjl⟨Ψ0|A† +kAlΦ|Ψ0⟩ , +(5.7) +where gijgjk = δi +k. When (5.7) is replaced in expression (5.4), we find an explicit solution of +the desired equation (5.2). +We emphasize that the necessary ingredient to arrive at (5.7) was the linear independence +of the states Ai|Ψ0⟩, which is equivalent to the statement that there is no non-vanishing +operator in A which annihilates the state |Ψ0⟩. We discuss this condition in the following +subsection. +5.1 +On the consistency of the defining equations +Before checking that the operators ˆΦ defined by (5.2), or equivalently via (5.4),(5.7), have the +desired properties, we need to check that equations (5.2) are self-consistent linear equations. +The only possible source of inconsistency is the following: if there was an element A ̸= 0 of +the time-band algebra A such that A|Ψ0⟩ = 0, this could potentially be a problem since we +would then have A|Ψ0⟩ = 0, while in general AΦ|Ψ0⟩ ̸= 0. Then the equation (5.2) would +imply 0 = A|Ψ0⟩ = AΦ|Ψ0⟩ ̸= 0 which is a contradiction. Relatedly, gij defined in (5.3) would +not be invertible and we would not be able to get to (5.7). +We will now show that this situation does not arise, that is +A|Ψ0⟩ ̸= 0 +∀A ∈ A , A ̸= 0 . +(5.8) +We will prove this by first proving that at large N (5.8) is true and then we will argue that +1/N corrections cannot change the conclusion. +We have been working under the assumption that the time-band is short enough, which +means that in the bulk there will be a region which is space-like relative to the time band. In +the large N limit, where gravitational backreaction is turned off, operators inside that region +(for example usual HKLL operators) commute with all elements of the algebra A, including +the appropriately normalized asymptotic charges qi. Hence, in the large N limit the algebra +A has a non-trivial commutant A′. We want to argue that this commutant continues to exist +when 1/N corrections are taken into account, provided that the state |Ψ0⟩ has non-vanishing +variance of O(N2) under the asymptotic charges. +Assuming that at large N the theory in the bulk behaves like usual QFT on a curved +background, we expect that an analogue of the Reeh-Schlieder theorem will hold for the +– 39 – + +commutant A′, which means that we can generate the code subspace H0 by acting on |Ψ0⟩ +with elements of A′. +Suppose now that there was an element A of the time-band algebra A which annihilated +the state |Ψ0⟩. Then for any element a′ ∈ A′ we have +Aa′|Ψ0⟩ = a′A|Ψ0⟩ = 0 . +(5.9) +Since states of the form a′|Ψ0⟩ generate H0 we conclude that the operator A has vanishing +matrix elements in H0 at large N. From this we can not immediately conclude that A = 0 as +an operator when 1/N corrections are included. For example, for |Ψ0⟩ = |0⟩ the normalized +SO(2, d) generators qi = Qi +N have vanishing matrix elements at large N, since they annihilate +|0⟩ and commute with all other operators. +However they are non-vanishing operators at +order 1/N. If A is a non-vanishing operator which has vanishing matrix elements at large +N on H0 then it means that it acts as a central element at large N. +Here we make an +additional assumption, that the only central elements are the SO(2, d) generators qi and +their functions36. Since, by assumption, the state |Ψ0⟩ has non-trivial variance under these +generators, we conclude that it cannot be annihilated by a non-trival A. +Let us assume now that we have a state of the form A|Ψ0⟩ which has finite (i.e. O(N0)) +positive norm at large N. Including 1/N corrections will generally modify the norm of this +state, but it will do so by corrections suppressed by powers of 1/N. +Since the previous +argument established that the leading large N norm of the state A|Ψ0⟩ is a finite positive +number, perturbative 1/N corrections cannot make it vanish. Hence we expect property (5.8) +to be true to all orders in 1/N perturbation theory. +We emphasize that the fact that we cannot annihilate the state by the time-band algebra +A relies on the fact that we have restricted our attention to small products of single-trace +operators. As discussed in a related context [20,25], if we consider the full algebra of operators +in the time-band we can find sufficiently complicated combinations which can annihilate the +state37. +Finally, as should be clear from the above, if the state |Ψ0⟩ has very small or vanishing +variance in the asymptotic charges then (5.8) fails and it is not possible to define operators +obeying (5.2). +5.2 +Proof that ˆΦ has the desired properties +Having established that equations (5.2) are consistent, we argue that the operator ˆΦ has the +desired properties. +First it is obvious by (5.2) that the operator ˆΦ has vanishing commutators with elements +of A. To see that consider A1 ∈ A and a general state in the code subspace which can be +written as A2|Ψ0⟩, with A2 ∈ A. Then we have +[ˆΦ, A1]A2|Ψ0⟩ = ˆΦ(A1A2)|Ψ0⟩ − A1(ˆΦA2|Ψ0⟩) = A1A2Φ|Ψ0⟩ − A1A2Φ|Ψ0⟩ = 0 , +(5.10) +36We believe this assumption to be quite weak, but it would be interesting to prove it more thoroughly. +37For example, consider a state |Ψ⟩ with ⟨Ψ0|Ψ⟩ = 0. Then the (complicated) operator |Ψ⟩⟨Ψ| annihilates +|Ψ0⟩. +– 40 – + +where in the second equality we used (5.2). Since this is true for all A2, we find +[ˆΦ, A1] = 0 +∀A1 ∈ A , +(5.11) +where it should be understood that this equation holds on the relevant code subspace. +Second, we will show that to leading order at large N, the operator ˆΦ acts like the HKLL +operator Φ. To see this, consider an arbitrary matrix element on the code subspace. Two +general states of the code subspace can be written as A1|Ψ0⟩, A2|Ψ0⟩. Then we have +⟨Ψ0|A† +1 ˆΦA2|Ψ0⟩ = ⟨Ψ0|A† +1A2Φ|Ψ0⟩ = ⟨Ψ0|A† +1ΦA2|Ψ0⟩ + ⟨Ψ0|A† +1[Φ, A2]|Ψ0⟩ . +(5.12) +In the first equality we used (5.2). Now, the operator A2 is some combination of single-trace +operators in the time band, as well as the normalized SO(2, d) generators qi. All of these +operators have commutators with Φ which are suppressed by powers of 1/N. Hence the last +term in the equation above is suppressed. All in all, we find +⟨Ψ0|A† +1 ˆΦA2|Ψ0⟩ = ⟨Ψ0|A† +1ΦA2|Ψ0⟩ + O(1/N) , +(5.13) +which establishes the desired result. This ensures that large N correlators of ˆΦ are the same +as Φ. +We emphasize that the operators defined in this section are not exactly the same as the +operators (4.2) discussed earlier. For example, unlike (4.2) the operators (5.2) were defined to +act only on the code subspace H0 of |Ψ0⟩ and not on the code subspace HT for T = O(N0). +Also, the commutator of (4.2) with the Hamiltonian is of order e−N2 while it is exactly zero, +within the code subspace, for the operators (5.2). +6 +Examples +In this section we consider various examples. Our primary focus will be on examining the +validity of equations (3.18), (3.21),(3.22), on which the construction of our operators relies. +6.1 +Coherent states +In general, we are interested in time-dependent semi-classical geometries. +Many of these +states can be thought of as bulk coherent states. We will discuss the overlap of these states +closely following [74]. In the CFT, these states are prepared by a Euclidean path integral +|Ψ⟩ = Te− +� +tE<0 dtEdd−1x φb(tE,x)O(tE,x) |0⟩ , +(6.1) +where O is a single-trace operator dual to a supergravity field, and the source is scaled +appropriately so that it leads to states with non-trivial gravitational backreaction, i.e. the +expectation value of the energy and variance of this state will scale like (3.1) and (3.3). +In the large N limit the overlap of two such states can be computed by a Euclidean +gravitational path integration which in the semi-classical limit can be approximated by a +saddle point computation. For example, the norm of the state is +⟨Ψ|Ψ⟩ ≈ e−Igrav(λb) , +(6.2) +– 41 – + +where λb is the following boundary condition for the bulk field +λb = +� +φb(tE, x), tE < 0 +φ⋆ +b(−tE, x), tE > 0 , +(6.3) +and Igrav(λb) is the on-shell gravitational action in the presence of the sources specified above. +Generalizing to two states |Ψ1⟩ and |Ψ2⟩, the normalized inner product between them is +R = +|⟨Ψ1|Ψ2⟩|2 +⟨Ψ1|Ψ1⟩⟨Ψ2|Ψ2⟩ , +(6.4) +which at large N can be computed by a supergravity saddle-point computation +R ≈ exp +� +−2 Re(Igrav(˜λ)) + Igrav(λ1) + Igrav(λ2) +� +, +(6.5) +where the supergravity solutions have the boundary sources ˜λ, λ1 and λ2 which take the +following form +˜λ = +� +φ2(tE, x), tE < 0 +φ⋆ +1(−tE, x), tE > 0, +λi = +� +φi(tE, x), tE < 0 +φ⋆ +i (−tE, x), tE > 0, +(6.6) +where i = 1, 238. +Notice that in each of the terms of (6.5), the gravitational on-shell action is proportional +to +1 +GN ∼ N2. +Since quantum mechanically we need R ≤ 1, we find that the following +inequality has to be satisfied +2 Re(Igrav(˜λ)) ≥ Igrav(λ1) + Igrav(λ2) , +(6.7) +for the on-shell value of solutions of the Einstein plus matter equations, for any choice of +sources of the form (6.6). If the two sources are different, we expect a strict inequality. It +would be interesting to explore this inequality directly from the gravitational point of view. +We discuss this further in the discussion. +We now move on to the computation of the return probability for states of the form (6.1) +after a small (not N-dependent) time evolution. That is, we take the time-evolved state, +|Ψ(T)⟩ = e−iHT |Ψ⟩, and consider the following quantity +R(T) = +|⟨Ψ(0)|Ψ(T)⟩|2 +⟨Ψ(0)|Ψ(0)⟩⟨Ψ(T)|Ψ(T)⟩ . +(6.8) +To apply the general formalism described above, we need to analyze how the Euclidean sources +φ0 preparing the state |Ψ(0)⟩ need to be modified to φT , in order to prepare |Ψ(T)⟩. From +38The sources φ2(tE, x) and φ⋆ +1(−tE, x) should decay sufficiently fast at the t = 0 surface such that the states +are normalizable. This also implies that the bra and ket preprations of different states can be smoothly glued +to each other. +– 42 – + +a technical point of view computing φT in terms of φ0 is not straightforward, as it requires +a solution of the Einstein equations. Nevertheless, we can in principle compute the return +probability using (6.4) and (6.5) with a modified source +˜λ = +� +φT (tE, x), tE < 0 +φ⋆ +0(−tE, x), tE > 0, +λT = +� +φT (tE, x), tE < 0 +φ⋆ +T (−tE, x), tE > 0 . +(6.9) +Thus we get +R(T) = exp +� +−2 Re(Igrav(˜λ)) + Igrav(λ0) + Igrav(λt) +� +, +(6.10) +and this is exponentially suppressed in the semi-classical limit because of the 1/GN ∼ N2 +coefficient in the gravitational action and the condition (6.7). +6.2 +Thermofield double state +We now consider the thermofield double state +|TFD⟩ = +1 +� +Z(β) +� +n +e− βEn +2 +|En⟩L ⊗ |En⟩R , +(6.11) +where the |En⟩’s are the energy eigenstates and Z(β) is the partition function at inverse +temperature β. In the strong coupling limit, for temperatures below the Hawking-Page tem- +perature, the state is dual to two entangled thermal AdS geometries, while for temperatures +higher than the Hawking-Page temperature, it is expected to be dual to the eternal black hole +in AdS [101]. This geometry has two asymptotically AdS boundaries, on the ”left” and the +”right”, hence the asymptotic symmetry group is SO(2, d)L × SO(2, d)R. The state (6.11) is +invariant under certain combinations of the asymptotic charges, for example we have +(HR − HL) |TFD⟩ = 0 +but +(HR + HR) |TFD⟩ ̸= 0 +(6.12) +and similarly for the other charges. In this case we can generalize the return probability to +include all possible large diffeomorphisms on the two sides +R(g1, g2) = |⟨TFD| UL(gL) UR(gR) |TFD⟩|2 +, +gL/R ∈ SO(2, d)L/R . +(6.13) +In this case we expect R(gL, gR) to rapidly decay along certain directions but remain constant +along others due to the symmetries of the state (6.11). +In what follows we focus on a particular class of deformations, corresponding to evolving +with HL + HR. This gives what is usually called the spectral form factor (SFF) defined as +R(t) = |⟨TFD|e−i T +2 (HL+HR)|TFD⟩|2 = +���� +Z(β + iT) +Z(β) +���� +2 +, +(6.14) +which was introduced in the context of the eternal AdS black hole in [95] and studied in detail +in [97]. +– 43 – + +We are interested in studying (6.14) above the Hawking-Page temperature for small times, +i.e, T ∼ O(1). One way to proceed is by computing Z(β) and then analytically continuing +β → β + iT. If we are above the Hawking-Page temperature Z(β) can be estimated by the +Euclidean AdS-Schwarzschild black hole saddle point +Z(β) ≈ e−IBH(β) , +(6.15) +where IBH(β) is the on-shell action on the Euclidean black hole background. For example, +we find +IBH(β) = − +π2 +2GNβ (for AdS3) +IBH(β) = +β +GN +g(rH) (for AdS5) , +(6.16) +where we have set the AdS radius ℓAdS = 1 and rH is the horizon radius, while +g(rH) = V3 +8π(−r4 +H + r2 +H) +(6.17) +where V3 is the dimensionless volume associated with the metric on a unit sphere. For the +AdS5 case, rH ≈ π/β for small real β. A detailed discussion of the action can be found +in [102]. The central charge of the CFT2 is c = 3/2GN and the rank for the gauge group of +the dual four dimensional SU(N) N = 4 super Yang Mills theory is given by N2 = π/2GN. +For small T the complexified partition function Z(β + iT) will be given in terms of the +analytic continuation of the above actions. Thus for T ≪ β, one gets the following for AdS3, +R(T) ≈ e +− 2π2 +β3 c T 2 +, +(6.18) +which is exponentially small in the large central charge limit39. Similarly for AdS5, we find +that Z(β) ∼ e +πN2 +β3 +in the high temperature limit. Again for T ≪ β, we have +R(T) ≈ e +− 12π +β5 N2T 2 +, +(6.19) +As T becomes larger and approaches T ∼ β, the dominant saddle point will no longer be the +black hole, as the analytically continued action can start to compete with thermal AdS. In +addition the analytically continued black hole saddle point corresponds to a geometry with a +complex metric, and as T ∼ O(β) this metric becomes ’unallowable’ according to the criteria +of [103], see also [104]. Interestingly, thermal AdS becomes the dominant saddle point before +the metric becomes not allowable [98]. +An exponential decay of R(T) in N is to be expected even when T ∼ β, since in this case +the thermal AdS saddle dominates and, |Z(β + iT)|2 ∼ e˜g(T)/β3 where ˜g is O(N0) periodic +function of time. Thus, the numerator of (6.14) |Z(β + iT)|2 is N0 while the denominator is +O(eN2) leading to an exponentially suppressed R(T). +39There will be additional terms suppressed in T 2/β2 which will not affect the exponential decay in the large +c limit as long as t is smaller than β. +– 44 – + +6.3 +Weakly coupled, large N gauge theories +It is interesting to consider the behavior of the SFF at small, or even vanishing ’t Hooft +coupling λ. In this case the bulk dual is stringy and moreover at λ = 0, the spectrum of the +dual CFT is (half)-integer-spaced and thus not chaotic at all. Nevertheless the decay (3.18) +is still valid for a certain time-scale, even in the free theory. This was discussed in detail +in [98]. For concreteness, we consider the partition function of free N = 4 SYM on S3 × R, +where the sphere has unit radius. It has the form [105,106] +Z(β) = +� +DU e +� +R +�∞ +m +1 +m zR +m(β)χR(Um) , +(6.20) +where DU is the invariant Haar measure on the gauge group normalized to one, χR is character +in the representation R and +zR +m(β) = +� +Ri,B=R +e−mβEi + (−1)m+1 +� +Ri,F =R +e−mβEi , +(6.21) +where the first sum is over bosonic states and the sum in the second term is over fermionic +states. +The behavior of the SFF +��� Z(β+iT) +Z(β) +��� +2 +, as well as of the microcanonical analogue YE,∆E(T), +based on the analytic continuation of (6.20) was discussed in [98]. +Even at λ = 0 the SFF obeys (3.18), though in this case the Poincare recurrence time +is very short, i.e. 4π.40 While in this limit the bulk theory does not admit a semiclassical +gravitational description, we could still apply the procedure (4.2) to identify operators with +vanishing commutators with the Hamiltonian to all orders in 1/N, though now they do not +have a nice bulk interpretation.41 In doing so, we would need to be careful to take t∗ to be a +short O(1) time-scale which is less than 4π. +Here we notice that similar results have been derived for the analytically continued super- +conformal index [107], which can be thought of as the SFF for 2-sided eternal supersymmetric +AdS black holes. +6.4 +Perturbative states around empty AdS +We now briefly discuss the return probability for perturbative states around empty AdS. We +want to consider states which have a large number of particles, but still small enough so that +we can ignore gravitational backreation. We can get some useful estimates by considering a +thermal gas of particles in AdSd+1. These are dual to a gas generated by single-trace operators +in the CFT. Suppose we have low-lying single-trace operators with conformal dimension ∆i. +For simplicity we consider only scalars and we take the radius of AdSd+1 to be 1. Then +40In our conventions conformal dimensions in the free theory are half-integers. +41To start with, the HKLL procedure cannot be implemented at subleading orders in 1/N due to the many +stringy fields present in the bulk. Therefore, the issue of non-commutativity with the Hamiltonian does not +stand out like it does in the case of Einstein gravity. +– 45 – + +the partition function of single-particle states z(β) and the multi-trace Fock-space partition +function are respectively +z(β) = +� +i +e−β∆i +(1 − e−β)d +, +Z(β) = exp +� ∞ +� +n=1 +1 +nz(nβ) +� +. +(6.22) +It is now straightforward to do the analytic continuation +Z(β + iT) = exp +� ∞ +� +n=1 +� +i +e−(nβ+inT)∆i +(1 − e−nβ+inT )d +� +. +(6.23) +For scalar BPS operators dual to SUGRA modes, ∆i is integer. Then it is obvious that the +SFF R(T) = +��� Z(β+iT) +Z(β) +��� +2 +has periodicity T = T + 2π, as expected. What we want to estimate +is the decay rate of the SFF at early times, and how close to 0 the SFF drops between the +recurrences. +First we notice that the partition function factorizes to a product over ∆i. Hence we can +study the behavior of a given ∆i and we drop the sum over i. If we first take the small β +limit, before analytically continuing, we find +Z(β) ∼ exp +� +ζ(d + 1) 1 +βd +� +. +(6.24) +Using this approximation we find that for early times +R(T) ∼ e +− d(d+1)ζ(d+1) +βd+2 +T 2 +. +(6.25) +As expected the decay is controlled by the variance of H. +Of course if we use the high +temperature approximation (6.24) to perform the analytic continuation, then we do not see +the recurrences. At high temperature the SFF starts decaying quite rapidly, stays close to +zero for a while and then goes back to 1 every T = 2π × integer. To find an estimate of how +closely it approaches zero it is convenient to evaluate it at T = π. Suppose that the conformal +dimension is an even integer. Then we find +R(π) = +exp +� +2 �∞ +n=1 +1 +n +e−nβ∆ +(1−(−1)ne−nβ)d +� +exp +� +2 �∞ +n=1 +1 +n +e−nβ∆ +(1−e−nβ)d +� +∼ e +−(2−2−d)ζ(d+1) 1 +βd − 1 +2d log β∆ +2 +(6.26) +So we see significant suppression at small β, though of course, the suppression does not scale +like e−N2. +We expect a similar qualitative behavior for R(T) for generic pure states of similar energy +as the states studied above (namely high energy states whose energy scales as O(N0)): they +will have recurrences every 2π, but the return probability will quickly decay to small values +for 0 < t < 2π. If we use (4.2) for such states, with t∗ ∼ O(1) < π, then the commutator +with H will be suppressed by a factor of the order of (6.26) rather than e−N2. Note that this +is not good enough, since the commutator we are trying to cancel is O(1/N), which in the +large N limit is much smaller than the suppression controlled by (6.26). +– 46 – + +6.5 +LLM geometries +An interesting class of semiclassical states with AdS5 × S5 asymptotics in type IIB super- +gravity are the LLM geometries [81]. These are dual to 1 +2-BPS states in N = 4 SYM. While +these geometries do not break all of the asymptotic symmetries, they do provide a useful toy +model where we can study in detail the behavior of the return probability as a function of +time. +The 1 +2-BPS states in N = 4 SYM on S3 × R are states that preserve 16 of the 32 super- +symmetries of the theory in addition to the bosonic symmetries SO(4) × SO(4) × R where +R corresponds to the Hamiltonian H − ˆJ where H is the Hamiltonian and ˆJ an R-symmetry +generator. +These states correspond to operators that lie in the (0, J, 0) representation of +the SU(4) ∼ SO(6) R-symmetry and they saturate a unitarity bound for their conformal +dimension. It is illuminating to consider the N = 1 vector and three chiral multiplet decom- +position of the N = 4 theory. In this case the scalars of the chiral multiplets are organized +into Zj = φj + iφj+3, where j = 1, 2, 3, which are in the adjoint representation. We will +focus on j = 1 from now on without loss of generality. Then the states we are interested in +correspond, via the state-operator map, to single-trace operators of the form Tr(Zni), as well +as multi-trace operators of the form Πi(Tr(Zni))ri [81,108,109]. +Since these operators saturate the unitarity bound ∆ = J, they correspond to the lowest +Kaluza-Klein mode of Z on S3. This mode has a harmonic oscillator potential due to its +conformal coupling to the curvature of S3. Thus we are interested in gauge invariant states of +the matrix Z in a harmonic potential [109]. The ground state, corresponding to empty AdS, +is given by a Gaussian wave function +Ψvac = Ce− 1 +2 N2tr(Z2) , +(6.27) +where C = (π/N)−N2/4 and we introduce the notation trZ = 1 +N TrZ = 1 +N +�N +i νii. Fluctua- +tions with operators with ∆ = J ≪ N will be small excitations around the ground state. As +discussed in [110] excitations with ∆ = J ∼ N2 will be other coherent states which are given +by +Ψ = C[J(Z)]1/2e−N2tr( 1 +2 φ(Z)2−iψ(Z)) , +(6.28) +parameterized by two functions φ(Z) and ψ(Z) which are monotonically increasing and ar- +bitrary functions of Z, respectively. J[Z] is the Jacobian given by det[∂φ(Z)ij/∂Zkl]. +It is well known that one can describe such a system by N fermions in a harmonic +potential [111]. In the large N limit, states in such a system can be thought of as droplets +in a two dimensional phase space, where for example a circular droplet corresponds to the +ground state of the system [111–113]. The precise connection between the functions φ(Z) and +ψ(Z) and the droplet picture on the phase space will be discussed in the next subsection. +In the bulk, the LLM solutions correspond to 10 dimensional geometries of asymptotically +AdS5 × S5 spacetimes, see appendix D, that are completely determined by a function z on a +two dimensional surface. In particular, specifying whether z takes value 1/2 or −1/2 at each +point on this plane completely specifies the full bulk solution. This is in parallel with the +– 47 – + +two dimensional fermionic phase space mentioned earlier where the fermion takes occupation +number 1 (black) or 0 (white) at each point in the phase space, giving droplet of a given shape. +For instance, in the fermionic picture the ground state is a circular droplet of a certain radius, +say r0. It corresponds in the bulk is to the empty AdS5 × S5. +Fluctuations with operators of ∆ = J ≪ N correspond to having ripples in the edge +of the circular droplet and corresponds to having gravitons propagating in the AdS5 × S5 +background. While operators of energy ∆ = J ∼ N correspond to giant gravitons in the +bulk. Operators with ∆ = J ∼ N2 correspond to other bulk geometries and different shapes +of droplets in the fermionic phase space [81, 114, 115]. +The geometries will not be time +translation invariant (rotational invariant in the fermionic picture) in general42, but they are +invariant under t → t + 2π. +The goal here is to consider a certain geometry that breaks time translation invariance and +compute its return probability for short time scales. In the fermionic picture this corresponds +to a droplet that breaks the rotational invariance, an ellipse for instance. +In the matrix +quantum mechanics picture it is easy to compute the return probability, evolving (6.28) with +the quadratic Hamiltonian and computing the square of the inner product. But first, we need +to review the dictionary between the two pictures. +6.5.1 +Computation of the return probability +The way the matrix quantum mechanics picture and the fermionic picture are related will +be obvious once we diagonalize the matrix Z and express it in terms of the eigenvalues (µi), +where the Jacobian becomes +J(Z) = +N +� +i +φ +′(µi) +� +i̸=j +φ(µi) − φ(µj) +µi − µj +, +(6.29) +which is 1 for the vacuum. In the large N limit, the Gaussian measure dZ exp +� +−N2tr(Z2) +� +will +reduce to the well known Wigner semi-circle distribution for the density of eigenvalues [116], +dϱ(µ) = 1 +π(2 − µ2)1/2Θ(2 − µ2)dµ. +(6.30) +Let us now introduce new variables to parameterize the coherent states in the large N limit, +w(µ) := dϱ(φ(µ))/dµ which is the density of eigenvalues and v(µ) := ψ(µ). These parameters +are canonical conjugates of one another43, that is their Poisson bracket is the Dirac delta +function. In the large N limit, the appropriately renormalized Hamiltonian (hcl) can also be +written in terms of w and v +′ = dv/dµ and thus an action can be written for these variables +[110,117–119]. In particular, +hcl = 1 +2 +� +dµ w(µ)(v +′(µ)2 + π3 +3 w(µ)2 + µ2). +(6.31) +42There are also static configurations, concentric circles for example [81] +43Note that the two variables are not totally independent and w(µ) has to satisfy a constraint, in particular +� +dµ w(µ) = 1. +– 48 – + +Coming back to the two dimensional phase space picture, we consider a blob centered at the +origin. We assume the horizontal direction (x-axis) represents the q variable of the phase +space, which we take to be the eigenvalues (µ). Consider a vertical line crossing the blob. +Assuming that the blob has a simple geometry without folds, this vertical line intersects the +boundary of the blob twice. We parametrize these points by p±(µ) respectively. Then, the +density of eigenvalues for any µ is proportional to (p+−p−)(µ). Computing the kinetic energy +of fermions for a given dµ by integrating p2/2 from p = p− to p = p+ and matching this to +the kinetic part of (6.31), we get +p± = ±πw + v +′. +(6.32) +This has also been mentioned in the context of c = 1 string theory in [120–123]. Note that +for the vacuum (i.e. the empty AdS5 ×S5 geometry), p± = ±(2−µ2)1/2Θ(2−µ2) and v′ = 0. +Since we are looking for a time dependent geometry, we need a blob in fermion phase +space that breaks the rotational symmetry. The simplest non trivial modification of (6.32) is +to take v to be quadratic44. In this case we have +p+(µ) = (2 − µ2)1/2Θ(2 − µ2) + 2µ. +(6.33) +This can be seen to be half of a tilted ellipse, which combined with an appropriate p− gives +the full elliptic blob. This will evolve non trivially under rotation and the corresponding +geometry will be a time dependent one. This geometry, together with the five form, can be +found using the mapping discussed earlier, by first solving for z(x1, x2, y) then inserting it +into (D.2), (D.3), (D.4) and (D.5). +Now we proceed with the computation of the return probability for this state. We go +back to (6.28) and consider a state Ψ(0) with φ = Z and ψ = v = Z2 and after evolving it, +compute the overlap +⟨Ψ(0)|Ψ(T)⟩ = +� +dZ Ψ(Z, 0)∗Ψ(Z, T) +(6.34) +The state we are interested has the form +Ψ(Z, 0) = +� π +N +�−N2/4 +e− 1 +2 N2(1−2i)tr(Z2) = +� +i,j +ϕ(νij) , where ϕ(ν) = +� π +N +�−1/4 +e− N +2 (1−2i)ν2 . +(6.35) +Since we are dealing with matrix quantum mechanics with a quadratic potential, each matrix +element evolves independently and governed by the usual harmonic oscillator propagator +ϕ(ν, T) = +� +dν +′K(ν +′, ν, T)ϕ(ν +′) , +(6.36) +where +K(ν +′, ν, T) = +� +N +2πi sinT exp +� iN +2sinT ((ν2 + (ν +′)2)cosT − 2νν +′) +� +. +(6.37) +44Translated circle blobs will not correspond to physical geometries when the gauge group is SU(N), since +the centre of the blob is fixed by imposing the condition Tr(Z)=0. +– 49 – + +for t < π. We can then compute the overlap +⟨Ψ(0)|Ψ(T)⟩ = [z(T)]N2, +z(T) = +� +dν ϕ⋆(ν, 0)ϕ(ν, T) +(6.38) +Following (6.36) we find +ϕ(ν, T) = +� N +πX +�1/4 +e−NYν2 and, +Ψ(Z, T) = +� N +πX +�−N2/4 +e−N2Ytr(Z2) , +(6.39) +where X and Y are periodic functions of time given by, +X(T) = (cosT + (2 + i)sinT)2 +Y(T) = 1 +2 +�(1 − 2i) cosT + i sinT +(i + 2) sinT + cosT +� +. +(6.40) +Thus, +z(T) = +� +dν ϕ⋆(ν, 0)ϕ(ν, T) = A1/2 , +(6.41) +where +A = +1 +3i sinT + cosT . +(6.42) +It can be checked that z(T) is 1 when T = 0 and +|z(T)|2 = |A| = +� +1 +9 sin2T + cos2T +�1/2 +. +(6.43) +Since |z(T)|2 ≤ 1, R(T) is an exponentially decaying function in the large N limit, for small +times, but a periodic function in T = π. That is +R(T) = |⟨Ψ(0)|Ψ(T)⟩|2 = e−N2F(T) , +(6.44) +where the function F(T) = −2log|z(T)| is zero at T = 0, and increases to the local maximum +F(T = π/2) = log 9 and goes back to zero at T = π. Thus in the time scales we are interested +in, in particular T < π/2, the square of the inner product (6.34) which is the return probability +of a given LLM semi classical geometry in the large N limit, is exponentially suppressed in N2 +as expected. Note that the return probability is periodic in π, which is due to the symmetry +of the particular state considered. In general, the period will be 2π. +We can also compute the overlap of states in different code subspaces built upon Ψ(Z, 0) +and Ψ(Z, T). The simplest is the inner product of the states Ψ(Z, 0) and Tr(Z2n)Ψ(Z, T) +which can be written as +⟨Ψ(0)| tr(Z2n) |Ψ(T)⟩ ≡ +� +dZ Ψ⋆(Z, 0)tr(Z2n)Ψ(Z, T) += +� +π +NX 1/2 +�−N2/2 � +dZ tr(Z2n) e− S +2 N2 tr(Z2) +(6.45) +– 50 – + +where S = (1 + 2i) + 2Y. Following (3.23), we can rewrite the above integral as +⟨Ψ(0)| tr(Z2n) |Ψ(T)⟩ = ⟨Ψ(0)|Ψ(T)⟩ +� +dZ tr(Z2n) e− S +2 N2 tr(Z2) +� +dZ e− S +2 N2 tr(Z2) +(6.46) +The second factor corresponds to an expectation value in a Gaussian matrix model. Keeping +only planar diagrams at large N we find +⟨Ψ(0)| tr(Z2n) |Ψ(T)⟩ ≃ ⟨Ψ(0)|Ψ(T)⟩ Cn +Sn +(6.47) +where Cn = +1 +n+1 +�2n +n +� +are the Catalan numbers. +Similarly, for multi-trace operators the overlap can be computed and using large N fac- +torization we get +⟨Ψ(0)| +k +� +i +tr(Z2ni) |Ψ(T)⟩ ≃ ⟨Ψ(0)|Ψ(T)⟩ +�k +i Cni +Sn +, +(6.48) +where n = n1 + ... + nk. +Thus, as long as n does not scale with N the correlator will still be exponentially sup- +pressed, otherwise the periodic coefficient can spoil the exponentially decaying behaviour. +This is to be expected since in such cases the dimension of the multi-trace operators will be +order N and they will not be just small fluctuations of the background and can, in principle, +evolve the state back in time to T = 0. In any case, our code subspace is constructed by the +action of multitrace operators whose dimension is finite in the large N limit, i.e, n is an O(1) +number. +6.6 +Kourkoulou-Maldacena states in SYK model +The SYK model is a quantum mechanical model of N Majorana fermions interacting with +random interactions which is given by the Hamiltonian +H = +� +iklm +jiklm ψiψkψlψm , +(6.49) +where ψi are the Majorana fermions {ψi, ψj} = δij, and the coupling jiklm has drawn from +the distribution +P(jiklm) ∼ exp +� +−N3j2 +iklm/12J2� +, +(6.50) +leading to disorder average of +jiklm = 0, +j2 +iklm = 3!J2 +N3 . +(6.51) +In a particular realization of the couplings, we consider pure states which are obtained +by using the Jordan-Wigner transformation and combining pairs of Majorana fermions into +qubit like operators and choosing states with definite eigenvalues for the σ3 components of +– 51 – + +all qubits. These states are denoted by |Bs⟩, where s = (s1, s2, ..., sN/2) with sk = ±1, and +they satisfy the relations below +Sk |Bs⟩ = sk |Bs⟩ , +(6.52) +where Sk = σk +3/2 ≡ 2i ψ2k−1ψ2k is the spin operator. By choosing all possible combinations +of the {sk}’s we get a basis of the Hilbert space whose dimension is 2N/2 (N is an even integer +number). We further evolve these states over some distance l in Euclidean time in order to get +low energy states |Bs,l⟩ = e−lH |Bs⟩ which we will refer to as Kourkoulou-Maldacena (KM) +states. To stay in the low-energy regime where the SYK model exhibits conformal invariance +we take 1 ≪ lJ ≪ N [76]. +As discussed in [76] the KM states can be thought of as a toy model of pure black hole +microstates which are out of equilibrium and which contain excitations behind the horizon. +Hence they are states which exhibit time-dependence and our general formalism should be +applicable. We start by discussing the behavior of the return probability for these states. +6.6.1 +Analytical computation of the return probability at large N +We start with the normalization of the KM states. In the large N limit, due to the approxi- +mate O(N) symmetry of the theory it can be shown [76] that +⟨Bs,l |Bs,l⟩ = ⟨Bs| e−2lH |Bs⟩ = 2−N/2Z(β) , +(6.53) +where β = 2l [76]. The return probability then in the large N limit is given by +R(T) = +���⟨Bs,l|e−iHT |Bs,l⟩ +⟨Bs,l |Bs,l⟩ +��� +2 += +���Z(β + iT) +Z(β) +��� +2 +. +(6.54) +In a low temperature expansion, the partition function can be estimated [124] using the +Schwarzian approximation to be +Z(β) ∝ e2 +√ +2π2αS N +βJ +(βJ)3/2 +. +(6.55) +Using (6.55) we find for the return probability +R(T) = +1 +(1 + T 2 +β2 )3/2 e +−(4 +√ +2π2αS +N +Jβ3 )T 2 +, +(6.56) +which is compatible with (3.18), after we take into account the different N-dependence in the +SYK model vs N = 4 SYM. +We can now try to test the more general decay of the inner product between states in +time-shifted code subspaces (3.21). Let us denote the unit-normalized KM states as +| �Bs,l⟩ = +|Bs,l⟩ +� +⟨Bs,l|Bs,l⟩ , +(6.57) +– 52 – + +and denote their time-dependence as | ˆBs,l(T)⟩ = e−iHT | �Bs,l⟩. We consider an operator A(t) +which is a simple combination of the fermions, so that the state A(t)| ˆBs,l⟩ is in the code +subspace. Then we write +⟨ �Bs,l(0)|A(t)| �Bs,l(T)⟩ = ⟨ �Bs,l(0)| �Bs,l(T)⟩ × ⟨Bs,l(0)|A(t)|Bs,l(T)⟩ +⟨Bs,l(0)|Bs,l(T)⟩ +. +(6.58) +Let us focus on the last ratio. We can rewrite it as +⟨Bs,l(0)|A(t)|Bs,l(T)⟩ +⟨Bs,l(0)|Bs,l(T)⟩ += ⟨Bs|e−(l+i T +2 )HA(t − T +2 )e−(l+i T +2 )H|Bs⟩ +⟨Bs|e−(l+i T +2 )He−(l+i T +2 )H|Bs⟩ +, +(6.59) +which depends holomorphically on l + i T +2 , so we can evaluate if by analytic continuation. All +in all we find +⟨ �Bs,l(0)|A(t)| �Bs,l(T)⟩ = ⟨ �Bs,l(0)| �Bs,l(T)⟩ × +� +⟨ ˆBs,l(0)|A(t − T +2 )| ˆBs,l(0)⟩ +� +l→l+i T +2 +. +(6.60) +At large N and for flip-invariant operators [76] we can also write this as +⟨ �Bs,l(0)|A(t)| �Bs,l(T)⟩ = ⟨ �Bs,l(0)| �Bs,l(T)⟩ × ⟨A(t − T +2 )⟩β|β→β+iT , +(6.61) +where in the last term we first compute the thermal 1-point function ⟨A(t− T +2 )⟩β as a function +of β and then analytically continue β. +As an example, we consider the case where A = ψk(t)ψk(t′) (no summation over k +implied). Following [76] we have for real time and large N +⟨ �Bs,l(0)|ψk(t)ψk(t′) | �Bs,l(0)⟩ = Gβ(t − t′) , +(6.62) +where, for t > t′, we have +Gβ(t − t′) = π1/4 +√2βJ +e−iπ/4 +� +sinh[π(t − iϵ)/β] +, +(6.63) +Therefore, using (6.60) we get +⟨ �Bs,l(0)|ψk(t)ψk(t′)| �Bs,l(T)⟩ = ⟨ �Bs,l(0)| �Bs,l(T)⟩ Gβ+iT (t − t′) , +(6.64) +where the last term can be computed as the analytic continuation of (6.63). +Similarly for A = ψ2k−1(t)ψ2k(t′)Sk we have [76] +⟨ �Bs,l(0)|ψ2k−1(t)ψ2k(t′)Sk| �Bs,l(0)⟩ = −2iskGβ(t)Gβ(t′) + O(1/N), +(6.65) +hence +⟨ �Bs,l(0)|ψ2k−1(t)ψ2k(t′)Sk| �Bs,l(T)⟩ = ⟨ �Bs,l(0)| �Bs,l(T)⟩× +(6.66) +× +� +−2iskGβ+iT (t − T +2 )Gβ+iT (t′ − T +2 ) + O(1/N) +� +. +(6.67) +The examples (6.64) and (6.66) are consistent with our general expectations, see (3.21) and +(3.22). +– 53 – + +β +5 +10 +15 +20 +0.05 +0.10 +0.15 +0.20 +(a) N = 14 +β +5 +10 +15 +20 +0.05 +0.10 +0.15 +0.20 +0.25 +(b) N = 20 +β +5 +10 +15 +20 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +(c) N = 24 +Figure 2: The blue lines are the numerical results for the variance of Hamiltonian as a +function of β while the yellow ones are the Schwarzian approximation ∆H2 = 0.396N/β3. +6.6.2 +Some numerical checks +In this subsection we perform some simple numerical checks of (3.21) and (3.24), as well as +the behavior of the operators (4.2) for KM states in the SYK model. The first step is to select +an appropriate value for the inverse temperature β = 2l. The early time decay of the return +probability is +R(T) = e−∆H2T 2 . +(6.68) +Earlier we used the Schwarzian approximation to compute the partition function (6.55) from +which we can also get the variance +∆H2 = 4 +√ +2π2αS +N +β3 = 0.396 N +β3 . +(6.69) +We compare this result with a numerical computation of the variance ∆H2 for a KM state +constructed from |Bs⟩ = |+ − −...−⟩. This is shown in Figure 2. In Figure 3, we show the +value of the plateau for the KM state, as defined in (3.19) for various values of N and β. +For the range of values of N we are interested in, we can take the inverse temperature to be +β = 5, which is the value we will use in what follows. +l +R +5 +10 +15 +20 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +N=14 +N=16 +N=18 +N=20 +N=20 +N=24 +Figure 3: The plateau height ¯R as a function of l = β/2 . +In Figure 4 we can see the return probability as a function of t for different values of N +for the corresponding KM state. As discussed in subsection 3.6, we expect that the overlap +– 54 – + +JT +R(T) +20 +40 +60 +80 +100 +0.2 +0.4 +0.6 +0.8 +1.0 +N=14 +N=16 +N=18 +N=20 +N=20 +N=24 +Figure 4: Return probability as a function of T for different values of N +between any state in the code subspace at t = 0 will and the one at t = T will also decay +exponentially fast. We can encode the overlap between all such pairs of states by +Rcode(T) = +1 +dcode +Tr[PT P0] . +(6.70) +For the numerical computation we need to make some choice about the code subspace. One +condition is that the dimension dcode of the code subspace should satisfy dcode ≪ 2N/2. As an +example, and for the purpose of the numerical computation, we can define the code subspace +as +Hcode = span{Oi1 +1 ...Oik +k |Bs⟩; ij = 0, 1} , +(6.71) +for some choice of the operators Oi. Here Dcode = 2k the value of k should be such that +D ≪ 2N/2. Note that the states in (6.71) are generally not orthonormal but it is easy to write +a projector on the code subspace in terms of elements of this basis, see [125] for a related +discussion. +In Fig. 5, we see plots of the behavior of Rcode(T) as a function of time for some specific +choices of such a code subspace: +• a : the dimension of the code subspace is D = 8 and the operators are chosen to be +O1 = ψ1(t = 0), +O2 = ψ1(t = 0.1), +O3 = ψ1(t = 0.5). +• b : the dimension of the code subspace is D = 8 and the operators are chosen to b +O1 = ψ1(t = 0), +O2 = ψ1(t = 0.1), +O3 = h. +• c : the dimension of the code subspace is D = 16 and the operators are chosen to be +O1 = ψ1(t = 0), +O2 = ψ1(t = 0.1), +O3 = ψ1(t = 0.5), +O4 = ψ1(t = 1). +where in case (b) the operator h is the normalized Hamiltonian +h = +1 +√ +N +(H − ⟨H⟩). +(6.72) +– 55 – + +0 +20 +40 +60 +80 +100 +0.2 +0.4 +0.6 +0.8 +1.0 +R (T) +code +JT +0 +20 +40 +60 +80 +100 +0.2 +0.4 +0.6 +0.8 +1.0 +N=14 +N=16 +N=18 +N=20 +(a) +0 +20 +40 +60 +80 +100 +0.2 +0.4 +0.6 +0.8 +1.0 +N=14 +N=16 +N=18 +N=20 +0 +20 +40 +60 +80 +100 +0.2 +0.4 +0.6 +0.8 +R (T) +code +JT +(b) +R (T) +code +JT +0 +20 +40 +60 +80 +100 +0.2 +0.4 +0.6 +0.8 +1.0 +N=14 +N=16 +N=18 +N=20 +(c) +Figure 5: Rcode(T) as a function of T for three different examples of codesubspaces in the +form of (6.71). +0 +20 +40 +60 +80 +100 +0.2 +0.4 +0.6 +0.8 +1.0 +N=14 +N=16 +N=18 +N=20 +R (T) +code +JT +(a) +● +● +● +● +● +■ +■ +■ +■ +■ +2 +4 +6 +8 +0.1 +0.2 +0.3 +0.4 +0.5 +● +■ +boundary-dressed +state-dressed +Jt +(b) +Figure 6: Results for the code subspace (6.73). (a) Rcode(T) as a function of T. (b) The blue +line is ⟨ψ3(0)ψ3(t)⟩ as a function of t, while in the case of the yellow line, ψ3(0) is replaced +by the dressed operator obtained from our proposal. Here N=20. +We finally check that the operator (4.2) has similar correlators as the boundary-dressed +operator. We take the code subspace as +Hcode = span{|Bs⟩, O1|Bs⟩, ...Ok|Bs⟩, h|Bs⟩, hO1|Bs⟩, ...hOk|Bs⟩}, +(6.73) +where the dimension of the code subspace is dcode = 2(k + 1) ≪ 2N/2. In Fig. 6, we plot the +result for the case of k = 5 and where the operators chosen to be +O1 = ψ1(t = 0), +O2 = ψ1(t = 2), +O3 = ψ1(t = 4) +O4 = ψ1(t = 6), +O5 = ψ1(t = 8) +for N = 20 (dcode = 12 ≪ 210) are plotted. One can see from Fig.6b that the state-dressed +operator for ψ3 has approximately the same correlation function as the original one. +6.7 +Holographic boundary states +The KM states discussed in the previous section can be thought of as certain a-typical black +hole microstates in the context of SYK/AdS2. +Interesting analogs in higher dimensional +examples of AdS/CFT can be found by considering boundary states in CFTs [126–128]. A +boundary state characterizes boundary conditions which can be imposed on a boundary of +space-time on which the CFT lives. For each allowed boundary condition, we can evolve the +– 56 – + +state along the Euclidean time to suppress the high-energy contributions and obtain a state +of finite energy which is called a regularized boundary state of the CFT. +For holographic theories, the CFT path integral maps onto the gravity path integral. +Therefore, we will be able to make use of the AdS/CFT correspondence to deduce the cor- +responding geometries if we can choose a state for which we can understand a gravity pre- +scription for dealing with the boundary condition at the initial Euclidean time. As discussed +in [129], we can describe boundary states by starting with the TFD state of two CFTs labeled +by L and R +|TFD(β/2)⟩ = 1 +Z +� +i +e−βEi/4 |Ei⟩L ⊗ |Ei⟩R , +(6.74) +and then project the TFD state onto some particular pure state |B⟩ of the left CFT. As a +result we obtain a pure state of the right CFT given by +|ΨB,β⟩ = 1 +Z +� +i +e−βEi/4⟨B |Ei⟩ |Ei⟩ . +(6.75) +If the temperature is high enough, the TFD state is dual to the maximally extended +AdS-Schwarzschild black hole in the bulk. The geometry which is dual to these regularized +boundary states is expected to contain a significant portion of the left asymptotic region. +Therefore, in a holographic CFT, this class of regularized boundary states can be regarded +as microstates of a single-sided black hole. These black hole microstates can be thought of +as black holes with end of the world (EOW) branes on the left side.45 Generally the EOW +brane configuration is time-dependent at the macroscopic level. Hence these are states with +energy and energy variance compatible with (3.1) and (3.3), so we expect to be able to apply +our construction and define operators (4.2). As we will discuss in the next section, one way +to think of them is that the gravitational dressing has been moved over to the EOW brane. +6.7.1 +Computation of the return probability and correlators +First we define unit-normalized boundary states +| �Ba(0)⟩ = +e− βH +4 |Ba⟩ +� +⟨Ba| e− βH +2 |Ba⟩ +. +(6.76) +Then we want to show that return probability of a boundary state +R(T) = |⟨ �Ba(0) | �Ba(T)⟩ |2 , +(6.77) +decays exponentially fast at early time. For boundary states in holographic 2d CFTs we have +(E.11) +G(β) = ⟨Ba|e− βH +2 |Ba⟩ ≃ e +π2c +6β . +(6.78) +45Proving from first principles that boundary states dual to EOW branes exist is far from trivial. It has been +investigated from a bootstrap perspective in [130], where it was suggested that such boundary states must be +extremely fine-tuned. In [131], the full classification of boundary states in large N symmetric orbifolds was +carried out, and typical boundary states are not of this form. +– 57 – + +where we have taken the CFT to be defined on a spatial circle of length 2π. For small T we +have +R(T) = |G(β + 2iT)|2 +|G(β)|2 +≃ e +− 4π2c +3β3 T 2 +. +(6.79) +The energy variance of the boundary state can be easily computed from (6.78) and we find +∆H2 = ⟨H2⟩ − ⟨H⟩2 = 4π2c +3β3 , +(6.80) +so the initial decay (6.79) is, not surprisingly, consistent with (3.16), (3.18) and (6.80). +In higher dimensional cases we can read from (E.13) +G(β) = e +αd +βd−1 , +(6.81) +thus +R(T) = |G(β + 2iT)|2 +|G(β)|2 +≃ exp +� +− αd +βd+1 4d(d − 1)T 2 +� +. +(6.82) +We can again check that +∆H2 = ⟨H2⟩ − ⟨H⟩2 = +αd +βd+1 4d(d − 1), +(6.83) +which is compatible with (6.82). +We now proceed with checking that the other states in the code subspace around a +boundary state are orthogonal to the time evolved code subspace. Consider for example the +state O(t, x)| �Ba⟩. Following similar reasoning as in subsection 6.6 we can show that +|⟨ �Ba(0)|O(t, x)| �Ba(T)⟩|2 = ⟨ �Ba(0)|O(t, x)| �Ba(T)⟩ ⟨O(t − T +2 , x)⟩β→β+2iT . +(6.84) +where ⟨ �Ba(0)|O(t, x)| �Ba(T)⟩ = Ga(I,β+2iT) +Ga(I,β) +. More generally +⟨ �Ba(0)|O(t1, x1)O(t2, x2)...O(tn, xn)| �Ba(T)⟩ = +⟨ �Ba(0)|O(t, x)| �Ba(T)⟩⟨O(t1 − T +2 , x1)O(t2 − T +2 , x2)...O(tn − T +2 , xn)⟩β→β+2iT . +(6.85) +Thus, as long as the analytical continuation of the correlation function in β does not introduce +any surprising N-dependent factors we will get the expected behavior (3.22). We now check +this condition for low-point functions in 2d boundary states. +Here we assume that for a holographic CFT, and if we are working in the large N limit, +the 1-point function of light conformal primaries can be computed by a method of images. +Then for a 1-point function of a scalar primary O with dimension ∆ on a boundary state we +have +⟨ �Ba(0)|O(t, x)| �Ba(0)⟩ = +AO +( β +π cosh[ 2π +β t])∆ . +(6.86) +– 58 – + +for some constant AO which depends on the boundary state a and the operator O. After the +analytic continuation necessary for (6.84) we find +⟨O(t − T +2 , x)⟩β→β+2iT = +AO +( (β+2iT) +π +cosh[ +2π +(β+2iT)(t − T +2 )])∆ . +(6.87) +Hence we notice that the results (6.84),(6.87) are consistent with our general expectations +(3.21),(3.22). +We can also check 2-point functions, which we can compute in the large N limit. First +we compute the 2-point function on the boundary state, using the method of images +⟨ �Ba(0)|O(t1, x1)O(t2, x2)| �Ba(0)⟩ = ++∞ +� +n=−∞ +1 +�� β +π sinh +� +π +β[(x1 − x2 + 2πn) − (t1 − t2)] +���2∆ ± +1 +�� β +π cosh +� +π +β[(x1 − x2 + 2πn) − (t1 + t2)] +���2∆ , +(6.88) +After the analytic continuation necessary for (6.85) we find from (6.88) that we do not notice +any unexpected behavior of this part of the correlator as T increases, so the result (6.85) +is dominated by the decay of the return probability, and is consistent with our expectations +(3.21),(3.22). +7 +Black Hole microstates +One question which is particularly interesting is whether we can apply our construction to +black hole microstates. +We have already mentioned in section 3.1 that there are various +classes of black hole microstates, some of which have macroscopic time dependence and some +of which do not. We will now discuss these various cases in more detail and interpret our +operators for these types of states. +7.1 +States with macroscopic time-dependence +We will start with the simplest situation: states with macroscopic time-dependence. This +can be visible outside the horizon, for example black holes in the presence of infalling matter. +Alternatively it can be that the geometry appears to be static outside the horizon but there +is no corresponding Killing isometry in the interior. As the first case is more straightforward, +we focus on the second case. Two examples of such states are boundary states of the CFT, +corresponding to end-of-the-world branes inside the horizon, which have already been dis- +cussed in the previous section. A second example is states prepared by the Euclidean path +integral on some surface of higher topology. The dual geometries have topology behind the +horizon, and are often referred to as geons [80, 132, 133]. It is worth re-emphasizing that +both of these states are usually prepared by the Euclidean path integral and are in fact very +a-typical states, even if the CFT 1-point functions are very close to those in a thermal state +– 59 – + +(or said differently, even if the classical geometry is exactly that of a black hole outside the +horizon). +Both of these examples involve pure states |Ψ0⟩ that have a large energy variance, of order +N2, such that the return probability will decay as (3.17). We can thus apply our construction +to build local operators that are not dressed to the boundary CFT. The interpretation is that +the operators are dressed with respect to the time-dependence of the interior. Consider for +example the genus-2 geon in d = 2, which is prepared by the Euclidean path integral on half +of a genus-2 surface [80,134]. Microscopically, the state can be described by +|Ψ0⟩ ∼ +� +i,j +Ciije−Eiβi/2−Ejβj |Ej⟩ , +(7.1) +where ∼ indicates that we have not been careful about the parametrization of the genus- +2 surface, but βi,j are related to the moduli of the surface. +The un-normalized overlap +of this state corresponds to a genus-2 partition function in the dumbbell channel, where +βj parametrizes the length of the two handles, and βi parametrizes the length of the neck +between them. +It is not straightforward to write down a metric that covers the entire space-time of such +states. Outside the horizon whose size is controlled by βj, they look exactly like the BTZ +geometry. Inside the horizon, they have macroscopic time-dependence. A nice coordinate +patch that covers the Wheeler-de Witt patch of the t = 0 slice of the geometry can be written +down in a very simple form +ds2 = −dt2 + cos2 t dΣ2 +2 , +(7.2) +where dΣ2 +2 is the constant negative curvature metric on half of a genus-2 surface. +This +coordinate patch covers the entire t = 0 slice of the geometry, which is precisely half of a +genus-2 surface. The neck corresponds to the horizon, and there is topology (one handle) +behind the horizon. From this metric, we explicitly see the time dependence of the geometry, +even if a metric for the full spacetime is hard to write down. +The interpretation of our +operator is that the dressing is to the time-dependence of the geometry that sits inside the +horizon. For end-of-the-world brane geometries, the situation is similar and the operator is +dressed to the end-of-the-world brane. +7.2 +Typical states +The question we would now like to ask is whether our prescription works in typical black hole +microstates. Contrary to states with end-of-the-world branes or topology behind the horizon, +it seems reasonable to expect that typical states should also look like the thermal state a +finite distance inside the black hole (see for example [135,136]). +Whether or not our prescription works depends on the definition of a typical black hole +microstate, and in particular on the energy spread we are choosing. One possibility is to +define typical states using an ensemble of energy eigenstates with spread O(N0) in energy +(recall that there are still eS with S ∼ O(N2) states in this energy band). In that situation, +– 60 – + +our prescription does not work, as the variance of energy is O(N0) and the return probability +will not decay fast enough. Another possibility is to consider typical states with an energy +spread similar to that of the canonical ensemble, that is +(∆E)2 ∼ O(N2) . +(7.3) +For such states, the return probability will decay following the behaviour (3.17). Therefore, +we can follow our prescription and define the operators in the same way and they will satisfy +the two properties of commuting with the Hamiltonian to all orders in 1/N and acting like +HKLL operators to leading order at large N. +While these operators are certainly diff-invariant, since they are operators defined in the +CFT, the bulk interpretation of their gravitational dressing on typical black hole microstates is +not entirely clear. When the gravitational configurations are macroscopically time-dependent, +our operators are dressed with respect to the features of the geometry. The typical states are +still time-dependent, but only microscopically, as it seems plausible to assume that macro- +scopically they are featureless. In some sense our operators are dressed to the microscopic +time-dependence of the state (the phases of the ci in (3.2)), but it is unclear exactly what +that means in the bulk. +Notice however, that if we start with a particular typical pure state |Ψ0⟩ and act with a +unitary made out of the operator (4.2), associated to that state, then the predictions for what +an infalling observer jumping into the black hole will see are unambiguous. For example, the +operators (4.2) will generally create an excitation in the bulk and the location in time relative +to that of the infalling observer who jumps from the boundary at a particular boundary time, +can be unambiguously computed for each state |Ψ0⟩ and corresponding operators (4.2). We +emphasize that for this interpretation it is important to remember that the operators (4.2) +are state-dependent and cannot generally be promoted to a single operator which acts in a +specific way globally on most typical states. +We briefly comment on black hole interior reconstruction. Suppose we start with a typ- +ical black hole microstate with energy spread of order (3.3). If we assume that the interior +geometry contains part of the left asymptotic region, then the possibility of removing the +dressing of the operators implies that we can deform the state behind the horizon by creat- +ing some particles there, in such a way that these excitations cannot be detected from the +boundary CFT by the measurement of single-trace correlators, including the Hamiltonian, in +the 1/N expansion. This was also discussed in [137, 138]. We emphasize that this does not +contradict the statements made in [25, 135, 136] that for typical states with microcanonical +energy spread, it is impossible to add excitations without affecting single-trace correlators. +7.3 +Two entangled CFTs +Similar considerations apply to geometries with two asymptotically AdS regions. Consider +two non-interacting CFTs with total Hamiltonian H = HL+HR. We take the full system to be +in a pure state |Ψ0⟩ which may be entangled, but we will assume the pattern of entanglement +– 61 – + +is generic. In particular, we do not consider states like the thermofield-double which have a +very fine-tuned structure of entanglement. We can imagine the state |Ψ0⟩ to be, for example, +UL |TFD⟩, where UL is a complicated random unitary acting on the left CFT. In this case we +can consider the following generalization of our construction. Let us consider the 2-parameter +family of time-shifted states +e−i(TLHL+TRHR)|Ψ0⟩. +We start with an HKLL operator Φ dressed with respect the to left system, which commutes +with HR but not HL. We now consider the following generalization of the operators (4.2) +�Φ = c +� +dTLdTRe−i(TLHL+TRHR)P0ΦP0ei(TLHL+TRHR) +(7.4) +using P0 = P L +0 ⊗ P R +0 and [Φ, P R +0 ] = 0 then +�Φ = c +� +dTLe−iTLHLP L +0 ΦP L +0 eiTLHL ⊗ +� +dTRP R +TR +(7.5) +The resulting operator commutes with both HL and HR on the relevant code subspaces. In +this case, the operator is not dressed with respect to the overall time-dependence of the full +system, but rather to the time dependence of the “left” subsystem. +There are states with special entanglement pattern such as the TFD state, which was +already discussed in section 6.2. The generalized return amplitude ⟨Ψ0|e−i(HLTL+HrTR)|Ψ0⟩ +which is a function of TL and TR does not decay in all directions for these special states. For +example, in the TFD state it is constant along the line TL = −TR. In those cases we cannot +set both commutators with HL, HR to zero. So we can move the dressing from one side to +another if we wish to, but there it is always dressed to one of the boundaries. This happens +because the TFD state has a symmetry, it is annihilated by HL − HR. +7.4 +Island discussion +Our prescription is also useful to resolve some paradoxes in the context of black hole evapo- +ration and islands. Consider a setup where a holographic CFT is coupled to a bath such that +the bulk description is given by an evaporating black hole. After the Page time, a non-trivial +quantum extremal surface appears in the bulk delimiting an island, i.e. a part of the interior +of the black hole that is encoded in the bath degrees of freedom rather than in those of the +CFT [11,139]. +There is an apparent tension in this context related to gravitational dressing [140]. If +we create an excitation in the island by acting with a local operator φisland, where does the +gravitational dressing go? It appears that the only place for the dressing to go is the boundary +CFT. But this implies that the local operator will have the property +[φisland, HCFT] ̸= 0 . +(7.6) +But this seems to be inconsistent, because since the operator is in the island, it should be +reconstructable from the bath degrees of freedom, and commute with the CFT degrees of +freedom. +– 62 – + +Our operators provide a way out of this paradox. We can apply our prescription above +in terms of two entangled systems with a generic pattern of entanglement (there is a subtlety +here since the bath and CFT are actually coupled rather than non-interacting, but we can +treat this interaction as weak). In that case, even if we did start with an operator that had a +non-trivial commutator (7.6), we would engineer a new operator which commutes with HCFT +up to exponentially small corrections. This new operator is now dressed with respect to the +radiation, rather than the boundary CFT. +The interpretation of the dressing is similar to that of the typical states. While it would +be tempting to imagine dressing the operator to the quantum extremal surface, the bulk +geometry only has extremely slow time-dependence so it is unclear if time-dependent features +of the geometry are sharp enough to dress with respect to them. It appears that that the +dressing is towards the microscopic time-dependence of the radiation. The story becomes less +subtle if we consider a doubly holographic model (see for example [10,141]). In that case, the +dressing to the bath can be directly geometrized in the higher-dimensional geometry. Our +operators can perhaps be thought as a counter-part of the operators in the doubly-holographic +setup, but in cases where the dressing cannot be so easily geometrized. +Finally, we would like to clarify the distinction between reconstruction and dressing. To +make things simple, let us consider the TFD state and consider an HKLL operator on the +left φL. This operator is dressed to the left CFT. Now we run our protocol, and as explained +above, we can move the dressing to the right. The operator ˆφL now commutes with HL but +no longer with HR [95]. This does not mean that it can be reconstructed from the right +degrees of freedom, but that it can be detected from the right CFT via the Gauss law tail. It +is still mostly built from the left CFT degrees of freedom, only its dressing has been pushed +to the right. +8 +Discussion +In this paper, we have investigated whether information can be localized in perturbative +quantum gravity, in the context of the AdS/CFT correspondence. The challenge at hand is +to construct local diff-invariant operators that are not dressed to the boundary where the +CFT lives. We have presented evidence that such operators exist, at least around high energy +states with a large energy variance. Such states include semi-classical geometries with features +that break the symmetries of the dual CFT and for such states, local operators can be dressed +to the features of the state. We have argued that there exist CFT operators that commute +with all single-trace operators in a narrow time-band to all orders in the 1/N expansion, +including the Hamiltonian and other charges that generate conformal transformations, while +at the same time act like standard HKLL operators to leading order at large N. +We have presented an explicit construction of such operators, and checked that they +commute with the Hamiltonian to all orders in the 1/N expansion, and act like HKLL oper- +ators to leading order. Technically the construction of such operators is made possible due +to the fact that different semi-classical states have exponentially small overlap. We have also +– 63 – + +discussed a generalization of our operators that would commute with all boundary charges of +the conformal group. Moreover, we presented a definition of operators that commutes with +all single-trace operators, not just conserved charges. The construction of these operators +is slightly less explicit, and we define them by specifying their action on the code subspace +around a semi-classical geometry. We argue that such operators commute with all single- +trace operators in a narrow CFT time-band, while also acting like HKLL operators to leading +order at large N. Acting with such operators creates excitations that are completely invisible +to CFT correlation functions in a narrow time-band, even if they become accessible at later +times when a lightray from the location where the bulk excitation was created reaches the +boundary. This suggests that information can be lozalized in perturbative quantum gravity, +to all orders in GN perturbation theory. We conclude with some open questions that we +raised along the way. +8.1 +The variance of the energy from semi-classical gravity +A quantity that played a primordial role throughout the paper is the variance of energy, which +controls the early time decay of the return probability through (3.16). One question that +would be interesting to understand better is how we can compute the variance ⟨Ψ0|∆H2|Ψ0⟩ +from semi-classical gravity. In appendix A we give an example that we can change the O(N2) +coefficient of the variance of the Hamiltonian without changing the semi-classical geometry. +This implies that the variance of the energy is not just a property of the geometry, but also +of the quantum state of the fields on top of that geometry. Of course, if the metric changes as +a function of time, this puts a bound on the variance through (3.5). This suggests that if we +start with some time-dependent semi-classical geometry with a matter QFT state with large +variance, it should not be possible to change the state in a way to make the variance decrease +to O(1) without changing the metric towards a time-independent solution. The mechanism +by which this would happen is unclear, and it would be interesting to pursue it further. +On a related note, we can ask how we can quantize the bulk mode associated to the Hamil- +tonian directly in gravity. The expectation value of the Hamiltonian is extracted through the +fall-off of the metric near the AdS boundary, as is standard in AdS/CFT, but this does not +capture its quantum 2-point function. If one computes the stress-tensor connected 2-point +function on the geometry, takes the relevant components and performs the spatial integrals, +one should obtain the variance. It would be desirable to have a more direct representation +of the variance in terms of the the bulk wavefunction of the non-propagating s-wave mode of +the graviton and also understand from this point of view the lower bound on the variance for +time-dependent geometries. +8.2 +Gravitational proof for the decay of the return probability +A central part of this paper was played by the decay of the return probability. The physical +interpretation of this decay for a semi-classical time-dependent geometry is that it computes +(the square) of an overlap between two distinct geometries, namely the original one and the +– 64 – + +time-evolved one. The general expectation is that the overlap of two distinct coherent states +should be given by +⟨λ1|λ2⟩ ∼ e−N2f(λ1,λ2) , +(8.1) +where f is some O(1) function whose real part is positive (we have assumed that the states +|λ1,2⟩ are normalized). The intuition is that N2 plays the role of 1/ℏ which controls the overlap +of coherent states, and from a gravitational stand-point, the on-shell action of any geometry +will be proportional to 1/GN. +However, this gravitational argument does not necessarily +imply that the real part of f is positive, which is required by reflection positivity of the CFT +dual. As we have seen in (6.7), interpreting geometries as quantum states implies constraints +on various on-shell actions. +It would be interesting to understand this problem directly in gravity. Can reflection +positivity be proven directly at the level of the gravitational path integral? This requires +proving (6.7) directly in gravity. A possible way to prove this is the following: we consider +two states λ1 and λ2 with fixed sources, and their associated geometries contributing to the +overlaps ⟨λ1,2|λ1,2⟩, with geometries g1 and g2 and on-shell actions I1 and I2. We start by +considering a gravitational configuration which is half of g1 (say the northern hemi-ball) and +half of g2 (the southern hemi-ball). This configuration has action +Itot = I1 + I2 +2 +. +(8.2) +Note that the geometry is off-shell at the gluing surface between g1 and g2, and there could be +another contribution Ijunction to the action coming from the gluing, which we will not include +for now. To find the smooth saddle-point geometry, we need to let this geometry relax by +modifying its configuration near the junction. One may be able to prove that this smoothing +of the glued geometry comes with a definite sign in the action, therefore proving (6.7). It +would be interesting to pursue this idea. +8.3 +Microscopically time-dependent states +We have seen that for any state with large energy and large energy variance, we can find +bulk local operators who commute with the time-band algebra. The interpretation of these +operators is that they are dressed with respect to features of the state (in particular the +time-depdence of the state), rather than to the boundary CFT. This intuitive picture is clear +when the state describes a semi-classical geometry that is macroscopically time-dependent, +as the time-dependence can be seen directly from the background metric which has features +with respect to which we can attach a gravitational dressing. +As we have discussed, our prescription also works for typical states with energy variance +of O(N2). In that context, the interpretation of the dressing is less clear. The dual geometry +is not macroscopically time-dependent. We can declare that the operator is dressed with +respect to the microscopic time-dependence, but it is unclear what that means. It would be +interesting to have a better physical understanding of the dressing for such type of states. +We hope to return to this question in the future. +– 65 – + +It is also important to note that our operators are state-dependent, even outside the +horizon. For a given typical state, we can use our construction to find the state-dressed local +operator. However, if we now pick a different typical state then the operator will not act in +the desired fashion. In this sense, our operators are similar to mirror operators [6], but they +can live outside the horizon. Nevertheless, we wish to emphasize again that independently +of questions surrounding the interpretation of these operators, an important message of this +paper is that these operators exist and that states created by acting on the corresponding +typical state with unitaries built from these state-dressed operators have identical correlators +of single-trace operators in a narrow time-band in the 1/N expansion as the original state. +Moreover, this can be done around any typical state once the state has been fixed. +8.4 +Microcanonical states and small energy variance +There are also typical states with a small energy variance, of O(N0). For example, when +one refers to the microcanonical ensemble, one often has in mind picking a state with spread +in energy which is O(N0). For such states, the return probability does not decay to values +which are exponentially small in N2 after an order one time, which means we cannot use our +construction to define state-dressed operators. The variance of the energy is a very coarse way +to define how time-dependent a state is, and for states with energy variance of size O(N0), +the state is not time-dependent enough to dress operators to it. Of course, all these states +look macroscopically time independent, and all the information is in the microscopic phases +of the state. It would be interesting to study this further, and have a better physical picture +of whether one can find state-dressed operators to these small variance states. +It is worthing mentioning that if the variance is O(Nc) for any 0 < c < 2, our pre- +scription does work. For typical states, this is some kind of intermediate regime between +canonical states and microcanonical states. +For coherent states that are macroscopically +time-dependent, this situation would occur if the profile of the fields are not O(1), but rather +scale with some positive power of GN. In that case, backreaction is small, but the return +probability still decays. It would be interesting to understand these regimes better, they +interpolate between coherent states of the bulk quantum fields propagating on a frozen AdS +background, and semi-classical geometries with a non-trivial metric. +8.5 +The AdS vacuum and low-energy states +For low-energy states like the AdS vacuum or states with an O(N0) energy above it, our +construction does not work. Therefore, the results of this paper do not contradict the claims +of [46], that for perturbative excitations on top of the AdS vacuum one can reconstruct the +state directly from the time-band. Technically, this happens because the return probability +does not decay to exponentially small values for such states. Physically, states like the AdS +vacuum have no features to which we can dress operators, so the only possible diff-invariant +way to specify a point is with relation to the boundary. Even classically, there are no diff- +invariant local observables in classical general relativity for the case of vacuum AdS. It thus +appears that the failure of constructing approximately local diff-invariant operators around +– 66 – + +the AdS vacuum happens because of the special nature of the state, rather than a fundamental +obstruction due to the non-locality of quantum gravity. +For excited states on top of the AdS vacuum, it is less obvious why local diff-invariant +states cannot be constructed. +One may imagine that if the VEV of a scalar field has a +quantum lump in some region of space-time, we could dress an operator to the location of +this lump. Technically, we see that at least our operators cannot achieve this goal. It would be +interesting to have a more physical understanding of why it is not possible to dress operators to +quantum profiles, rather than semi-classical ones. As we have seen in the previous subsection, +it is not completely related to backreaction. If we consider a coherent state on top of vacuum +AdS corresponding to a source which scales as N1/4, the return probability would decay fast +enough for our construction to work, even if backreaction can be neglected. Note however +that such a state is not really part of the low-energy EFT on top of vacuum AdS, since it has +energy that scales with some fractional power of the Planck scale. It would be interesting to +understand this better. +Acknowledgments +We are happy to thank Micha Berkooz, Jan de Boer, Monica Guica, Elias Kiritsis, Shota +Komatsu, Hong Liu, Olga Papadoulaki, Suvrat Raju, Erik Verlinde, Spenta Wadia, and +Sasha Zhiboedov for stimulating discussions. EB and NV would like to thank CERN-TH +for their hospitality during the preparation of this work and M. Bertolini for his invaluable +support during this work. The work of EB and NV is partially supported by INFN Iniziativa +Specifica - String Theory and Fundamental Interactions project. +A +Changing the variance of H +We would like to understand whether the variance of the energy is accessible within semi- +classical gravity, simply from the geometry, or whether it requires more knowledge and in +particular, the knowledge of the bulk quantum state for the fields propagating on the back- +ground. As we will see, knowledge of the quantum state seems to be required to extract the +variance. +The quantity we would like to compute is +⟨Ψ0|H2|Ψ0⟩ − ⟨Ψ0|H|Ψ0⟩2 ≡ ⟨Ψ0|H2|Ψ0⟩c . +(A.1) +This is a connected correlation function in holography, which usually would be compute from +the 2-point function of the associated propagating fields on the relevant background. This +2-point function is sensitive both to the geometry and to the bulk quantum state of the +propagating fields. However, here the situation is more subtle, because we are not studying +the local correlation function of an operator, but rather the 2-point function of the spatial +integral of a local operator. In this particular case, the situation is a lot more confusing +– 67 – + +because the dual bulk field would be the s-wave graviton, which is not a propagating degree +of freedom in gravity. +So what computes this variance? +We will not be able to answer this question, and +we believe it to be an interesting open problem which we hope to return to in the future. +Nevertheless, we will study some particular states that should be interpreted as adding an s- +wave graviton in the bulk. Even though this mode doesn’t propagate, we will see that adding +it can affect the CFT variance. We will consider two type of deformations of the thermofield +double (TFD) state, both of which are related to adding an integrated stress-tensor operator +on the cylinder that prepares the TFD state. Let us start with some basics. We consider the +TFD state +|TFD⟩ = +1 +√ +Z +� +i +e−βEi/2 |Ei⟩ |Ei⟩ . +(A.2) +We assume that the partition function has the usual large N behavior +Z(β) = exp +� +N2 +� +F0(β) + 1 +N2 F1(β) + ... +�� +, +(A.3) +from which we can compute +⟨Hn⟩β = (−1)n 1 +Z +dn +dβn Z . +(A.4) +where H is HL or HR. We have +⟨TFD| H |TFD⟩ = ⟨H⟩β = −N2F ′ +0 − F ′ +1 , +(A.5) +⟨TFD| H2 |TFD⟩ − ⟨TFD| H |TFD⟩2 = ⟨H2⟩β,c ≡ ⟨H2⟩β − ⟨H⟩2 +β . +(A.6) +We have +⟨H2⟩β,c = N2F ′′ +0 + F ′′ +1 . +(A.7) +Now, consider the following state +|ψ⟩ = H |TFD⟩ . +(A.8) +We now have +⟨ψ|ψ⟩ = ⟨H2⟩β +(A.9) +Let us now see how the energy and variance of the state have evolved. We have +⟨ψ| H |ψ⟩ +⟨ψ|ψ⟩ += ⟨TFD| H3 |TFD⟩ +⟨TFD| H2 |TFD⟩ = +⟨H⟩3 +β + 3 ⟨H2⟩β,c ⟨H⟩β + ⟨H3⟩β,c +⟨H⟩2 +β + ⟨H2⟩β,c +, +(A.10) +where we defined +⟨H3⟩β,c ≡ ⟨H3⟩β − 3 ⟨H2⟩β,c ⟨H⟩β − ⟨H⟩3 +β . +(A.11) +Large N factorization implies that we can expand this answer and we find +⟨ψ| H |ψ⟩ +⟨ψ|ψ⟩ += ⟨H⟩β + 2 +⟨H2⟩β,c +⟨H⟩β ++ · · · += −N2F ′ +0 − F ′ +1 − 2F ′′ +0 +F ′ +0 ++ · · · . +(A.12) +– 68 – + +We see that we obtain the TFD answer, up to a correction term, which is of size N0. This +means we have not changed the geometry classically, but only added a quantum particle on +top of the TFD state. Similarly, one can compute +⟨ψ| H2 |ψ⟩ +⟨ψ|ψ⟩ +− +�⟨ψ| H |ψ⟩ +⟨ψ|ψ⟩ +�2 += +⟨H⟩4 +β + 6 ⟨H2⟩β,c ⟨H⟩2 +β + · · · +⟨H⟩2 +β + ⟨H2⟩β,c +− +� +⟨H⟩2 +β + 4 ⟨H2⟩β,c + · · · +� += ⟨H2⟩β,c + · · · += N2F ′′ +0 + · · · +(A.13) +We see that that the energy has changed at N0, but the variance has not changed at order +N2, only at order N0. So this state modifies both the variance and the energy at subleading +order compared to the TFD. We will now build a state that modifies the energy at subleading +order, but the variance at leading order compared to the TFD. +Consider the state +|φ⟩ = (H − ⟨H⟩β) |TFD⟩ . +(A.14) +We now have +⟨φ|φ⟩ = ⟨H2⟩β,c , +(A.15) +and we can now compute the energy in this state: +⟨φ| H |φ⟩ +⟨φ|φ⟩ += +⟨H3⟩β − 2 ⟨H2⟩β ⟨H⟩β + ⟨H⟩3 +β +⟨H2⟩β,c += ⟨H⟩β + +⟨H3⟩β,c +⟨H2⟩β,c += −N2F ′ +0 − F ′ +1 − 2F ′′′ +0 +F ′′ +0 ++ · · · . +(A.16) +We see that this state modifies again the energy only at order N0, and in a slightly different +way than the previous state. In a similar way, we compute the variance and find +⟨φ| H2 |φ⟩ +⟨φ|φ⟩ +− +�⟨φ| H |φ⟩ +⟨φ|φ⟩ +�2 += ⟨H⟩2 +β + 3 ⟨H⟩2 +β,c + +2 ⟨H3⟩β,c ⟨H⟩β + ⟨H4⟩β,c +⟨H2⟩β,c +− +� +⟨H⟩β + +⟨H3⟩β,c +⟨H2⟩β,c +�2 += 3 ⟨H2⟩β,c + +⟨H4⟩β,c +⟨H2⟩β,c +− +� +⟨H3⟩β,c +⟨H2⟩β,c +�2 += 3N2F ′′ +0 + 3(F ′′ +0 )2F ′′ +1 − (F ′′′ +0 )2 + F ′′ +0 F ′′′′ +0 +(F ′′ +0 )2 ++ ... +(A.17) +One can see that the change in the variance is order N2 (it is three times the variance of the +TFD state), so this is a modification of the variance at the order we were looking for. +From this, we can conclude that the semi-classical geometry is not enough to extract the +variance of the energy. The quantum state of the bulk fields is equally important. For the +state |φ⟩, we have the same leading large N properties, but a different quantum state for the +graviton. The fact that it is the s-wave of the graviton that enters is still puzzling, and it +would be interesting how to propertly quantize this non-propagating degree of freedom. We +leave this for the future. +– 69 – + +B +Boosts in global AdS +As we have discussed in section 3, the conformal generators on the d-dimensional cylinder +R × Sd−1 organize themselves as time-translations, rotations, and 2d remaining generators +which correspond to boosts in the dual AdS geometry. The goal of this section is to discuss +whether there exist states that can preserve the boost symmetry. As we have seen throughout +the paper, symmetries that are broken by semi-classical states allow us to specify bulk points +by dressing the location of a bulk point to the feature of the state that breaks the symmetry. +It is important to understand which symmetries are broken, and which symmetries can be +preserved by semi-classical states. For time translations and rotations, this is straightforward, +but it is somewhat more subtle for boosts, which is the purpose of this section. +The 2d boost generators can be realized as d non-independent copies of SL(2, R) [83]. +For simplicity, we will study the case of AdS3, but the higher dimensional versions follow +in a straight forward manner. +In d = 2, the two copies of SL(2, R) are well-known and +correspond to the left and right moving sectors of conformal transformation. The generators +are given by L−1, L0, L1 and ¯L−1, ¯L0, ¯L1. Time-translations and rotations are obtained by +the combinations +H = L0 + ¯L0 , +J = L0 − ¯L0 . +(B.1) +The four residual generators correspond to boosts in AdS3. For explicit expressions, see [142]. +We would now like to analyze whether non-trivial states can be annihilated by these boosts. +As a starting point, notice that there are obviously CFT states which are annilitated by +L−1 and ¯L−1: primary states. However, we would like to consider generators that can be +exponentiated to norm-preserving group elements. +This means the generators should be +Hermitian. +The generators L−1 and ¯L−1 do not satisfy this property. +However, we can +assemble them into the combinations +L+ = L−1 + L1 +, +L− = i(L−1 − L1) +(B.2) +Using that L† +−1 = L1, we see that L± are hermitian operators and can thus be exponentiated +to form unitaries. +The question we would like to ask is whether there are states in the Hilbert space that +are eigenstates of L±. We will see that the only finite energy eigenstates of these operators +are those where the left-moving part of the CFT is in the vacuum. To see this, we consider +the commutator +[L+, L−] = 4iL0 +(B.3) +Suppose now that |ψ⟩ is a normalizable eigenstate of —say— L+. Computing the expectation +value of this equation we find +⟨ψ|L0|ψ⟩ = 0 +(B.4) +From the positivity of the energy spectrum this is possible only if L0|ψ⟩ = 0. The only states +with this property are states where the left moving sector of the CFT is in the vacuum. +– 70 – + +Non-trivial states will thus break boost invariance, which can be use to specify the radial +location of an operator. For the construction of operators presented in this paper, this would +require considering the states obtained by acting with the unitary operators on semi-classical +states |ψ0⟩ as +e−iγL± |ψ0⟩ , +(B.5) +and studying the generalized return probability +R(γ) ≡ +��⟨ψ0| e−iγL± |ψ0⟩ +��2 . +(B.6) +These return probabilities have not been studied but for semi-classical states, it is natural to +expect them to be exponentially small for γ ∼ O(1). +C +Early time decay of the return probability +We wish to estimate the early time decay of the return probability (3.15). We will see that +at very early times, namely t ∼ 1 +N , we can find the decay purely from large N factorization. +We will first recall a general property of coherent state overlaps which follows from large N +factorization, and then adapt the situation slightly to the return probability. +C.1 +Overlap of coherent states and large N factorization +Coherent states of quantum gravity in AdS/CFT can be described by states prepared by a +Euclidean path integral with sources turned on for single-trace operators. These states are +thus given by +|λ⟩ = e +� +x0<0 dxdλ(x)O(x) |0⟩ , +(C.1) +where we have not written the appropriate time-ordering which is left implicit. We will now +show that the overlap is given by +⟨λ1|λ2⟩ = e +� +Rd λ∗ +1(y)λ2(x)⟨O(y)O(x)⟩ + O(1/N) , +(C.2) +where it should be understood that y is integrated over the upper half plane while x is +integrated over the lower half plane. +We can explicitly expand out the integrals of the bra and the ket states, and use large +N factorization: this implies that the operators should be paired up and contracted using +Wick’s theorem, up to 1/N corrections. At a given power in the source, we will have a term +of the form +�� +dxdy +�k +1 +(k!)2 λ∗ +1(y)kλ2(x)k ⟨0| Ok(y)Ok(x) |0⟩ . +(C.3) +We can now apply Wick’s theorem and find +�� +dxdy +�k +1 +(k!)2 λ∗ +1(y)kλ2(x)k ⟨0| Ok(y)Ok(x) |0⟩ = 1 +k! +�� +dxdyλ∗ +1(y)λ2(x) ⟨0| O(y)O(x) |0⟩) +�k +, +– 71 – + +which we can re-exponentiate to find (C.2). Note that we have not written the normalization +of the states, which takes care of the Wick contraction between any two operators living both +in the lower half plane, or upper half plane. Similarly, terms which have a different powers +of upper and lower operators do not give contributions to leading order at large N because +we cannot pair the operators and use Wick’s theorem. +For this to work, we have implicitly assumed that λ ∼ O(N0). To see this, note that +the connected correlation functions of higher-point operators are suppressed by 1/N, but also +have more sources than lower-point functions. If we scale the sources as λ ∼ N1/2, which is +the correct scaling to induce O(1) back-reaction on the dual spacetime46, we have to be more +careful, as some of the terms we dropped involving connected correlators will be the same +size as the Wick contractions. For example, we have +λ∗ +1(y)λ2(x) ⟨O(y)O(x)⟩ ∼ N2 +(C.4) +(λ∗ +1(y)λ2(x))2 ⟨O(y)O(y)O(x)O(x)⟩c ∼ N2 . +(C.5) +This means that we cannot truncate to the sector of Wick contraction, and we must resum +the entire expansion. Note however that the contributions corresponding to loop diagrams +in AdS are still suppressed by 1/N, so we are resumming tree-level diagrams to build the +backreacted geometry. +The upshot of this analysis is that we can use large-N factorization to easily compute +the overlap of coherent states, but only if the sources are O(1), in which case the exponent +in the exponential is also O(1). If we try to make the sources scale with N, the exponent +will be of order N2 and then infinitely many contributions must be resummed. We will now +apply this logic to the return probability. +C.2 +The return probability +We can now apply the same logic as above, taking the operator e−iHT to be seen as an +imaginary Euclidean source for the Hamiltonian (which is the integral of the stress-tensor). +We want to compute +R(T) = ⟨Ψ0|e−iHT |Ψ0⟩⟨Ψ0|eiHT |Ψ0⟩ . +(C.6) +Applying the logic above, we would find that to leading order we have +R(T) = e−iT⟨Ψ0|H0|Ψ0⟩eiT⟨Ψ0|H0|Ψ0⟩ = 1 + O(1/N) . +(C.7) +So we see that the candidate leading term vanishes, and we must go to the next order. This is +due to the nature of the return probability, which is a square of overlaps. A quick expansion +of the exponentials shows that at order T 2, we have +T 2� +− ⟨Ψ0|H2|Ψ0⟩ + +� +⟨Ψ0|H|Ψ0⟩ +�2� += −T 2∆H2 . +(C.8) +46For operators that have unit 2-point function. +– 72 – + +For reasons similar to those explained above, this term can be exponentiated such that we +find +R(T) = e−T 2∆H2 + O(1/N) . +(C.9) +As in the previous section, we can only trust this approximation if the exponent is O(1). Be- +cause we are considering states that have ∆H ∼ N2, we see that we can trust this exponential +decay of the return probability for time-scales up to t ∼ 1/N. +For larger time-scales, it may still hold, but it cannot be justified based solely on large +N factorization. It is instructive to consider the case of the thermofield double state and the +spectral form factor, as we already discussed in section 6.2. For simplicity, we set d = 2 where +we have +Z(β) = e +c +12 +4π2 +β . +(C.10) +The spectral form factor then gives +R(T) = e +π2c +3 +� +1 +β+IT + +1 +β−iT +� += e +2π2c +3 +β +β2+T 2 . +(C.11) +We can expand this expression in T, as long as T ≪ β, to find +R(T) ≈ Z(β)2e +− 2π2c +3 +T 2 +β3 . +(C.12) +We find the exponential decay that goes like T 2. What is important is that even though T +must be much smaller than β, it is allowed to scale as N0. This cannot be justified solely +from large N factorization, but still holds in this particular context. We expect the return +probability to satisfy this property for holographic states more generally. +D +LLM solutions in the bulk +The LLM geometries correspond to solutions of type IIB supergravity with symmetry SO(4)× +SO(4)×R. We assume the axion and dilaton are constant and the IIB three forms are vanish- +ing. We introduce coordinates xµ = (t, y, x1, x2) and Ω3, ˜Ω3 for two 3-spheres corresponding +to the SO(4) isometries. We parametrize the five form as +F5 = Fµνdxµ ∧ dxν ∧ dΩ3 + ˜Fµνdxµ ∧ dxν ∧ d˜Ω3 , +(D.1) +where the self duality of the five form implies that the two forms F and �F are dual to each +other. +After demanding that the geometry preserves the Killing spinor in the presence of the +five form, we arrive at the following solution for the 1 +2-BPS bulk states [81] +ds2 = −(dt + Vidxi)2 +h2 ++ h2(dy2 + dxidxi) + yeGdΩ2 +3 + y +eG d˜Ω2 +3 , +(D.2) +– 73 – + +where every function in the metric is expressed in terms of a function z(x1, x2, y) and we +defined z = 1 +2tanh G, h−2 = 2y cosh G, and +y∂yVi = ϵij∂jz, +y(∂iVj − ∂jVi) = ϵij∂yz . +(D.3) +For the forms F, ˜F we have +F = dBt ∧ (dt + V ) + BtdV + d ˆB, +˜F = d ˜Bt ∧ (dt + V ) + ˜BtdV + d ˆ˜B , +(D.4) +where Bt = − 1 +4y2e2G and ˜Bt = − 1 +4y2e−2G. On the other hand, +d ˆB = −1 +4y3 ⋆3 d(z + 2 +y2 ), +d ˆ˜B = −1 +4y3 ⋆3 d(z − 2 +y2 ) , +(D.5) +where ⋆3 is the epislon symbol in the flat three dimensions. +The only free function, z, is constrained to solve the equation, +∂i∂jz + y∂y(∂yz +y ) = 0 . +(D.6) +We focus our attention on the plane y = 0. Since the product of the radii of the two 3-spheres +is y, there will be a conical singularity at y = 0 unless the function z has a special behaviour. +Let’s consider the case where R1 is kept finite, i.e, e−G → 0 as y → 0. Thus, one has, +z ∼ 1/2 − e−2G + ... . If one assumes that z = 1/2 at y = 0, then one gets the expansion, +z ∼ 1/2 − y2f(x1, x2) + ... for some positive function f, with our boundary conditions. Thus, +e−G ∼ yc(x1, x2) + ... and h2 ∼ c(x1, x2) + ... . Therefore, close to y = 0, the part of the +metric involving R2 will look like, +h2dy2 + R2d˜Ω2 +3 ≈ c(dy2 + y2d˜Ω2 +3) . +(D.7) +Thus the conical singularity is resolved. In the case where R2 is kept fixed, the same argument +goes through but now with the condition that z = −1/2 at y = 0. +With these boundary values of z at y = 0 as a source, one can solve the Laplace equation47 +(D.6) and compute z(x1, x2, y). In addition, Vi can also be expressed in terms of an integral +of z(x1, x2, 0) over the two dimensional space. +E +Notes on boundary states +Some useful references for this section are [79,143–145]. +47More precisely, it is a Laplace equation for z/y2. +– 74 – + +E.1 +Boundary states in 2D CFT +Boundary states in a 2d CFT need to satisfy [143] +(Ln − ˜Ln) |B⟩ = 0. +(E.1) +In any Verma module, one can find a simple solution to these conditions as +|Ih⟩ = +� +⃗k +|⃗k, h⟩L ⊗ |⃗k, h⟩R , +(E.2) +where |⃗k, h⟩L is a linear combination of Virasoro descendants of the primary state |h⟩ char- +acterized by an infinite dimensional vector ⃗k = (k1, k2, ...) with non-negative integer compo- +nents. We identify these states by starting with descendants of the form +...LKn +−n...LK1 +−1 |h⟩L . +(E.3) +and forming an orthonormal basis selected such that L⟨⃗k, h|⃗k′, h⟩L = δ⃗k,⃗k′. +The state |Ih⟩ is called the Ishibashi state for the primary state |h⟩L, where the states +|⃗k, h⟩ are the descendant on top of the primary labeled by h. It can be seen easily that +Ln|Ih⟩ = ˜Ln|Ih⟩ . +(E.4) +It is clear that the Ishibashi states have maximal entanglement between the left-moving and +right-moving sectors. Linear combinations of the Ishibashi states satisfy the constraint (E.1) +as well. +Physical boundary sates are given by special linear combinations of Ishibashi states which +are called Cardy states +|Ba⟩ = +� +h +Ca,h |Ih⟩ . +(E.5) +Physical boundary states should satisfy a consistency condition of the partition function on +a finite cylinder related to open-closed duality [143]. +The Cardy states are singular because the norm of the Ishibashi states is divergent. One +can define regularized boundary states by evolving in Euclidean time as +|Ba,β⟩ = e− β +4 Hc |Ba⟩ , +(E.6) +where β is a positive constant and Hc = L0 + ˜L0 − c +12. Since [L0 − ˜L0, Hc] = 0, the state (E.6) +is still space-translational invariant on the circle, but it is time-dependent. +Ishibashi states are orthogonal to each other. The amplitude of Euclidean time evolution +by β/2 between two such states is computed as +⟨Ik|e−βHc/2|Il⟩ = δklχk(e−β/2) . +(E.7) +– 75 – + +χk is the character for the primary k. On the other hand, the Cardy states are not orthogonal +to each other but satisfy the open-closed duality relation as follows +⟨Ba|e− β +2 Hc|Bb⟩ = +� +k +N(k) +a,b Trk[e− 4π2 +β Ho] +(E.8) +where Ho = Lo− c +24 denotes the Hamiltonian in the dual channel, characterized by the bound- +ary conditions a, b. On the right hand side, Trk[...] denotes a trace in the sector associated +to a primary k as well as its descendants. Moreover, N(k) +a,b counts the degeneracy of sectors +which belong to the primary k with boundary conditions a and b. +In the high temperature limit β → 0, we find that +⟨Ba|e− β +2 Hc|Bb⟩ ≃ N(km) +a,b +e− 4π2 +β (h(min) +a,b +− c +24 ) , +(E.9) +where km is the lightest primary among those satisfy N(km) +a,b +̸= 0, whose conformal dimension +is denoted as h(min) +a,b +. +We can estimate the inner products between two normalized boundary states in this limit +as +⟨ψa|e− β +2 Hc|ψb⟩ = +⟨Ba|e− β +2 Hc|Bb⟩ +� +⟨Ba|e− β +2 Hc|Ba⟩⟨Bb|e− β +2 Hc|Bb⟩ +≃ δa,b + N(km) +a,b +e− 4π2 +β h(min) +a,b +. +(E.10) +Note that N(0) +a,a = 1. In this way, a large gap in the open string channel leads to a large +exponential suppression of off-diagonal elements of inner products. +In holographic BCFT, the inner product between two boundary states can be computed +by evaluating the gravity action on the dual background. When we consider the gravity dual +of a cylinder, there are two candidates of classical gravity solutions depending on whether +the end of the word brane is connected or disconnected which are called connected and +disconnected solutions. When we consider the overlap for an identical boundary condition +a, then both the connected and disconnected solution are allowed. In the limit β → 0, the +connected solution is favored and one can find that +⟨Ba|e− β +2 Hc|Ba⟩ ≃ e +π2c +6β . +(E.11) +We will use it later to calculate the return probability for boundary states. +In addition +to it, one can find the inner product between two boundary states with different boundary +conditions. In this case, only the disconnected solutions are allowed and +⟨Ba|e− β +2 Hc|Bb⟩ ≃ e +cβ +12 +S(a) +bdy+S(b) +bdy, +(E.12) +where S(i) +bdy, i = a, b are the boundary entropies [79]. +– 76 – + +E.2 +Boundary states in higher dimensions +One can generalize to higher dimensions and define a boundary state |Ba⟩ as a state associated +to a (d − 1)-dimensional boundary in d-dimensional CFT [79,146]. Taking the boundary to +be a torus Td−1, the inner product between two boundary states in a holographic BCFT can +be computed as a partition function on a d-dimensional open manifold Iβ/2 ×Td−1 where Iβ/2 +is a length β/2 interval. As in the 2d case, there are two bulk solutions, a connected and a +disconnected one. In the β → 0 limit the connected solution is dominant and one can find +the inner product between two identical boundary states using the gravity solution as +⟨Ba|e− β +2 Hc|Ba⟩con ≃ eαd/βd−1 , +(E.13) +where +αd = (4ζ(T))d Rd−1 +16GN +Ld−1 , +(E.14) +where R is the AdS radius, L is the length of the compactified spatial directions and ζ(T) is +a function of tension which is defined when T < 0 as +ζ(T) ≡ Γ(1/d)Γ(1/2) +Γ(1/d + 1/2) +R|T| +d(d − 1)(1− R2T 2 +(d − 1)2 )1/d−1/2F(1, 1/d, 1/2+1/d; 1− R2T 2 +(d − 1)2 ) , (E.15) +and when T > 0, ζ(T) = 2π +d − ζ(−T). The tension takes values in the range |T| < d−1 +R . For +d > 2, ζ(T) non-trivially depends on T and there is an upper bound of the tension T < T∗ +which T∗ > 0 and ζ(T∗) = 0 [79]. +E.3 +Correlation functions in BCFTs +Let us first start with the simplest case where the CFT is defined on the upper half plane +and the boundary state |B⟩ is placed along the real axis. We consider the 1-point function of +a local operator placed at z in the upper half plane. In the case of a CFT on the plane, the +1-point function of a primary operator in the vacuum is required to vanish by the symmetries. +These are partly broken in a BCFT. The remaining symmetries constraint the 1-point function +to have the form +⟨O(z)⟩UHP = +AO +(2 Im(z))∆ , +(E.16) +where AO is determined by the details of the theory and the precise boundary state in +question. One could think of this as the boundary providing a source for the operator O. +The 2-point function of a primary operator in a BCFT is more complicated than the +case with no boundaries where it is exactly fixed by the symmetries. Non-trivial information +about the operator content and OPE coefficients is necessary to compute the 2-point function +exactly in a BCFT. We assume that for large N holographic CFTs the large N 2-point +function takes the form +⟨O(z1)O(z2)⟩UHP = ⟨O(z1)⟩UHP ⟨O(z2)⟩UHP + ⟨O(z1)O(z2)⟩ ± ⟨O(z1)O(z∗ +2)⟩ , +(E.17) +– 77 – + +where +⟨O(z1)O(z2)⟩ = +1 +|z1 − z2|2∆ , +(E.18) +where the contribution from an image insertion placed at z∗ +2. The sign of the last term is +governed by the boundary conditions, being either Dirichlet (−) or Neumann (+). +Mapping the z coordinate to a new coordinate w by +w → z = exp(2πw/β + i2π/4) , +(E.19) +we can map the upper half plane to the a strip of width β/2, where the positive (negative) +real axis is mapped to the lower (upper) edge of the strip. +Since primary operators continue to transform in the usual way, the correlation functions +now transform to +⟨O(w)⟩strip = +AO +( β +π cos[ 2π +β τ])∆ +⟨O(w1)O(w2)⟩connected +strip += +1 +| β +π sinh[ π +β(w1 − w2)]|2∆ ± +1 +| β +π cosh[ π +β(w1 − ¯w2)]|2∆ , +(E.20) +where the second line is only the connected piece of the large N 2-point function [126]. Higher +order correlation function can be found through large N factorization. +Correlation functions on a state defined on a circle by +|Bβ⟩ = e−βH/4 |B⟩ , +(E.21) +can be thought of as correlation function on a cylinder of width β/2 where the boundary state +is placed on both sides. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Universit`a di Milano - Bicocca,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' I-20126 Milano,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Italy eInstitute of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Ecole Polytechnique F´ed´erale de Lausanne,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' CH-1015 Lausanne,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Switzerland fTheoretical Physics Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' CERN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' CH-1211 Geneva 23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Switzerland E-mail: ebahiru@sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='it, alexandre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='belin@unimib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='it, kyriakos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='papadodimas@cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='ch, gabor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='sarosi@cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='ch, nvardian@sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='it Abstract: Within the AdS/CFT correspondence, we identify a class of CFT operators which represent diff-invariant and approximately local observables in the gravitational dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Provided that the bulk state breaks all asymptotic symmetries, we show that these operators commute to all orders in 1/N with asymptotic charges, thus resolving an apparent tension between locality in perturbative quantum gravity and the gravitational Gauss law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The interpretation of these observables is that they are not gravitationally dressed with respect to the boundary, but instead to features of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We also provide evidence that there are bulk observables whose commutator vanishes to all orders in 1/N with the entire algebra of single-trace operators defined in a space-like separated time-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This implies that in a large N holographic CFT, the algebra generated by single-trace operators in a short-enough time- band has a non-trivial commutant when acting on states which break the symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It also implies that information deep in the interior of the bulk is invisible to single-trace correlators in the time-band and hence that it is possible to localize information in perturbative quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='08753v1 [hep-th] 20 Jan 2023 Contents 1 Introduction 1 2 Aspects of locality in field theory and gravity 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1 Classical field theories 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2 Localization of information in QFT 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1 Comments on the split property 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2 Subtleties with gauge invariance 11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3 Classical and Quantum Gravity 13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1 On the initial value problem of general relativity 13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2 Diff-invariant observables in classical GR 15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3 State-dressed observables 16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4 A time-band in AdS 18 3 Holographic setup 19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1 Gravitional states in AdS, large diffeomorphisms and asymptotic symmetries 20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2 Locality in AdS 23 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3 The CFT description and the time band algebra 23 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4 Formulating the main goal 25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='5 Time-shifted states and return probability 26 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6 The return probability 27 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='7 Other asymptotic charges 30 4 State-dressed operators 31 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1 Vanishing commutator with H to all orders in 1/N 32 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2 Similar action as HKLL operators 33 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3 Interpretation and comments 34 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4 A similarity transformation 36 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='5 Other asymptotic charges 37 5 A more general argument for the commutant 37 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1 On the consistency of the defining equations 39 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2 Proof that operators have the desired properties 40 6 Examples 41 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1 Coherent states 41 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2 Thermofield double state 43 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3 Weakly coupled, large N gauge theories 45 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4 Perturbative states around empty AdS 45 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='5 LLM geometries 47 – i – 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1 Computation of the return probability 48 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6 Kourkoulou-Maldacena states in SYK model 51 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1 Analytical computation of the return probability at large N 52 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2 Some numerical checks 54 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='7 Holographic boundary states 56 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1 Computation of the return probability and correlators 57 7 Black Hole microstates 59 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1 States with macroscopic time-dependence 59 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2 Typical states 60 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3 Two entangled CFTs 61 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4 Island discussion 62 8 Discussion 63 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1 The variance of the energy from semi-classical gravity 64 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2 Gravitational proof for the decay of the return probability 64 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3 Microscopically time-dependent states 65 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4 Microcanonical states and small energy variance 66 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='5 The AdS vacuum and low-energy states 66 A Changing the variance of H 67 B Boosts in global AdS 70 C Early time decay of the return probability 71 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1 Overlap of coherent states and large N factorization 71 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2 The return probability 72 D LLM solutions in the bulk 73 E Notes on boundary states 74 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1 Boundary states in 2D CFT 75 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2 Boundary states in higher dimensions 77 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3 Correlation functions in BCFTs 77 1 Introduction It is generally believed that in quantum gravity, space-time locality is an emergent notion which becomes accurate and useful in certain limits of the underlying theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This perspective is realized in the AdS/CFT correspondence [1]: bulk locality becomes precise in the large N, – 1 – strong coupling limit and when probing the theory with simple enough operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Moreover, a large number of proposals aiming to resolve the black hole information paradox rely on a certain amount of non-locality [2–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' A natural question is to understand whether non- local features of quantum gravity are visible only in the non-perturbative regime, or whether remnants of non-locality are also visible at the perturbative level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Even in classical general relativity it is not entirely straightforward to formulate the concept of locality, as it is non-trivial to define local observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Physical observables need to be diff-invariant and, in order for them to also be local, they have to be associated to points in space-time which have to be specified in a diff-invariant way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' If the space-time has a boundary, a standard approach is to define points relationally with respect to the boundary or by completely fixing the gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We say that these observables are gravitationally dressed with respect to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' However, the resulting observables, while diff-invariant, are not strictly localized and have non-vanishing Poisson brackets at space-like separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' A particular aspect of this difficulty is related to the gravitational Gauss law: in gravitational theories defined with asymptotically flat or AdS boundary conditions, the Hamiltonian, and other asymptotic symmetry charges, are boundary terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Acting with a candidate local, diff-invariant observable in the interior of space will generally change the energy of the state, which is immediately measurable at space-like separation due to Gauss’s law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Despite these difficulties, at the classical level, there are ways of defining local and diff- invariant observables in the neighborhood of a state, provided that the state is sufficiently complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' A class of such observables introduced a long time ago [14–16] will be reviewed in sub-section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3, see also [17–19] for more recent discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' These observables respect the causal structure of the underlying space-time, in the sense that their Poisson brackets at space- like separation vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In particular, provided that the state we are considering is complicated enough, the action of these observables is not visible by the boundary Hamiltonian, as these observables only rearrange energy in the interior of space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The price we have to pay is that these observables are not defined globally on the phase space of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' They have desired properties only for certain states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' A natural question is to what extent can such local diff-invariant observables be defined at the quantum level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' As mentioned above, we do not expect to be able to find exactly local diff-invariant observables at the non-perturbative level, however it may be possible to do so in perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This question is important in order to be able to quantify departures from locality in quantum gravity and to understand if there is a way to generalize the structure of algebras of observables of quantum field theory to situations where gravity is included perturbatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It is useful to formulate these questions in the context of the AdS/CFT correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We consider a CFT state |Ψ0⟩ that is dual to a semi-classical asymptotically AdSd+1 geometry in global coordinates and a short time-band near the boundary as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We consider the algebra A of observables in semi-classical gravity which are localized in this time band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This algebra includes the Hamiltonian and other asymptotic charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' From the point of view of the dual CFT, it is natural to identify the algebra A with the algebra generated by – 2 – Figure 1: The single-trace operators localized in the time band t ∈ (−ϵ, ϵ) × Sd−1 (dark blue region on boundary) form an algebra A which is conjectured to be dual to the causal wedge of the region (light blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' If the state |Ψ0⟩ of the system breaks all symmetries, then the causal diamond in the middle (light red), which is spacelike separated from the time-band, corresponds to the commutant A′ of the algebra A when acting on the code subspace of the state |Ψ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' single-trace operators localized in this time-band, we will call it the ”single-trace algebra”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The expectation is that the single-trace algebra A corresponds to the causal wedge of the time-band [20]1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Notice that here we have causal-wedge reconstruction and not entanglement wedge reconstruction, as we are looking only at the single-trace subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In the CFT the notion of a time-band algebra only makes sense at large N, since large N generates a natural hierarchy between operators that are small combinations of single-trace operators and arbitrarily complicated operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For finite N there is no such hierarchy and the time-slice axiom would imply that A is the full CFT algebra2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Algebras of single-trace operators in holographic CFTs have been discussed in [6,25,26] and more recently in [27–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' If the time-band is short enough, then there is a region in the bulk which is space-like with respect to the time-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We will refer to this region as the ”diamond”3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' If we were able to define diff-invariant observables localized in the diamond, they should commute with the algebra A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' As already mentioned, the question is non-trivial as these observables must be gravitationally dressed and if we use the boundary to dress them, then they will not commute with A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For example, it appears that since the Hamiltonian H is an element of A it would be able to detect any excitation added in the interior of the diamond using the gravitational Gauss law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' To summarize, the question we want to examine: 1A different approach for studying time-bands based on gravitational entropy and minimal surfaces was initiated in [21–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It would be interesting to understand possible connections between those ideas and the results presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 2We do not include in the single-trace algebra elements like eiHtO(t = 0, x)e−iHt with t = O(N 0) and large enough to exit the time-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Such ”precursor” operators are complicated from the point of view of operators in the time-band and go beyond the semi-classical description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 3For now we assume that the state has simple topology and there are no black hole horizons in the interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 3 – Does the algebra A, when acting on the state |Ψ0⟩ and small perturbations around it, have a non-trivial commutant in the 1/N expansion?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' As we will discuss later, we need to refine the question by demanding that the commutant acts non-trivially within the code-subspace of the state, in order to avoid obvious but uninteresting constructions4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We emphasize that we do not expect the algebra to have a commutant at finite N [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' A closely related question is that of localization of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' According to AdS/CFT the quantum state of the CFT at any moment in time contains the full information of the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In particular, if we had considered the full algebra of all operators in the time-band, as opposed to the algebra generated by few (relative to N) single-trace operators, then we would be able to reconstruct the interior of the diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Suppose however, that we only have access to the algebra A of single-trace operators in the time band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Can we then reconstruct the information of whatever is hidden inside the diamond?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This can also be rephrased as follows: Given a state |Ψ0⟩, can we find another state |Ψ0⟩′ such that the correlators of the single-trace algebra A in the time-band, evaluated on these two states agree to all orders in 1/N, but correlators of single-trace operators differ at O(N0) outside the time-band?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The intuition here is that we want to find a state |Ψ0⟩′ which contains an additional excitation relative to |Ψ0⟩ in the interior of the diamond which becomes visible by single-trace operators only after a light-ray has reached the boundary i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' in the future or past of the time-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' If the algebra A had a commutant then we could take |Ψ0⟩′ = U(A′)|Ψ0⟩ for some unitary U built out of operators A′ in the commutant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We will provide evidence that the answer to the two aforementioned questions is positive, provided that the state |Ψ0⟩ is complicated enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The reasoning was first outlined in [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In this paper we extend the construction in a few ways and provide additional arguments and examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Standard approaches to bulk reconstruction lead to observables which are relationally defined with respect to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This is the case for the HKLL reconstruction [33–39], as well as approaches based on the Petz map [40, 41] or modular reconstruction [42, 43], as they all require some sort of boundary dressing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For concreteness we start with a standard HKLL operator given by Φ(t, r, Ω) = � bdry dt′ dΩ′ d−1K(t, r, Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' t′, Ω′)O(t′, Ω′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1) 4 For example, for a complicated state with energy of O(N 2), a unitary which rotates the phase of a single energy eigenstate will have commutators of O(e−N2) with all elements of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' However, this would not be an interesting example, as this operator is generally ”invisible” from the bulk point of view and does not create excitations inside the diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 4 – Here K is a particular Green’s function which depends on the background metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Implicit in this expression is a gauge-fixing scheme in a particular coordinate system, which is uniquely determined by making use of the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' If we pick the point (t, r, Ω) to be in the diamond, the operator (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1) commutes with all single-trace operators in the time band at large N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' At subleading orders multi-trace corrections need to be added to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1) to ensure vanishing commutators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' However the commutator with the Hamiltonian and other asymptotic charges, which is nonzero at order 1/N, cannot generally be corrected by multi-trace corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The physical reason is that the operator (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1) is gravitationally dressed with respect to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The non-vanishing commutator with H appears to be an obstacle in identifying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1) as an element of the commutant of A [44,45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In this paper we present a way to find operators which commute with the asymptotic charges to all orders in 1/N, while at the same time create excitations in the interior of the diamond similar to those of the HKLL operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' These operators can be defined provided the state |Ψ0⟩ that we are considering breaks all asymptotic symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' These operators correspond to observables gravitationally dressed with respect to features of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' A crucial starting observation is that if a state |Ψ0⟩ is dual to a bulk geometry which breaks the asymptotic symmetries, then the overlap ⟨Ψ0|U(g)|Ψ0⟩ g ∈ SO(2, d) , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2) is generally exponentially small, of order O(e−aN2) with Re(a) > 0 provided that the element g is sufficiently far from the identity5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Here SO(2, d) represents the asymptotic symmetry group of AdSd+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We will quantify this statement more precisely in the later sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In fact, we will provide evidence that if we introduce the code subspace around the state |Ψ0⟩, defined as H0 = span{|Ψ0⟩, O(t, Ω)|Ψ0⟩, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=', O1(t1, Ω1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='On(tn, Ωn)|Ψ0⟩} , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3) and similarly Hg for the state U(g)|Ψ0⟩ then any inner product between unit normalized states of H0, Hg will also be of order O(e−aN2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Starting with a standard HKLL operator Φ we consider the operator �Φ = c � B dµ(g)U(g)P0ΦP0U(g)−1 , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4) where P0 denotes the projector on (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3) and dµ(g) is the Haar measure on SO(2, d) and B is a reasonably sized neighborhood of SO(2, d) around the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The overall normalization constant c will be specified later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The main claim, which will be discussed in section 4, is that operators (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4) have the desired properties: their commutators with the asymptotic symmetry charges Q of SO(2, d) are exponentially small [Q, ˆΦ] = O(e−N2) , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='5) 5But not too far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The state may return to itself in compact directions of the conformal group or approxi- mately back to itself due to Poincare recurrences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 5 – when acting on the code subspace, while at the same time, the leading large-N action of ˆΦ on the code subspace (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3) is the same as that of the corresponding HKLL operator Φ, that is ⟨Ψ1|ˆΦ|Ψ2⟩ = ⟨Ψ1|Φ|Ψ2⟩ + O(1/N) ∀ |Ψ1⟩, |Ψ2⟩ ∈ H0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6) The interpretation is that by performing the integral (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4) we have removed the gravitational dressing of the operators from the boundary and moved it over to the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This is only possible on states where (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2) decays sufficiently fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The operators (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4) have vanishing commutators with the asymptotic charges to all orders in 1/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This demonstrates that the apparent obstacle to identifying a commutant due to Gauss’s law can be overcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In order to find a true commutant we need to ensure vanishing commutators to all orders in 1/N with all single-trace operators in the time-band algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It would be interesting to explore whether a formula achieving this goal and similar to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4) can be derived, possibly by integrating over the unitary orbits generated by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We provide an alternative formal argument supporting the idea that the algebra A has a nontrivial commutant when acting on the code subspace Hcode of a complicated state |Ψ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' To see that we consider an operator ˆΦ defined by ˆΦA|Ψ0⟩ = AΦ|Ψ0⟩ ∀A ∈ A , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='7) where again Φ is a standard HKLL operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This represents a set of linear equations, one for each A ∈ A, which define the action of ˆΦ on H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' A sufficient condition for the consistency of these equations is that for all non-vanishing operators A ∈ A we have A|Ψ0⟩ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In section 5 we provide evidence that this is true in the 1/N expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Given that these equations are consistent, we will show in section 5 that the operators ˆΦ defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='7) obey the following properties: i) by construction they commute with operators in A and ii) to leading order at large N act like HKLL operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This provides evidence that the algebra A has a commutant in the 1/N expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' As mentioned earlier, a commutant is not expected at finite N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Indeed, at finite N it is possible to find complicated operators in the time-band which annihilate the state |Ψ0⟩ and equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='7) do not have a consistent solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' If we take the state |Ψ0⟩ to be the vacuum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' empty AdS, then the previous construction fails: since the vacuum is invariant under the asymptotic symmetries we no longer have the decay of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6) fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Also (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='7) fails because there are operators in the time-band, in particular H, which annihilate the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We emphasize that this failure is not a limitation of our particular construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Instead the interpretation of this failure is that since empty AdS has no bulk features, the only way to specify a point in the bulk is by dressing it to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Hence any bulk diff-invariant operators acting around the vacuum will not commute with the asymptotic charges [44,45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This can also be seen from the fact that even classically, the local diff-invariant observables cannot be defined properly in the vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We emphasize that the results of this paper do not contradict the claim of [46] that specifically for perturbative states around empty AdS, it is possible to reconstruct the state from correlators in the time-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' However we notice that interesting states, that is, states – 6 – which have bulk observers capable of performing physical experiments, are expected to be of the form where the symmetries are broken and the construction presented in this paper can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' If the state |Ψ0⟩ corresponds to a black hole state, and if the variance of the asymptotic charges scales like N2 6 we find that using the operators (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4) we can create excitations behind the horizon which cannot be detected by correlators of single-trace operators in the 1/N expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Understanding how to diagnose these excitations from a CFT calculation remains an outstanding open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We emphasize that this does not contradict the fact that, generally, excitations created by unitaries on top of typical states with small energy spread can be detected by single-trace correlators [25,47,48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Such states with small energy spread are those for which our construction cannot be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The operators we identify provide evidence supporting the idea that locality is respected in perturbative quantum gravity and that information can be localized in subregions at the level of perturbation theory, provided that the underlying state is sufficiently complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It also suggests that it should be possible to associate algebras of observables to subregions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' However these observables have certain features of state-dependence, since both (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='7) give operators which are defined only on the code-subspace of the original state |Ψ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It is certainly possible to extend the domain of definition of our operators by combining together code subspaces of sufficiently different states, each one of which must break the asymptotic symmetries, thus partly eliminating the state-dependence of the operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' However the number of these states must not be too large, otherwise the small overlaps between the code subspaces start to accumulate and modify the correlators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This becomes particularly relevant for black hole states, where we do not expect to have operators with the desired properties defined globally for most microstates and some genuine state-dependence is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The plan of the paper is as follows: in section 2 we review background material about various aspects of locality in field theory and gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In section 3 we describe the setup in AdS/CFT and study the decay of the inner product (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In section 4 we introduce the operators (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4) and discuss their basic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In section 5 we provide an alternative argument for the existence of a commutant based on equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In section 6 we consider various examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In section 7 we consider aspects of our operators in the presence of black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Finally we close with a discussion of open problems in 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 2 Aspects of locality in field theory and gravity In this section, mostly addressed to non-experts, we review some background necessary to explore the question of localizing information in different regions of space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' A closely related question is the association of algebras of observables to subregions and the factorization of the Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We start with non-gravitational field theories, where a non-dynamical background space-time can be used in order to define sub-regions and their causal relations, and then we consider the additional complications when gravity is taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 6For example, this is true for black hole states with energy spread similar to the canonical ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 7 – In relativistic theories we expect that signals and information cannot travel faster than light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We then want to address the following question: consider an initial space-like slice Σ and divide it into a compact subregion D and its complement D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We denote by J(D′) the domain of dependence of D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The question is the following: is it possible to modify the state7 in region D without affecting the state in J(D′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' If the answer is positive then an observer initially in D′, and confined to move in J(D′), cannot reconstruct information about the interior of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Then we say that information can be localized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1 Classical field theories At the classical level this question can be addressed by studying the initial value problem: we specify initial data C on a spacelike slice Σ and then look for a solution in the entire space-time, or at least a neighborhood of the slice Σ, compatible with the initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The initial data will typically include the values and time-derivatives of various fields of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The theories we will be considering have gauge invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' One of the implications is that the existence of a solution is guaranteed only if the initial data satisfy certain constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In relativistic field theories theories the dynamical equations are hyperbolic, which ensures that signals propagate forward from Σ at most at the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' On the other hand the constraint equations for initial data are of elliptic nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This makes the question of being able to specify the initial data independently in region D and its complement D′ non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It is thus convenient to divide the question formulated above in two steps: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Localized preparation of states: for given initial data C1 on Σ satisfying the constraints, to what extent can we deform to other initial data C2, also satisfying the constraints, such that C1, C2 agree on D′, possibly up to a gauge transformation, but differ essentially8 on D?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' No super-luminal propagation: suppose we are given two initial data C1, C2 which satisfy the constraints, which agree on D′ and differ on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We then want to show that the two corresponding solutions agree on J(D′), possibly up to a gauge transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We will return to the classical problem in theories with gauge invariance in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For now we briefly consider the simplest example of a free Klein-Gordon field in flat space obeying □φ = m2φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We consider initial data on the slice Σ corresponding to t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The initial data on this slice are parametrized by C = {φ(t = 0, x), ∂0φ(t = 0, x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In this case condition A mentioned earlier is clearly satisfied: the initial data do not need to obey any constraint, so we can simply select the functions φ, ∂0φ to have any smooth profile with 7Either classical state, or quantum density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 8i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' cannot be matched by a gauge transformation on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 8 – features strictly localized in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Notice that this requires the use of non-analytic initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Condition B is also satisfied, see [49] for a basic review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2 Localization of information in QFT In non-gravitational QFT we can associate algebras of observables to space-time regions [50–52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Locality is exact, and is expressed by the condition that algebras corresponding to space-like separated regions commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' An analogue of the initial value problem in QFT is expressed by the condition of primitive causality or relatedly the time-slice axiom which postulates that the only operators commuting with the algebra generated by operators in a time-band are proportional to the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Moreover a local version of these statements postulates that the algebra of operators in a subregion coincides with the algebra of operators in the causal domain of dependence of the subregion [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' An intuitive way to see that that information can be localized in QFT is as follows: suppose |Ψ0⟩ is a state in the Hilbert space of the QFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Consider a unitary operator UD constructed out of observables localized in D and the new state |Ψ⟩ = UD|Ψ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The unitary UD modifies the state by creating an excitation in region D which encodes the desired infor- mation in that region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For any observation OD′ in region D′, and more generally in J(D′), we have ⟨Ψ|OD′|Ψ⟩ = ⟨Ψ0|U † DOD′UD|Ψ0⟩ = ⟨Ψ0|OD′|Ψ0⟩ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1) where we used [UD, OD′] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Hence states |Ψ⟩, |Ψ0⟩ are indistinguishable by measurements in J(D′) and the excitation created by UD in D is invisible in J(D′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1 Comments on the split property More generally we would like to know whether it is possible to independently specify the quantum state in space-like separated regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The question is non-trivial since in most quantum states these regions will be entangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It is believed that, as long as the regions in question are separated by a finite buffer region, then the answer should be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This is related to the split property of quantum field theory [52,54–56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The split property can be defined as follows: consider the causal diamond whose base is a ball D1 and the corresponding operator algebra AD1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Consider a slightly larger ball D2, containing D1, with corresponding operator algebra AD2 in its causal diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The split property is satisfied if we can find a type I von Neumann algebra of operators N such that AD1 ⊂ N ⊂ AD2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It has been shown that quantum field theories with a reasonable thermody- namic behavior, as expressed in terms of nuclearity conditions (see [52] for an introduction), satisfy the split property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Using the algebra N we can have strict localization of quantum information which is completely inaccessible from J(D′ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 9In the case of non-relativistic theories, for example the heat equation, which is first order in time and hence not hyperbolic, we are able to specify the initial data in subregions independently but the speed of propagation is unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Hence the heat equation obeys condition A but not B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 9 – Equivalently, the split property can be defined by the existence of state |φ⟩ which is cyclic and separating for the algebra AD1∪D′ 2 and such that ⟨φ|a b|φ⟩ = ⟨0|a|0⟩⟨0|b|0⟩ ∀ a ∈ AD1, b ∈ AD′ 2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2) where |0⟩ is the Minkowski vacuum and D′ 2 denotes the complement of D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In the state |φ⟩ the mutual information between regions D1 and D′ 2 is vanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Such a state is not uniquely defined, since for any unitary U ∈ A(D′ 1∩D2) a state of the form U|φ⟩ will also satisfy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Starting with a split state |φ⟩ we can construct more general states by exciting the two regions D1 and D′ 2 acting with localized operators in the corresponding algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Since there is no entanglement between D1 and D′ 2 in the split state |φ⟩ the two algebras act independently and we can arbitrarily approximate an excited state in D1 and another state in D′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' An interesting question is to estimate the energy of a split state10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We do not expect a split state to be an energy eigenstate, so in general it will have non-vanishing energy variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Here we provide some very heuristic arguments about the expectation value of the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' As a starting point, let us consider a CFT on R1,d−1 with coordinates x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=', xd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We define two regions to be the causal domains of two slightly displaced Rindler wedges with bases x0 = 0, x1 < −ϵ and x0 = 0, x1 > ϵ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The two wedges are separated by the buffer region −ϵ < x1 < ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In this case the total energy of the split state will be infinite due to the infinite planar extension of the regions in the transverse directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' However, we expect to have a finite energy per unit area E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Since we are dealing with a CFT then the only scale in the problem is the size ϵ of the buffer region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Hence by dimensional analysis the energy per unit area will scale like E = s ϵd−1 where s is a constant depending on the CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' If we now consider a more general compact region D1 of typical size R, which is separated by a small buffer region of typical size ϵ from D′ 2 then we would expect that a split state with respect to D1, D′ 2 will have energy which in the ϵ → 0 limit will scale like E = s A(∂D1) ϵd−1 + O( ϵ R) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3) where A(∂D1) is the area of the boundary of D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This is a heuristic estimate and it would be interesting to investigate it more carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' As mentioned above, this is the expectation value of the energy and it would be interesting to understand the spectral decomposition of a split state in the energy basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Notice that a split state does not respect the Reeh-Schlieder property with respect to the algebra AD1 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This implies in particular that the split state should have non-compact support in energy, since otherwise the Reeh-Schlieder property would have to hold for D1, see for example [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 10Since the split state is not unique, a reasonable question might be finding the lowest possible expectation value for the energy of a split state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 11Since there is no entanglement between D1 and D′ 2 we cannot create excitations in region D′ 2 by acting with operators in D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 10 – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2 Subtleties with gauge invariance Consider U(1) gauge theory minimally coupled to a charged scalar with Lagrangian L = − 1 4FµνF µν − (Dµφ)∗Dµφ , Dµφ = ∂µφ − igAµφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The system has U(1) gauge invariance Aµ → Aµ + ∂µΛ, φ → eigΛφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The dynamical equations are ∂νFµν = ig(φ∂µφ∗ − φ∗∂µφ) − 2g2Aµφ∗φ □φ = ig(∂µAµ)φ + 2igAµ∂µφ + g2AµAµφ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4) In this case the initial data are C = {Aµ(t = 0, x), ∂0Aµ(t = 0, x), φ(t = 0, x), ∂0φ(t = 0, x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Here we encounter the subtleties mentioned for gauge systems: initial data related by a gauge transformation are physically equivalent and initial data are admissible (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' lead to a solution) only if the obey a constraint, the Gauss law, which is the µ = 0 component of the first equation in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4) ∂i(∂0Ai − ∂iA0) = ig(φ∂0φ∗ − φ∗∂0φ) − 2g2A0φ∗φ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='5) We now revisit the two properties mentioned in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The fact that the dynamical part of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4) obey condition B follows from general properties of hyperbolic equations of this type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Let us now examine question A in this theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='5) we see that if we try to deform the initial data in region D, then we may be forced to change them in D′ too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For example if we turn on a profile for the scalar in region D with total non-zero charge, then the gauge field has to be turned on in region D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The Gauss law constraint (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='5) is of the familiar form ∇ · ⃗E = ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This imposes the constraint that � ∂D ⃗E · d⃗S = QD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' However it is clear that once we make sure that the initial data in D′ are compatible with the Gauss constraint from the total charge QD enclosed in D, there are many ways of rearranging the initial data in region D keeping those in D′ fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In other words there are deformations of the constraint equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='5), which are not gauge-equivalent, and which have compact support localized in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This means that theory under consideration obeys condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Moving on to the quantum theory, we can consider U(1) gauge theory weakly coupled to matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' As in the classical theory the total charge Q enclosed in a region can be measured on its boundary and the total charge of the entire state can be measured at space-like infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' At the quantum level we can get information not only about the expectation value of the charge but all the higher moments ⟨Ψ|Qn|Ψ⟩ , n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='. (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6) To proceed it is useful to consider observables in this theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Physical observables must be gauge invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In a U(1) gauge theory there are several examples of such observables which are also local, for example local operators constructed out of Fµν(x) or φ∗(x)φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Other inter- esting gauge invariant operators which are not completely local, but can be contained in com- pact regions are closed Wilson loops eig � C Aµdxµ or bilocals of the form φ∗(x)eig � y C,x Aµdxµφ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 11 – All these operators are neutral and do not change the electric charge of the region D, if they are entirely contained in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We can use such operators localized in region D to construct unitaries UD which can be used to modify the state inside D leaving all correlators outside invariant, as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' So information can be localized in this theory if we work with neutral operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' But what if we want to create an excitation in region D which has non-zero charge?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We already know from the classical problem that it will not be possible to add a charge in D without affecting the exterior due to Gauss law (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The same is true at the quantum level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' A charged operator φ in D is not gauge invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It can be made gauge invariant by dressing it with a Wilson line extending all the way to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We can think of the Wilson line as a localized tube of electric flux ensuring that Gauss law is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It may be energetically better to smear the Wilson line in a spherically symmetric configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The important point is that the dressed operator Φ(x) = eig � x ∞ Aµdxµφ(x) is no longer a local operator, though it is gauge invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' If we act with a unitary made out of this operator, we will modify correlators outside D and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1) will fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This means that the addition of the charge in D can be detected immediately outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This is not surprising, as the same thing is already visible at the classical level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' However, looking a bit more carefully, we run into certain somewhat surprising features of the quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Suppose we have several charged fields φi, labeled by a flavor index i, with the same electric charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We construct the corresponding dressed operators Φi(x) = eig � x ∞ Aµdxµφi(x), using some particular prescription for the Wilson line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' These obey [Q, Φi(x)] = g Φi(x) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='7) where Q = � S2∞ ∗F is the charge operator which can be measured at space-like infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Suppose the point x = 0 is inside D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We create a charged excitation of type i in region D by acting on |0⟩ with a unitary Ui = eiϵΦi(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Then we study correlators in region D′ in the state Ui|0⟩ in perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Consider a correlator of Q and Φj(x) in region D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' ⟨0|U † i Φj(x)QUi|0⟩ = ⟨0|Φj(x)|0⟩ + iϵ⟨0|[Φj(x)Q, Φi(0)]|0⟩ + O(ϵ2) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='8) where to leading order in the perturbative expansion the second term is ⟨0|[Φj(x)Q, Φi(0)]|0⟩ = g⟨0|Φj(x)Φi(0)|0⟩ ∝ δij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='9) Hence by measuring correlators of all φj(x) and Q in D it seems that we can detect not only the presence of a charge in D, which is expected by Gauss’s law, but we can even identify the flavor of the charged particle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' the value of the index i in the interior of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' A similar argument in the gravitational case was discussed in [25, 48] for black hole states and in [58] around empty AdS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The reason we were able to get information beyond the total charge in D is that in the vacuum the fields have non-trivial entanglement, on which the non-vanishing 2-point function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='9) depends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' When we act with the unitary containing the Wilson line, the Wilson – 12 – line disturbs the pattern of entanglement in such a way that it breaks the symmetry between the fields φi and we can detect from D′ the flavor of the excitation in D This suggests a way to avoid the issue and succeed in hiding the flavor of charge in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We start with the analogue of a split state in the U(1) gauge theory, see the discussion in [59], and then create the charged excitation in D by acting with the same unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In that case there is no entanglement bewtween D and D′ and hence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='9) will vanish making it impossible to tell from measurements in D′ what is the type of charged excitation in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='12 This requires creating the charged excitation on top of the split state, with typical energy scaling like (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3), rather than the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3 Classical and Quantum Gravity First we notice that in non-perturbative quantum gravity we do not expect to be able to localize information in space: holography and AdS/CFT suggest that the fundamental degrees of freedom in quantum gravity are not local, but rather lie at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Moreover there is strong evidence that an ingredient towards the resolution of the black hole information paradox is that the naive factorization of the Hilbert space in space-like separated subregions may not be true in the underlying theory of quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' On the other hand at the classical level in General Relativity we do have an exact notion of locality and information can be localized, as we will discuss below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' An interesting question, which is the main focus of this paper, is to understand the fate of locality at the level of perturbative quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1 On the initial value problem of general relativity In General Relativity the initial value problem is formulated by starting with a spacelike slice Σ and specifying the data C = (hab, Kab) where hab is the intrinsic metric and Kab the extrinsic curvature of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' If we have matter then the values of the fields and their normal derivatives need to be specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Initial data related by spatial diffeomorphisms on the slice Σ are gauge-equivalent and have to be physically identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In general relativity there is one more subtlety: even if we have two initial data on the slice Σ which are not related by a spatial diffeomorphism, they may still correspond to the same physical solution in space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This is related to the freedom of choosing the initial slice Σ in space-time and diffeomorphism invariance in full space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Admissible initial data, which can be extended into a solution of the Einstein equations must obey the following constraints R + (Ka a)2 − KabKab = 16πGρ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='10) ∇aKab − ∇bKc c = −8πGJb , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='11) 12A more mundane way to hide the charge is to add ”screening charges” in the buffer region, but here we want to discuss how information can be localized even though a Wilson line extends all the way to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 13 – where R is the Ricci scalar of hab on Σ, the covariant derivatives are with respect to hab on Σ, na is the unit normal to Σ and ρ = Tabnanb and Jb = −hc bTcana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We now want to address the question of localization of information in classical general relativity, as formulated in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' A theorem, see for example [49,60], settles question B for pure general relativity: if we have two admissible initial data which agree, up to spatial diffeomorphism, on a part D′ of Σ, then the corresponding solutions will agree, up to a space- time diff, on the development of D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This continues to be true in the presence of matter provided certain reasonable conditions are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This shows that in general relativity signals propagate at most at the speed of light: if we modify the initial data only in the region D, then the signals will propagate in the causal future of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Then we come to question A, that of localizing information on compact regions on Σ: to what extent is it possible to find two initial data satisfying the constraints (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='10), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='11), which agree on D′ but differ on D?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='13 The equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='10) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='11) are non-linear and of elliptic nature, though underdetermined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Understanding the space of solutions of the constraint equations is an interesting problem which has been studied extensively in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Here we summarize some relevant points: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Gravitational Gauss law: in asymptotically flat or AdS space-times, the energy and other conserved charges are defined at space-like infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The constraints of general rel- ativity relate these asymptotic charges to contributions from excitations in the interior of space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For example, in the Newtonian limit the constraint equations reduce to the gravitational analogue of Gauss’s law □φ = 4πGρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' As in electromagnetism this implies that the initial data in region D′ know about the total mass enclosed in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Existence of localized deformations: it is possible to find many solutions of the constraint equations which look the same in the domain D′ but differ on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For example, if we restrict our attention to spherically symmetric solutions, Birkhoff’s theorem implies that there is a large number of solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='10) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='11) which all look like the Schwarzschild metric of mass M in D′ but differ in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Examples include static, interior, star-like geometries supported by matter or more generally spherically symmetric, time- dependent collapsing geometries of mass M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' More generally, it has been shown [61] that under reasonable conditions a compact patch D of a solution of the constraints (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='10) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='11) can be glued to a boosted, Kerr solution in D′ of appropriate mass, angular momentum, momentum and center of mass position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The existence of a large number of solutions, which all look exactly the same in D′ demonstrates that it is possible to localize information in classical general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 13Here we need to keep in mind that even if the initial data differ on D they may correspond to the same solution in space-time, as they may correspond to two different choices of the slice Σ in the same space-time solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 14 – 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Comments on the vacuum: For asymptotically AdS geometries, if a solution looks like empty AdS in D′14 then it is guaranteed to be empty AdS in D as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In other words, starting with the vacuum it is not possible to modify the initial data in D into a new solution, without at the same time modifying the solution in D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2 Diff-invariant observables in classical GR We now consider the question of defining local diff-invariant observables in gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This is a long-standing problem which is subtle even at the classical level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Let us consider general rel- ativity, possibly coupled to other fields, defined with certain asymptotic boundary conditions at infinity (for example asymptotically flat or AdS) or on a closed manifold of fixed topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We denote by X the space of solutions of the equations of motion, in any possible coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' On this space we have the action of the group Diff of diffeomorphisms15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Solutions related by a diffeomorphism are physically identified and we introduce X = X/Diff .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='12) We can think of a diff-invariant observable as a function which has definite values on points of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' However, we do not demand an observable to be necessarily defined on the entire space of solutions X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Instead we will allow observables to possibly have a limited domain of definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Hence a diff-invariant observable is a map A : U ⊂ X → R , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='13) where U is an open subset of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Such observables can also be expressed as functions on X which must obey A(s) = A(f∗s), where s denotes a solution in some coordinate system and f∗ the action of a diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In order for a diff-invariant observable to be local we need to impose additional conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' To formulate these conditions it is useful to introduce the Peierls bracket {A, B} between two diff-invariant observables [62], which is a covariant generalization of the Poisson bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' To compute the value of {A, B} we consider a modification of the action as S → S + ϵA and compute the difference of the first order change of observable B on the perturbed solutions with advanced (+) and retarded (−) boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The Peierls bracket is defined as16 {A, B} = δ− AB − δ+ AB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='14) It can be shown that the Peierls bracket has similar properties as the Poisson bracket, for example linearity, antisymmetry and the Jacobi identity, and in fact coincides with the Poisson bracket if a Hamiltonian formalism is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' One of the advantages of the Peierls bracket 14Here we assume that D is compact so D′ includes the region near space-like infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 15If the space-time is non-compact along space we only consider small diffeomorphism, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' those which become trivial fast enough at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 16The first order solutions are not unique due to diffeomorphism invariance, however the ambiguity drops out when computing the change of the diff-invariant observable B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 15 – is that we do not need to pass to the Hamiltonian formalism which is somewhat complicated due to the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Notice that to define the Peierls bracket of two observables A, B they must have a common domain of definition on X and the bracket will be generally a non-trivial function on this overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We would like to define diff-invariant observables which can be associated to points in space-time with the property that if two such observables are associated to space-like sep- arated points the corresponding Peierls bracket must vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The difficulty in doing this is that in order to define an observable we need to define it at least in an open neighborhood around a state as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='13), so we need some prescription for following ”the same point”, on which the candidate diff-invariant observable will be localized, as we move on the space of solutions X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' General covariance implies that there is no canonical way to keep track of the point as we change the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' If the space-time has a well-defined boundary we can find prescriptions which select a point in space-time for each solution in X relationally with respect to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For example in AdS we can define a diff-invariant observable which seems to be localized at a point by considering a radial geodesic at right angle from a specific point on the boundary, moving a fixed regularized distance along it and measuring the value of a scalar quantity, for example a scalar field or a scalar combination of the curvature, at the resulting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This gives a map from the space of solutions X to R, so it is a diff-invariant observable which could potentially be local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Notice however that the location of the resulting point depends on the entire geometry along the geodesic, all the way from the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Changing the metric anywhere along this geodesic will move the resulting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Hence the value of the observable will not strictly depend on local data near the point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Similarly, if we act with one of the asymptotic symmetries the boundary starting point will move and also the resulting bulk point will move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This implies that the Peierls brackets of this candidate observable with the boundary charges, or other observables along the geodesic will be non-zero, even though these regions are space-like separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Hence this relational observable is not really local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Another way to to define candidate local diff-invariant observables is to consider a com- plete gauge fixing scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Then observables in the particular gauge labeled by a space- time coordinates are automatically diff-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' However they will generally have non-local Peierls brackets, since the assignment of a coordinate value to a point in space-time in the particular gauge, will generally depend on the solution everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Additional difficulties arise in space-times without boundaries, for example in de Sitter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' A boundary is an (asymptotic) part of the spacetime where gravity is not dynamical anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This is why we can for example anchor geodesics to the boundary, and define relational diff-invariant observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Without a boundary, there is no part of the space-time where gravity is turned off, and consequently no place to anchor geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3 State-dressed observables If we consider a solution that is sufficiently complicated it is possible to specify points, and hence define local diff-invariant observables, by using features of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We emphasize that – 16 – these observables will not have all the desired properties over the entire space of solutions X, so these observables have certain aspects of state-dependence as discussed around (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' One approach based on this idea was studied by DeWitt [16], building on [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For a D-dimensional space-time we start by identifying D scalar quantities Za, a = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=', D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' These can be combinations of curvature invariants and other scalars formed by the fields of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We could try to fix a coordinate system by using these D-scalars as coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We can use this intuition to introduce candidate local diff-invariant observables of the form φ(Za 0) = � dDx φ(x) δD(Za − Za 0) det∂Z ∂x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='15) Here Za are the D scalar quantities introduced above and φ is any other scalar combination of the fundamental fields of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Similar constructions can be done for fields with tensor indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Some comments are in order: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For a general space-time which is in-homogenous, and for certain choices of the values Za 0, the delta function in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='15) will click on a finite number of points, so the quantity above is well-defined and finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In symmetric space-times it will either not click at all, hence the observable will be zero, or an infinite number of times so the observable will be ill-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This shows that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='15) is a quantity which is defined only on part of the phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This is in accordance with our expectation that state-dressed observables have to be state-dependent (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Suppose that the observable (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='15) is well defined on a state s and a neighborhood U of the space of solutions X around it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It is clear that, at least at the classical level, this observable is diff-invariant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' a well defined map φ(Za 0) : U ⊂ X → R and hence a good observable according to the definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' One can show that under certain conditions, observables (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='15) are also local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' If we have a state s on which two such observables φ(Za A), φ(Zb B) are well defined, with the property that the delta functions click at single points A, B and that these points are space-like separated with respect to the metric of s, then the corresponding observables have vanishing Peierls brackets {φ(Za A), φ(Zb B)} = 0, see [63] for a review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This follows from the causality properties of linearized Green’s functions appearing in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='14) around the solution s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Notice that if two points A, B are spacelike separated on a solution s, then there is a small enough neighborhood of s in which they remain space-like separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Hence their Peierls bracket will vanish in this entire neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This shows that, as long as we accept that observables may be defined only locally on the phase space of solutions, it is possible to find local, diff-invariant observables in classical general relativity around states which are complicated enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' These are also the interesting states, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' those containing bulk observers who want to study physics in their environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 17 – 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Similar ideas are useful in cosmology, where the value of a scalar field can be used as as clock [64–66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The next question is whether it is possible to define similar observables at the quantum level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Aspects of this question were discussed in [17] and [18], where it was argued that there is a quantum version of these observables which retain their locality properties to all orders in the ℏ expansion, even though they are not expected to be local at the non-perturbative level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Various difficulties are encountered at the quantum level including the question of the renormalization of the composite operators (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='15), establishing diffeomorphism invariance at the quantum level and the role of Poincare recurrences which will generally introduce infinite copies where the delta function will have support [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In this paper we provide support in favor of this conjecture by finding observables with certain similarities in spirit to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='15) directly in CFT language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This has the advantage that any object built directly in the CFT is by construction diff-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4 A time-band in AdS We now specialize to a setup that will allow us to make contact with AdS/CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We consider geometries that are asymptotically AdSd+1 and we consider a short time-band T−ϵ,ϵ on the boundary in global coordinates, defined as the set of points (−ϵ, +ϵ)×Sd−1 , ϵ > 0, where the first interval refers to the time coordinate t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Near the boundary we can select a Fefferman- Graham coordinate system where the fields, for example the metric and a scalar of mass m2, have the behavior ds2 = dr2 r2 +r2(−dt2 +dΩ2 d−1)+r2−dgµν(r, x) dxµdxν gµν(r, x) = g(0) µν (x)+g(2) µν (x)r−2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' φ = r−∆(φ(0)(x) + φ(2)(x)r−2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=') , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='16) where x = (t, Ωd−1) and ∆ = d 2 + � d2 4 + m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Here we consider normalizable states so the growing modes, which would be dual to sources in the CFT, are set to zero17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The Fefferman- Graham coefficients g(0) µν (x), φ(0)(x) are diff-invariant observables and are labelled by boundary coordinates18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This set of observables includes the asymptotic charges, for example the ADM Hamiltonian can be computed as H = 1 const � Sd−1 dΩd−1g(0) 00 (x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='17) We focus on these Fefferman-Graham observables restricted in the time band T−ϵ,ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This set of observables is closed under Peierls brackets and form a Poisson algebra A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Notice that in this algebra we do not include observables which would be finite distance under Poisson flow, 17We only assume that the sources are zero in the time band T , they could be turned on in the far past in order to prepare a state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 18The subleading coefficients are fixed by the equations of motion in terms of the leading ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 18 – otherwise flowing by finite distance with H would take us out of the time-band, see also the discussion in [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Starting with the classical theory, we ask whether we can find observables localized deep in the interior of AdS which are space-like with respect to the time-band and which have vanishing Peierls brackets with observables in the time-band algebra A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' These candidate observables are to be defined as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='13), in particular they need to be defined on a neigh- borhood U ⊂ X of a solution s ∈ U and not necessarily on the entire space of solutions X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It is clear that observables defined relationally with respect to the boundary, or with a gauge fixing condition which makes use of the boundary, do not satisfy these conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Due to their gravitational Wilson lines they will have non-vanishing Peierls brackets with the Hamiltonian and other charges on the boundary [44,45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Such observables generally change the energy of the state, which due to the gravitational Gauss law can be measured in the time band T−ϵ,ϵ by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Another point of view is that such observables identify a point in the bulk, and in particular a moment in time, relationally with respect to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Thus an infinitesimal motion in time of the starting point on the boundary is translated via the relational prescription into an infinitesimal time motion of the corresponding bulk point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Then the Peierls bracket of the candidate bulk observable with H generates time-derivatives of the point in the bulk and is non-vanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The discussion of the previous subsection implies that if we start with an asymptotically AdSd+1 solution s of the bulk equations which is complicated enough, then we can define diff-invariant observables of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='15) in a neighborhood of s so that they have vanish- ing Peierls bracket with all elements of the time-band algebra A including charges like the Hamiltonian (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Such observables do not change the total energy of the state but instead they rearrange the energy, ”absorbing” from the background solution the amount of energy they themselves create.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' These observables select a point in the bulk, and a moment in time, by using features of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In what follows we will provide evidence that the same conclusions are true in perturbative quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We will proceed by translating the question in CFT language and using the AdS/CFT correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 3 Holographic setup In this paper, we will study the question of locality in quantum gravity in the context of the AdS/CFT correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' A question we would like to understand is how certain bulk subregions are encoded in the boundary CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' There are cases where this is well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For example, the bulk dual of a boundary subregion is known as the entanglement wedge, which is the bulk region extending between the boundary subregions and the relevant Ryu- Takayanagi surface extending in the bulk [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This correspondence between parts of the boundary and bulk is known as subregion-subregion duality [42,69,70], and it is worthwhile to mention that in general, the entanglement wedge of a boundary subregion is much larger – 19 – than its causal wedge (the part of the bulk contained by lightrays shot from the causal developments of the boundary subregion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Subregion-subregion duality and entanglement wedge reconstruction utilizes the organi- zation and entanglement of CFT degrees of freedom organized spatially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We will be interested in rather different bulk subregions, which lie deep down in the bulk and never extend to the boundary CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' What is the CFT dual of a causal diamond located deep near the center of AdS?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The answer to this question remains elusive, and in particular it is understood that in general, these bulk regions do not correspond to the entanglement wedge of any bound- ary subregion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' There have been previous attempts to understand the CFT mapping of such regions, see for example [21–24] which attempt to assign a meaning to the entropy of a gen- eral closed codimension-2 spatial curve in AdS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Here we will follow a different approach by focusing on the algebra of single-trace operators [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We will start by reviewing some basic but relevant features of AdS/CFT, before turning to a discussion of the class of states that we will be considering throughout this paper and their salient properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1 Gravitional states in AdS, large diffeomorphisms and asymptotic symme- tries We will be interested in gravitational solutions which are asymptotically AdSd+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We have in mind an embedding in a top-down setup with a holographic dual CFT, like N = 4 SYM at strong coupling, on S3 × R and the N-scaling we indicate in most of the paper refers to this theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' However for most of the discussion the details of the embedding in string theory, the extra fields, as well as the presence of a compact internal manifold are not important unless explicitly stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Solutions to the bulk equations of motion can be thought of as states in the dual CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' If we think of a bulk geometry described by a Penrose diagram, the diagram really represents the entire time-history of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We can take the state to live at t = 0 on a boundary Cauchy slice, and the portion of the geometry relevant to describing the state is an initial data surface given by a bulk Cauchy slice (or the Wheeler-de Witt patch associated to the boundary Cauchy slice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' To view these geometries as states of the dual CFT, it is important that the bulk fields have a fall-off corresponding to normalizable modes with vanishing CFT sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='19 We want to consider semi-classical solutions with non-trivial bulk geometries, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' where backreaction is strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The corresponding CFT states |Ψ0⟩, which we take to be pure, have large energies which scale as ⟨Ψ0| H|Ψ0⟩ ∼ O(N2) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1) and as we will see, they will generally also have an energy variance of the same order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We will also consider perturbative excitations of the quantum fields on top of the background 19If these states are prepared by a Euclidean path-integral [71–74], sources can be turned on in the Euclidean past which prepares the state, but it is important that they vanish as tE → 0 for the geometries to be interpreted as states in the undeformed CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 20 – geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' These excitations add/subtract quantum particles which change the energy by an O(N0) amount, and whose backreaction on the geometry is thus generally small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Geometries of this type will often be macroscopically time-dependent, such that the initial data on a bulk Cauchy slice changes as we perform time-evolution of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This has consequences for the variance of the energy, as we will now see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Any state |Ψ0⟩ can be expanded in the basis of CFT energy eigenstates as |Ψ0⟩ = � i ci|Ei⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2) The time-dependence of the bulk geometry implies that such states will have energy variance (∆H)2 ≡ ⟨Ψ0|H2|Ψ0⟩ − ⟨Ψ0|H|Ψ0⟩2 ∼ O(N2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3) To see this, consider the inequality 1 2|⟨[H, A]⟩| = 1 2 ⟨∂tA⟩ ≤ ∆H · ∆A , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4) where in the first equality we assumed that the operator A is not explicitly time-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Then we have ∆H ≥ 1 2 ⟨∂tA⟩ ∆A ∼ O(N) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='5) where we have used large N factorization for the operator A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This shows that provided there is macroscopic time-dependence (the classical vev of A changes at leading order), the variance of the energy scales at least as N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='20 Some bulk geometries we will consider are macroscopically time-dependent, but only inside the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In this case, we cannot use the argument above, but we still expect the variance to be of order N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It is interesting to ask whether the variance is a quantity that can be extracted from the semi-classical geometry alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In general, we expect that the quantum state of the fields in the bulk is important as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We discuss this further in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' There are various types of explicit constructions of states of this kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' There are states prepared by Euclidean path integral with sources for single-trace operators [71–74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' These states should be interpreted as coherent states of the quantum gravitational dual, which are labelled by phase-space points corresponding to initial data21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' There are also states prepared by a boundary state of the CFT, further evolved by some amount of Euclidean time [76–79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The bulk interpretation of these states is that they correspond to black hole geometries with End-of-the-World branes sitting behind the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This is an example where the bulk geometry is macroscopically time-dependent, but only behind the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Similarly, for two-dimensional CFTs, we can construct pure states by performing the path 20Note that if the variance is parametrically larger than O(N 2), the state may no longer have a good semi-classical interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' An example would be a superposition of black holes of different masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 21It appears that one may not construct arbitrary initial data this way, see [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This will not affect our construction and for states prepared by a Euclidean path integral, we should simply keep in mind that we have access to a restricted class of initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 21 – integral over a surface of higher topology, for example half a genus-2 surface, see [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' These geometries are also macroscopically time-dependent behind the horizon, but instead of having a brane behind the horizon, they have topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Finally, it is worth noting that there are semi-classical geometries that also preserve supersymmetry, the most famous of which are the LLM geometries [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In these cases, one can obtain a better understanding of the dual CFT states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We will come back to these geometries in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' As usual in gravity, we should identify solutions which are related by small diffeomor- phisms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' diffeomorphisms that vanish near the AdS boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' There is also a class of large diffeomorphisms, which are compatible with the boundary conditions imposed in the definition of our theory of AdS gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This set of diffeomorphisms forms what is called the asymptotic symmetry group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In the case of AdSd+1, d ≥ 3 this is the conformal group SO(2, d), while for d = 2 it gets enhanced to the Virasoro group [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' When acting on a given bulk solution these large diffeomorphisms will generally transform the geometry into a new state, which is physically distinguished from the previous one, unless of course the original state happens to be invariant under the symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We will later also discuss solutions with two asymptotic boundaries, such as the eternal black hole in AdS, in which case the asymp- totic symmetry group is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Let us now discuss the various elements of the asymptotic group/conformal group: Time translations: One particular class of states we will discuss are those with semi- classical time-dependence in the bulk, for example a state corresponding to the gravita- tional collapse of a star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In this case large diffeomorphisms corresponding to asymptotic time translations transform the state as |Ψ0⟩ → e−iHt|Ψ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The initial data correspond- ing to |Ψ0⟩ is not the same as that of e−iHt|Ψ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Our end goal will be to provide local operators whose gravitational dressing is done towards a feature of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' If the state is time-dependent then we can select a moment in time by using the features of the state, as opposed to the boundary time coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' On the other hand if the state is static, then the only way to identify a moment in time is by dressing to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This is why it will be important for us to consider time-dependent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' SO(d) rotations: If the state breaks SO(d), then asymptotic rotations transform it to a new state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In this case we can use the features of the state to identify the angular location of a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' On the other hand, if the state is SO(d) invariant it will generally not be possible and at best we can obtain an operator smeared over the bulk angular coordinates, or alternatively we can fix the angular location by dressing to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' AdS boosts: The Lorentzian conformal group acting on Sd−1 × R has another 2d generators which correspond to boosts in various directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' These can be realized as d non-independent copies of an SL(2, R) algebra, see for example [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Any state with finite energy cannot be annihilated by Hermitian combinations of these generators, which we show in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The only state which is annihilated by these generators is the global vacuum and any other state will necessarily transform under the action of – 22 – these boosts22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Therefore, in any non-trivial state, we can fix the radial position of an operator without referring to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In a top-down setup, the gravity dual may have an internal manifold, like the S5 in the context of N = 4 SYM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In such cases, we would need to break the R-symmetry to localize a bulk operator in the internal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In this paper, we will mostly restrict to a bottom up construction without an internal manifold but it would be an interesting generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2 Locality in AdS We are now ready to discuss locality in quantum gravity with asymptotically AdS boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We would like to understand whether one can define local observables and whether we can localize information deep in the center of the AdS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The presence of the AdS boundary allows us to define one natural class of diff-invariant observables: The fields in AdS can be expanded in a Fefferman-Graham expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The coefficients of this expansion are themselves diff-invariant observables, which are dressed to the boundary since the Fefferman-Graham gauge is chosen with respect to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Let us call these observables FG-observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For example, the AdM Hamiltonian is one particular observable in this class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In perturbative quantum gravity, we can also consider the expectation values of these observables as well as their higher-point correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' As we will discuss below, if we want to stay within the regime which can be described by semi-classical gravity we may need to restrict the complexity of the correlators (for example the number of operator insertions in the correlation function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We emphasize again that all these observables are dressed with respect to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In particular, they will generally not commute with the Hamiltonian or the other charges described in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The question we would like to address is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' If we start with a state with a semi-classical geometric description, is there a way to modify the state in the interior of AdS, without modifying any of the correlators of FG-observables localized in a short time-band of the boundary?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' If the answer is yes, this means we can localize information since an observer living near the boundary will have no way to know whether or not we modified the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Rather than trying to come up with bulk objects that achieve this goal, we will address this question directly in the dual CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This has the following advantage: any object built out of CFT degrees of freedom is necessarily diff-invariant and non-perturbatively well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Provided the object acts in the right away, we can be assured that the construction is fully consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3 The CFT description and the time band algebra Consider a large N holographic CFT which is dual to semi-classical general relativity coupled to matter fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In the large N limit, we can define the algebra A generated by single-trace operators in a time-band Dt1,t2, where we allow products of single-trace operators where the 22States with infinite energy like the AdS-Rindler vacuum could also potentially be annihilated by some boost generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 23 – number of factors is arbitrary but scales like O(N0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='23 This was originally discussed in [20], inspired by the earlier work [6, 25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In [20] it was proposed that the algebra A can be thought of as being dual to the causal wedge of the region Dt1,t2 in the bulk (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This picture also suggests that the algebra A has a commutant which can be idenfitied with a spacelike-separated causal diamond in the interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Algebras of this type have received attention recently [27–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The work [20] studied this setup for states which are small perturbations around the AdS vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The geometry of AdS is homogeneous and featureless since it is a maximally symmetric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' As already discussed in the previous section, this makes the definition of local diff-invariant observables challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We would like to revisit the time-band algebra, this time in cases where the bulk state has features, which in particular are time-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This means the state must be highly excited as can be seen for example from its energy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' At infinite N the problem can be understood in terms of QFT on a curved and in general time-dependent background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In particular, gravitational backreaction of the quantum fields can be ignored and one does not need to talk about gravitational dressing, which is a form of backreaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In this case, the existence of the commutant is obvious because we are in a QFT situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Note that if the Hamiltonian (which is always an element of the time band algebra) is normalized appropriately24, its commutator with the other single-trace operators is suppressed by 1/N and thus vanishes when N is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' At the level of 1/N corrections, the existence of the commutant is less obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Backreac- tion must now be taken into account and the gravitational Gauss law can spoil the commutator between H and the other operators of the time-band algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For example, the standard way to write bulk fields in terms of CFT operators is the HKLL construction [33–39] Φ(t, r, Ω) = � bdry dt′ dΩ′ d−1K(t, r, Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' t′, Ω′)O(t′, Ω′) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6) where K is related to a Green’s function of the Klein-Gordon operator on the appropriate bulk geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This operator is defined purely within the CFT so it is manifestly diff-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' To leading order at large N, it acts as a bulk field and commutes with other bulk fields at spacelike separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Notice however that in order to define the kernel K we have to choose a coordinate system in the bulk, which often is taken using Fefferman-Graham gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' As we already mentioned, this gauge choice is defined by making use of the asymptotic boundary, and an HKLL operator is thus dressed to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Because of this, the commutator between an HKLL operator and the Hamiltonian will not vanish at subleading orders in the 1/N expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The physical origin of this effect is the gravitational Gauss law: acting with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6) will generally create or destroy a particle in the bulk, thus changing the energy of the state, which 23Notice that at finite N the algebra in a time-band would be the same as the full algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In the large N limit, a natural hierarchy emerges between ”small products” of single-trace operators and the rest of the algebra, which allows us to consider the notion of a time-band algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 24A useful normalization is h = 1 N (H − ⟨Ψ0|H|Ψ0⟩), which ensures that ⟨Ψ0|h2|Ψ0⟩ ∼ O(N 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 24 – can be immediately measured at spacelike infinity by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' One can try to correct the HKLL operators at higher orders in 1/N by mixing it with other single- and multi-trace operators, see [39, 84, 85], but the commutator with the Hamiltonian is universal and generally cannot be removed in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It is also possible to think about the dressing in terms of (smeared) gravitational Wilson lines connecting the bulk operator to the boundary, which make it diff- invariant at the price of making it non-local [86–89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The commutator with H is nonzero because H picks up the contribution of the Wilson line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This raises the question of whether the algebra A still has a commutant at subleading orders in 1/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The main goal of this paper is to provide evidence for the existence of such a commutant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We will do so by identifying a class of operators that are gravitationally dressed with respect to features of the state, rather than dressed to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In particular, these operators will have vanishing commutators with the Hamiltonian, to all orders in 1/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In this paper, we will focus mostly on ensuring that bulk operators have a vanishing commutator with the Hamiltonian (and the other charges), but it would be important to extend our construction to all single-trace operators in Dt1,t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We given an alternative argument for the existence of a commutatant to all orders in 1/N in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The existence of a commutant for A in 1/N perturbation theory would imply that in- formation can be localized in regions of the bulk and is not visible from the boundary at the level of perturbative quantum gravity25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We are now ready to formulate the concrete goal that we will achieve in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4 Formulating the main goal Our goal is to improve the locality properties of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6) by moving the gravitational dressing from the boundary to the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' From a technical point of view, we will find CFT operators �Φ which obey two properties: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' [Qi, �Φ] = 0 to all orders in 1/N, for all asymptotic charges Qi ∈ SO(2, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The correlators of �Φ agree with those of ΦHKLL to leading order in the large N expansion, on the code subspace of |Ψ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In taking the large N limit it is important to track how various effects scale with N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' As we will see, our new operators �Φ have vanishing commutator with Qi to all orders in the 1/N expansion, but have a non-vanishing commutator at the level of e−N2 corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In what follows we will first focus on ensuring a vanishing commutator of ˆΦ with the Hamiltonian H to all orders in 1/N and then discuss the generalization to the other charges in SO(2, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' As we will see, our construction will not work for |Ψ0⟩ = |0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Technically, this is because the vacuum does not comply with the properties (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Physically, it is because 25See [46,58,59,67,90–94] for other discussions of localization of information in perturbative quantum gravity, with varying conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 25 – the AdS vacuum has no feature that we can use to attach the dressing of our local operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Note that this is in line with the results of [46], where a protocol to reconstruct the bulk state from correlators in the time-band was discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='5 Time-shifted states and return probability We will now present the main technical tool that will enable us to define state-dressed opera- tors: the return probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Let us start with a state |Ψ0⟩ satisfying the properties (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We define the following one-parameter family of states |ΨT ⟩ = e−iTH|Ψ0⟩ T ∈ R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='7) In the bulk, the states |ΨT ⟩ are related to |Ψ0⟩ by a large diffeomorphism, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' one that does not vanish near the boundary and induces a boundary time-translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It is important to emphasize that they are different quantum states, even though they are related by a symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' If we think about the phase space of gravity in AdS, the family of states correspond to different phase space points, just like a particle moves on phase space as a function of time in classical mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' From the bulk perspective, if |Ψ0⟩ was a coherent state, we can also think of |ΨT ⟩ as coherent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We would now like to consider the overlap of such states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In particular, we would like to study the overlap ⟨Ψ0|ΨT ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='8) Thinking of these states as coherent states is useful to gain intuition about such overlaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For the simple harmonic oscillator, the overlap of two coherent states is ⟨α|β⟩ = e− 1 ℏ f(α,β) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='9) for a very simple quadratic function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For states on the gravitational phase space, recalling that ℏ ∼ GN ∼ 1/N 2, we thus expect ⟨Ψ0|ΨT ⟩ = e−N2f0(T) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='10) for a function f0 whose real part is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In the gravitational setting, it is not straight- forward to directly compute f0(T) from the phase space information, see [47] for a discussion on nearby states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' There is a general way to compute f0(T) based on a Euclidean preparation of the states [74], but it requires some effort (in particular solving the non-linear Einstein equations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The computation of f0(T) directly from the information on an initial data slice, which specifies the point on phase-space, is an interesting problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='26 It is also instructive to think about the overlap from a microscopic point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In the CFT, the overlap is given by ⟨Ψ0|ΨT ⟩ = � i |ci|2e−iTEi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='11) 26Similarly, we do not know of a gravitational argument that guarantees that the real part of f0(T) is positive, which must be the case if the geometries have a state interpretation in the dual CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We comment on this further in the discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 26 – Note that there are eS(E) terms here, each of size e−S(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The suppression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='10) must therefore come from the summation over a large number of phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' If the bulk state has no periodicities in time, we expect the real part of f0(T) to increase as we increase T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' However, this increase will not continue forever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We will shortly give an estimate of the time-average of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='11), and argue that the decay will saturate at some point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Physically, the non-trivial overlaps (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='11) imply that it is not correct to think that all the states |ΨT ⟩ are independent, see also [47, 95, 96] for related discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In particular, even if the bulk state is not macroscopically periodic, there will still be a microscopic periodicity of the state due to Poincare recurrences, that will happen at very large T ∼ O(eeN2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Throughout this paper, we will be interested in much earlier time scales so it will be sufficient for us to treat the states |ΨT ⟩ as quasi-orthogonal since all overlaps will be exponentially small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We will also need to define the notion of code subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Starting with the state |Ψ0⟩ we define the code subspace as H0 = span{|Ψ0⟩, O(t, Ω)|Ψ0⟩, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=', O1(t1, Ω1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='On(tn, Ωn)|Ψ0⟩} , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='12) generated by acting on |Ψ0⟩ with a small number (n ≪ N) of single-trace operators27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It will also be useful to define the projector P0 on this subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Similarly, a code subspace can be defined for each of the time-shifted states HT = span{|Ψ⟩T , O(t, Ω)|ΨT ⟩, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=', O1(t1, Ω1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='On(tn, Ωn)|ΨT ⟩} , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='13) with the corresponding projector PT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The projectors P0 and PT are simply related by time- evolution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' we have PT = e−iTHP0eiTH , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='14) and in particular, we emphasize again that PT ̸= P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In what follows, it will be convenient to work with real quantities rather than the overlap (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='8), and we are now ready to define the return probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6 The return probability We now ready to examine the T-dependence of the overlap (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='11) in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' As explained above, it is more convenient to work with a real quantity so let us define the return probability R(T) := |⟨Ψ0|e−iTH|Ψ0⟩|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='15) It is similar to the spectral form factor (the two coincide when |Ψ0⟩ = |TFD⟩ and H = HL + HR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Recently, the spectral form factor has been extensively discussed in connection to the black hole information paradox and quantum chaos, see for example [97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The time-scales of interest in that context are again late times such as t ∼ eN2 (note this is much shorter than 27To be precise, we should also give a small smearing to the single-trace operators in order to avoid UV divergences of operator insertions at coincident points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We will leave it as implicit in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 27 – the Poincare recurrence time which is doubly exponential).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Here again, we will be interested in much earlier time-scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In general, it is difficult to compute (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' As we mentioned above, the overlaps can be computed from time-shifted coherent states in gravity but the best known technology to do so uses the Euclidean path integral and involves solving the non-linear Einstein’s equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Nevertheless, we can compute the very early time dependence using large N factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We present this calculation in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' At early times, we have R(T) = e−(∆H)2T 2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='16) which is generally valid for times up to T ∼ O(N−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For the purposes of this paper, we want to understand how the return probability behaves at time-scales T ∼ O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Here, the decay does not follow from large N factorization and it is in general not an easy task to compute it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In Appendix C, we review that for the TFD state, the return probability (which is the spectral form factor) decays as RTFD(T) = e−N2fTFD(T), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='17) where fTFD(T) is O(N0) and for early times T ∼ O(N0) ≪ β behaves like fTFD(T) ≈ αT 2, where α is an O(N0) constant which depends on the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This is an extremely fast decay, much faster than thermalization where the prefactor in the exponent is of order N0, and shows that thermofield double states at different times orthogonalize exponentially fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We expect similar behaviour for many other semi-classically time-dependent states, that is for timescales of T ∼ O(1), we expect R(T) ∼ e−N2 ˜f0(T) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='18) for a positive and O(N0) function ˜f0(T) which depends on the state |Ψ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We expect that for small T the function ˜f0(T) starts quadratically, as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Note that this fast decay is not even a consequence of quantum chaos, as it can occur at weak coupling or even in free theories, provided they have a large number of degrees of freedom (see [98] for a study of this question in weakly coupled N = 4 SYM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The difference between a free theory and a holographic one will manifest itself in the time-scale during which the exponentially small overlap remains valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For free N = 4 SYM, the spectrum is integer spaced and so the return probability will be periodic with period 2π, while in a chaotic theory it will take doubly exponentially long for the signal to return to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The average late-time value of the signal is also highly dependent on whether the theory is chaotic or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For a system with no degeneracies,28 R = lim t∗→∞ 1 2t∗ � t∗ −t∗ dT R(T) = � i |ci|4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='19) 28Systems like N = 4 SYM will have degeneracies due to superconformal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For example, for every primary, there are towers of descendants with degenerate energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Nevertheless, the number of degenerate states is exponentially smaller than the number of all states, at least in the high-energy sector of the theory, so the degeneracy only contributes a subleading effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 28 – For the type of states we are considering, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' those with a large energy variance, this is exponentially small, and scales as e−α′N2, where α′ is an O(1) constant which depends on the particular |Ψ0⟩ we have picked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This value is often referred to as the plateau, especially in the context of the spectral form factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Between the initial decay (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='17) and the plateau (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='19), there can be other regimes, which are particularly interesting in connection to quantum chaos [99,100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For example, in the spectral form factor, the plateau is preceded by a ramp where the signal grows linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' These effects will not be important for the present work, as we will only consider O(1) timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The crucial point we will exploit throughout the paper is that the signal is already exponentailly small in N2 at those timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The overlap (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='8) obeys the property ⟨Ψt0|Ψt0+T ⟩ = ⟨Ψ0|ΨT ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='20) This may appear trivial, but it means that even if the bulk geometry appears to be static at the semi-classical level, the return probability may still decay following (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='17) if the state had a period of manifest bulk time-dependence in the far past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Said differently, the variance in energy which determines the decay is unchanged under time-evolution, so even if the 1-point functions have stabilized, the variance remains large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This observation is particularly relevant in the case of a black hole formed by gravitational collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The exponential decay (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='17) can be extended to more general correlators of the form ⟨Ψ0|O(t1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' O(tn)|ΨT ⟩, where O are single-trace operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We expect ⟨Ψ0|O(t1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' O(tn)|ΨT ⟩ = F(T)⟨Ψ0|ΨT ⟩ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='21) where F(T) is finite in the large N limit and satisfies F(0) = ⟨Ψ0|O(t1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' O(tn)|Ψ0⟩ , dkF(T) dT k |T=0 = O(N0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='22) To see the exponential decay we write (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='21) as ⟨Ψ0|O(t1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' O(tn)|ΨT ⟩ = ⟨Ψ0|O(t1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' O(tn)|ΨT ⟩ ⟨Ψ0|ΨT ⟩ ⟨Ψ0|ΨT ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='23) The second term in this product is really responsible for the decay of the correlator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The first term is hard to evaluate from first principles, but in holography its meaning is clearer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In the bulk theory, it is computed by computing a correlation function on a background dictated by the Euclidean path integral with different sources on the northern and soutern hemisphere (corresponding to |Ψ0⟩ and |ΨT ⟩, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This correlator is O(1) and a smooth function of the background, which will generally change slowly with T, so we expect its time derivatives not to scale with N as indicated in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We check this statement in a few examples in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 29 – To sum up, any state in the code subspace (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='12) has an exponentially small overlap with any state in the code subspace (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This can be summarized by the relation Rcode(T) = 1 dcode Tr[PT P0] = O(e−N2 ˜f(T)) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='24) where dcode is the dimensionality of the code subspace, and for the time-scales we have dis- cussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The decay (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='24) can be used in combination with other useful inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For example, for a Hermitian operator O with eigenvalues λi, and if [P0, O] = 0, we have |⟨Ψ0|O|ΨT ⟩|2 ≤ � Tr[O4] � Tr[PT P0] and |⟨Ψ0|O|ΨT ⟩|2 ≤ max(λ2 i ) Tr[PT P0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='7 Other asymptotic charges More generally we can consider the change of the state by large diffeomorphisms corresponding to the other asymptotic symmetries of the theory, in the case of AdSd+1 the conformal group SO(2, d) with the generators we discussed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This leads us to define a natural generalization of the return probability R(g) = |⟨Ψ0|U(g)|Ψ0⟩|2 , g ∈ SO(2, d) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='25) where U(g) is the unitary realizing the conformal transformation of the CFT on Sd−1 × time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' What can we expect for these overlaps?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' To start, let us suppose the state |Ψ0⟩ breaks rotational SO(d) symmetry at the classical level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' By this, we mean that bulk dual geometry breaks the symmetry, which would be the case for some spherically asymmetric lump of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Take J to be the angular momentum generator, then we expect that the variance of J will be of O(N2) for such a state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Hence we expect that for small values of a rotation angle φ dual to J we will have R(φ) = e−(∆J)2φ2 = e−κN2φ2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='26) for κ ∼ O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For more general angles, we expect R(φ) = e−N2frot(φ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='27) However, because angular momentum is quantized, we have R(φ + 2π) = R(φ) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='28) hence the function frot(φ) has period 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In this direction of the conformal group the return probability has a very short Poincare recurrence equal to 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' All in all we find that as we increase φ away from 0 the return probability R(φ) very quickly dips down to exponentially small values and stays there until the Poincare recurrence at φ = 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' As we see from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='27), for any fixed φ which is in the range (0, 2π), we have R(φ) being exponentially small in the large N limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Of course if the state respects spherical symmetry then the return probability will not decay in the corresponding SO(d) directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It is worthwhile to discuss several distinct – 30 – scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In the simplest case, the state preserves the symmetry and is thus annihilated by the generators of rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The second simplest situation is the case where the symmetry is manifestly broken at the classical level (for example an asymmetric lump of matter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In this case, the breaking of the symmetry is manifest, and would be visible in the 1-point function of single-trace operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' There are also more subtle situations where the state breaks the symmetry classically in the bulk, but this may be invisible in the 1-point functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' An example of this are states by prepared by the path integral on higher genus surfaces in d = 2, and have topology behind the horizon [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='29 Finally as discussed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1, we expect that semi-classical states also break the other conformal symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We can get some intuition by considering a state dual to a conformal primary of dimension ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In this case the return probability along one of the conformal boost directions is determined by a group theoretic computation R(s) = |⟨∆|e−isK|∆⟩|2 = � 1 cosh2 s �2∆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='29) For primary states with ∆ ∼ O(N2), we get exponential decay of the form e−N2f(s) for any non-zero s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Notice that for the conformal boosts we do not expect any Poincare recurrence for large s, which in the case of primaries is obvious from the formula above, since such a transformation monotonically increases the energy of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In the case of AdS3 the asymptotic symmetry group is enhanced to Virasoro and similar statements hold for the flow of the state under more general large diffeomorphisms generated by Ln, Ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' To summarize, if we start with a state |Ψ0⟩ which breaks all conformal symmetries at the level of the semi-classical geometry we expect that R(g) defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='25) will decay exponentially fast in all directions away from the identity element on the conformal group manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 4 State-dressed operators We are now in a position to introduce operators ˆΦ which satisfy the two properties described in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4, namely their commutator with the Hamiltonian and other asymptotic charges is zero to all orders in the 1/N expansion and they act like HKLL operators to leading order at large N on the code subspaces {HT , T ∈ (−t⋆, t⋆)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Here t∗ is an order one (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' N0)) time of our choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We define the HKLL operator Φ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6), in the N → ∞ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In this limit the bulk is described by a quantum field theory on a curved spacetime and code subspaces for different T will be strictly orthogonal to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In addition, Φ is a local bulk operator which commutes with all the boundary single-trace operators in the time band algebra, including 29The thermofield double also has this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It breaks rotational symmetry of each CFT individually, but the breaking is invisible in 1-point functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It would be interesting to understand if this type of breaking always requires a horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 31 – the appropriately normalized Hamiltonian [36,84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' But it will no longer be commuting once 1/N corrections are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In particular, we will have [Φ, H − ⟨H⟩ N ] = O(1/N) ̸= 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1) Again, the physical reason behind this is that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6) is a diff-invariant operator that is dressed to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Note that for the naive HKLL operator (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6), the commutator with other single- trace operators will also be non-zero at order O(1/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For almost all single-trace operators, this can be removed order by order in 1/N by adding the appropriate corrections to Φ [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' However, these modifications will not be able to remove the non-vanishing commutator with the Hamiltonian (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Thus, to remove the gravitational dressing to the boundary CFT, a more sophisticated procedure is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We start by focusing on setting the commutator with the Hamiltonian to zero and discuss the extension to other asymptotic charges later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' To this end, we introduce the following operator30 �Φ = c � t∗ −t∗ dT e−iTHP0ΦP0eiTH , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2) where t∗ is an O(N0) timescale of our choice, and c is an overall normalization constant c−1 = � t∗ −t∗ dT⟨Ψ0|PT |Ψ0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3) As we will see, the projector P0 will be key and will make �Φ act appropriately on the code subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The range (−t⋆, t⋆) determines the set of code subspaces on which �Φ acts in the desired fashion, and ultimately cannot be taken to be bigger than the time range where the exponential decay of the return probability (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='17) is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' To make the operator (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2) have the desired properties on as many states as possible, we can take this range to be the time range where the return probability decays exponentially, though this is not strictly necessary and a t∗ of O(N0) is sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We also provide an alternative presentation of the operators in subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In the following subsections, we will study the action of these operators in the relevant code subspaces, and will be particularly interested in their commutator with the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1 Vanishing commutator with H to all orders in 1/N We now show that the operator (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2) has vanishing commutator with H to all orders in 1/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We start by rewriting the commutator as [H, �Φ] = −i d ds � eisH �Φe−isH���� s=0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4) 30Recall that P0 is the projector on the code subspace of |Ψ0⟩, and thus [Φ, P0] = 0 in that code subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Therefore, we could have defined operators with the same action on the code subspace as (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2), using a single projector on the left (or right) of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Even though the resulting operators would act in the same way on the relevant code subspace, the operators would not be exactly identical: they would have additional non-zero matrix elements associated to subspaces orthogonal to H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 32 – and performing a change of variables, we find [H, �Φ] = −i d ds � c � t∗−s −t∗−s dT e−iTHP0ΦP0eiTH���� s=0 = ic(Pt∗Φt∗Pt∗ − P−t∗Φ−t∗P−t∗) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='5) where we defined Φt∗ = e−iHt∗ΦeiHt∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Using the decay of the return probability through (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='24), we see that the commutator inserted inside a correlator of a small number of single- trace operators and evaluated on the state |ΨT ⟩ will give an exponentially small answer, since each of the two terms in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='5) give exponentially small numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This is valid for any T as long as |T| < t⋆ and |T| − t⋆ ∼ O(N0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Thus, [H, �Φ] = O(e−γN2) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6) where γ is positive and O(N0), proving property 1, defined in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4, for these opera- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Note (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6) is true for our set of code subspaces with T constrained as above, but not for all states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For example, the commutator is not exponentially suppressed in the state |Ψt∗⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2 Similar action as HKLL operators A vanishing commutator with the Hamiltonian is necessary but not sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' There are many CFT operators that commute with the Hamiltonian up to exponentially small corrections in N2, but they will not have the same effect as acting with a local bulk operator, see for example footnote 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Therefore, we also need to show that the operator ˆΦ behaves in the same way as the HKLL operator (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6) to leading order at large N inside correlation functions of single-trace operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For that we consider ⟨Ψ0|O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='�Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='O|Ψ0⟩ = =c � t∗ −t∗ dT ⟨Ψ0|O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='e−iTHP0ΦP0eiTH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='O|Ψ0⟩ =c � t∗ −t∗ dT ⟨Ψ0|O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='P0PT (e−iTHΦeiTH)PT P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='O|Ψ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='7) In the last line, we have inserted two projectors P0, which we are free to do since the correlators is evaluated in the state |Ψ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The integrand above corresponds to TrPT P0, up to some operator insertions that do not affect its general structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='24) we see that the integrand will be exponentially suppressed as |T| increases (and is not O(1/N)) because of the exponentially small overlap of the code subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We can thus evaluate the integral by a saddle-point method controlled by the large N limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The dominant contribution comes from T = 031.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='21) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3) we have ⟨Ψ0|O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='�Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='O|Ψ0⟩ = ⟨Ψ0|O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='O|Ψ0⟩ + O(1/N), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='8) 31One might worry about the possibility of rapidly oscillating phases, such as the one in ⟨Ψ0|ΨT ⟩ displacing the location of the saddle point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Notice however that from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='21),(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='22) it follows that such rapidly oscillating phases cancel between the bra and ket contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 33 – as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The 1/N corrections can be thought of coming from corrections to the leading saddle-point, and would be sensitive to the more detailed form of F(T) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Notice that if we apply the operator �Φ to one of the time-shifted states, then as long as |T| < t∗, we find ⟨ΨT |O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='�Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='O|ΨT ⟩ = ⟨ΨT |O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='(e−iTHΦeiTH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='O|ΨT ⟩ + O(1/N) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='9) Thus in the code subspace HT , ˆΦ acts as e−iTHΦeiTH to leading order at large N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' To make this more manifest, we can also write (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2) as �Φ = c � t∗ −t∗ dT PT (e−iTHΦeiTH)PT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='10) Since we have shown that, to leading order at large N, ˆΦ and Φ have the same matrix elements on the entire code subspace it follows that higher point functions of ˆΦ will also agree at large N with those of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Consider for instance, ˆΦi = c � t∗ −t∗ dT e−iTHP0ΦiP0eiTH (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='11) where Φi ≡ Φ(xi) is an HKLL operator located at a certain spacetime point xi, then in the large N limit ⟨Ψ0|O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='�Φ1�Φ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='�Φn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='O|Ψ0⟩ =cn � t∗ −t∗ dT1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='dTn ⟨Ψ0|O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='PT1(e−iT1HΦ1eiT1H)PT1PT2 (e−iT2HΦ2eiT2H)PT2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='PTn(e−iTnHΦneiTnH)PTn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='O|Ψ0⟩ ≈ ⟨Ψ0|O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='Φ1Φ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='Φn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='O|Ψ0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='12) In addition, this implies that the commutator of ˆΦi’s is the same as that of HKLL operators in the large N limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Two operators, ˆΦ(xi) and ˆΦ(xj), will have zero commutator at spacelike separated points whereas they have O(1) commutator if they are timelike-separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This is true even though these operators do not translate under commutation with the boundary Hamiltonian, up to exponentially small corrections in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Nevertheless, they still have bulk space-time labels and preserve the causal properties of HKLL operators in the large N limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3 Interpretation and comments We have just seen that to leading order in the large N limit, the operator (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2) acts like the HKLL operator (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6) in the appropriate code subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' However, it commutes with H to all orders in 1/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The existence of these operators provides strong evidence that the algebra of single-trace operators in a short time band can have a non-trivial commutant when acting on time-dependent states of high energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The vanishing of the commutator with H should be interpreted as (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2) being gravita- tionally dressed not with respect to the boundary, but instead with respect to features of – 34 – the bulk state, in particular its time-dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This can be seen by the fact that ˆΦ acts differently on different states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' On the time-shifted states |ΨT ⟩ and their code subspaces, it acts as e−iTHΦeiTH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For example, imagine that in the state |Ψ0⟩ we have a supernova explo- sion taking place at t = 0 and we chose the operator (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6) so that it acts right next to the explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In the state |ΨT ⟩ the explosion obviously takes place at t = −T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' From equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='9), we can see that the operator �Φ will act again right next to the supernova explosion, even though the supernova is now at t = −T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Therefore, one and the same operator �Φ knows how to always act at the correct moment (right next to the explosion) for the entire family of states |ΨT ⟩, as long as |T| < t∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The finiteness of t∗ indicates that there is still some residual boundary dressing, which however is not visible in pertubation theory32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The property of being dressed with respect to features of the state is also present in the local observables one defines in general relativity, discussed in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' These state dressed observables are defined at points where a set of D scalars, like the Ricci scalar or RµνρσRµνρσ where Rµνρσ is the Riemann tensor, ’click’ with a certain set of numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The observables are labeled by these values and they are evaluated precisely where the scalars take those values in each state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Locality of these observables requires them to be defined only in some neighbourhood of a classical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In the same spirit, the operators discussed in this section are also local for a certain family of code subspaces, see section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' As mentioned earlier, if the spacetime is so symmetric that the scalars take the same values throughout the spacetime, then these classical observables are not well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Since every point in the spacetime is physically equivalent, it is reasonable that local observables are ill defined for these solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For this reason, the observables are state dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Similarly, it is not possible to apply the same logic discussed in the previous subsections to empty AdS, or other static states, as there are no time-dependent features in the bulk that can be used as a ’clock’ to define a moment in time where the operator acts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Technically, the return probability for such states does not exhibit the rapid decay (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We thus see a nice parallel between the classical and quantum situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The definition of our operator gives a bulk operator which is dressed with respect to features of the state, but in an implicit manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Our construction does not permit us to extract the details of the dressing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Going back to our example of a supernova explosion, one might guess that the dressing is with respect to the supernova and that one could in principle define a gravitational Wilson line between the operator and the supernova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' But what if the state described instead two supernovas exploding at the same or different times?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' To which explosion would our operator be dressed to?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The construction does not give a definite answer, and the way to address this question would be to enlarge the set of code subspaces on which our operator correctly acts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For example, if our operator did not move under the time-translation of one of two supernovae, we would say that it is dressed to the other one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We hope to return to this question in the future, but see subsection 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3 for some related remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 32Similar remarks were made in [18] for the DeWitt observables in AdS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 35 – 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4 A similarity transformation We briefly mention a variant of operators with properties similar to those of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We first define the shifted Hamiltonian33 ˆH = H − ⟨Ψ0|H|Ψ0⟩I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='13) Then we introduce V = c √ 2 � t∗ −t∗ dTe−i ˆHT P0 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='14) with c given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We have V V † = c2 2 � t∗ −t∗ dT � t∗ −t∗ dT ′e−i ˆHT P0ei ˆHT ′ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='15) where we used P 2 0 = P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Following arguments similar to those of the previous subsection, we find that to leading order at large N, and when computing the matrix elements of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='15) within the code subspace, the two integrals in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='15) can be computed by a saddle point method, where the dominant saddle is T = T ′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We then find that in this class of states and at large N V V † ≃ I, and V †V ≃ I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='16) in the sense that, within the code subspace V behaves like a unitary, up to 1/N corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Then we start with a boundary-dressed operator Φ and define ˆΦ = V ΦV † .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='17) Following similar arguments as before we can show that the operator (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='17) satisfies properties 1 and 2 of subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' To check the commutator of ˆΦ with H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We write [H, ˆΦ] = −i d ds � ei ˆHsV ΦV †e−i ˆHs� |s=0 = −i d ds c2 2 ( � t∗−s −t∗−s dTe−i ˆHT )P0ΦP0( � t∗−s −t∗−s dT ′ei ˆHT ′)|s=0 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='18) which again localizes on boundary terms and is thus exponentially suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Second, to show that the leading large N correlators of ˆΦ are the same as those of Φ we follow exactly the same reasoning as in the previous subsection, but now we will have two time-integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Each one of these time integrals will lead to a sharply suppressed Gaussian around T = T ′ = 0 and can be evaluated by saddle-point at large N, reproducing the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 33This shift is useful in order to avoid rapidly oscillating phases in the discussion below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 36 – 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='5 Other asymptotic charges More generally we need to make (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6) commute with all boundary symmetry generators corresponding to asymptotic symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For asymptotically AdSd+1 space-times this is the conformal group SO(2, d) and we consider a generalization of the form �Φ = c � B dµ(g)U(g)P0ΦP0U(g)−1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='19) where now c−1 = � B dµ(g)⟨Ψ0|U(g)P0U(g)−1|Ψ0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='20) Above, dµ(g) is the Haar measure on SO(2, d) and B is a reasonably sized connected sub- manifold of SO(2, d) containing the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The commutator with conformal generators will then be given by operators in the code subspace of states U(g∗)|Ψ0⟩, where g∗ lies on the boundary ∂B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For the construction to work in this generalization we must make sure that the overlaps R(g) = |⟨Ψ0|U(g)|Ψ0⟩|2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='21) decay exponentially in the geodesic distance of g from the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' As discussed in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='7 we expect this to be true for states which break all symmetries at the semiclassical level34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The quantity R(g) is an interesting generalization of the return probability (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='15) that would be interesting to study further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 5 A more general argument for the commutant The operators (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='19) constructed in the previous section commute with the asymptotic charges to all orders in 1/N, however they commute with the other single-trace operators in the time- band generally only to leading order in 1/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' To identify a commutant for the time-band algebra A, the operators (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='19) have to be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In this short section we outline a somewhat different argument suggesting that it is indeed possible to find a commutant to all orders in 1/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We caution the reader that the argument that follows is based on certain assumptions which seem physically plausible, but for which a rigorous proof is still lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' A more careful treatment for the existence of a commutant (as well as a mathematically precise definition of the time-band algebra in the first place) would be desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Let us start with a standard HKLL operator Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We also introduce the notation qi = Qi−⟨Qi⟩ N for where Qi denotes any of the asymptotic SO(2, d) charges and Oj a general single- trace operator in the time-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Our goal is to find an operator ˆΦ which has the following properties: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' [ˆΦ, qi] = 0 and [ˆΦ, Oj] = 0 for all qi ∈ SO(2, d) and Oj ∈ A, to all orders in 1/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 34For compact symmetries, such as rotations, R(g) will have recurrences every 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Hence along the compact directions we take g∗ ∼ O(1) < 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 37 – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' To leading order at large N the correlators of ˆΦ with qj, Oi must be the same as those of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In particular this means that for single-trace operators Oi outside the time-band we generally expect [Oi, ˆΦ] = O(N0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The first condition is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The second condition is necessary in order to ensure that the operator ˆΦ acts in the expected way, at least to leading order at large N, and creates particles that can be detected with an O(1) effect by operators outside the time-band when light rays from the diamond hit the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Here we remark that in order for the two conditions to be mutually consistent, it is important that we impose the second condition only to leading order at large N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The point is that [qi, Φ] = O(1/N) hence when looking at leading order correlators it is indeed consistent to demand simultaneously that i) ˆΦ commutes with qi and that ii) ˆΦ acts like Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' However, when moving on to subleading corrections we have a non-vanishing commutator [qi, Φ] hence we cannot impose both conditions at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We choose to impose that our operators ˆΦ continue to commute with qi to all orders in 1/N, but we allow their correlators to depart from those of Φi at subleading orders in 1/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We now define the desired operators ˆΦ by specifying how they act on the code subspace H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Earlier we defined the code subspace as the space generated by acting on |Ψ0⟩ with single-trace operators, which are not necessarily restricted in the time-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' However, by an analogue of the Reeh-Schlieder theorem35 we expect that for reasonable bulk states |Ψ0⟩ the code subspace H0 can also be generated by acting on |Ψ0⟩ with only elements of the time-band algebra A H0 = span{A|Ψ0⟩} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1) We now define the action of the the operator ˆΦ on the code subspace by the following condi- tions ˆΦA|Ψ0⟩ = AΦ|Ψ0⟩ , ∀A ∈ A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2) This set of linear equations, one for every element of the small algebra A, defines the action of ˆΦ on the code subspace, in a way which satisfies the desired properties as we will see below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Notice that these equations can also be represented as follows: we first select a basis of linearly independent elements Ai of the algebra A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' then we define the matrix of 2-point functions gij = ⟨Ψ0|A† iAj|Ψ0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3) From (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1), it follows that the set of states |i⟩ = Ai|Ψ0⟩ form a (possibly over-complete) basis of the code subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Since ˆΦ is an operator on the code subspace it can be written as ˆΦ = Kij|i⟩⟨j| = KijAi|Ψ0⟩⟨Ψ0|A† j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4) 35This was discussed in [20] for the case of empty AdS and at large N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We believe that a similar result should hold for more general heavy states and even when taking 1/N corrections into account, but it would be interesting to develop a more careful proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 38 – for an appropriate choice of Kij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' To find the matrix K, we start with the desired relation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2) written as ˆΦAl|Ψ0⟩ = AlΦ|Ψ0⟩ , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='5) then we replace ˆΦ with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4) and multiply from the left with ⟨Ψ0|A† k to get gjl gki Kij = ⟨Ψ0|A† kAlΦ|Ψ0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6) If the set of states |i⟩ = Ai|Ψ0⟩ are linearly independent then the matrix gij is positive definite and invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In that case we can solve for K as Kij = gikgjl⟨Ψ0|A† kAlΦ|Ψ0⟩ , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='7) where gijgjk = δi k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' When (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='7) is replaced in expression (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4), we find an explicit solution of the desired equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We emphasize that the necessary ingredient to arrive at (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='7) was the linear independence of the states Ai|Ψ0⟩, which is equivalent to the statement that there is no non-vanishing operator in A which annihilates the state |Ψ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We discuss this condition in the following subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1 On the consistency of the defining equations Before checking that the operators ˆΦ defined by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2), or equivalently via (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4),(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='7), have the desired properties, we need to check that equations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2) are self-consistent linear equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The only possible source of inconsistency is the following: if there was an element A ̸= 0 of the time-band algebra A such that A|Ψ0⟩ = 0, this could potentially be a problem since we would then have A|Ψ0⟩ = 0, while in general AΦ|Ψ0⟩ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Then the equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2) would imply 0 = A|Ψ0⟩ = AΦ|Ψ0⟩ ̸= 0 which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Relatedly, gij defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3) would not be invertible and we would not be able to get to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We will now show that this situation does not arise, that is A|Ψ0⟩ ̸= 0 ∀A ∈ A , A ̸= 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='8) We will prove this by first proving that at large N (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='8) is true and then we will argue that 1/N corrections cannot change the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We have been working under the assumption that the time-band is short enough, which means that in the bulk there will be a region which is space-like relative to the time band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In the large N limit, where gravitational backreaction is turned off, operators inside that region (for example usual HKLL operators) commute with all elements of the algebra A, including the appropriately normalized asymptotic charges qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Hence, in the large N limit the algebra A has a non-trivial commutant A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We want to argue that this commutant continues to exist when 1/N corrections are taken into account, provided that the state |Ψ0⟩ has non-vanishing variance of O(N2) under the asymptotic charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Assuming that at large N the theory in the bulk behaves like usual QFT on a curved background, we expect that an analogue of the Reeh-Schlieder theorem will hold for the – 39 – commutant A′, which means that we can generate the code subspace H0 by acting on |Ψ0⟩ with elements of A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Suppose now that there was an element A of the time-band algebra A which annihilated the state |Ψ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Then for any element a′ ∈ A′ we have Aa′|Ψ0⟩ = a′A|Ψ0⟩ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='9) Since states of the form a′|Ψ0⟩ generate H0 we conclude that the operator A has vanishing matrix elements in H0 at large N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' From this we can not immediately conclude that A = 0 as an operator when 1/N corrections are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For example, for |Ψ0⟩ = |0⟩ the normalized SO(2, d) generators qi = Qi N have vanishing matrix elements at large N, since they annihilate |0⟩ and commute with all other operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' However they are non-vanishing operators at order 1/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' If A is a non-vanishing operator which has vanishing matrix elements at large N on H0 then it means that it acts as a central element at large N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Here we make an additional assumption, that the only central elements are the SO(2, d) generators qi and their functions36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Since, by assumption, the state |Ψ0⟩ has non-trivial variance under these generators, we conclude that it cannot be annihilated by a non-trival A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Let us assume now that we have a state of the form A|Ψ0⟩ which has finite (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' O(N0)) positive norm at large N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Including 1/N corrections will generally modify the norm of this state, but it will do so by corrections suppressed by powers of 1/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Since the previous argument established that the leading large N norm of the state A|Ψ0⟩ is a finite positive number, perturbative 1/N corrections cannot make it vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Hence we expect property (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='8) to be true to all orders in 1/N perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We emphasize that the fact that we cannot annihilate the state by the time-band algebra A relies on the fact that we have restricted our attention to small products of single-trace operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' As discussed in a related context [20,25], if we consider the full algebra of operators in the time-band we can find sufficiently complicated combinations which can annihilate the state37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Finally, as should be clear from the above, if the state |Ψ0⟩ has very small or vanishing variance in the asymptotic charges then (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='8) fails and it is not possible to define operators obeying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2 Proof that ˆΦ has the desired properties Having established that equations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2) are consistent, we argue that the operator ˆΦ has the desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' First it is obvious by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2) that the operator ˆΦ has vanishing commutators with elements of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' To see that consider A1 ∈ A and a general state in the code subspace which can be written as A2|Ψ0⟩, with A2 ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Then we have [ˆΦ, A1]A2|Ψ0⟩ = ˆΦ(A1A2)|Ψ0⟩ − A1(ˆΦA2|Ψ0⟩) = A1A2Φ|Ψ0⟩ − A1A2Φ|Ψ0⟩ = 0 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='10) 36We believe this assumption to be quite weak, but it would be interesting to prove it more thoroughly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 37For example, consider a state |Ψ⟩ with ⟨Ψ0|Ψ⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Then the (complicated) operator |Ψ⟩⟨Ψ| annihilates |Ψ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 40 – where in the second equality we used (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Since this is true for all A2, we find [ˆΦ, A1] = 0 ∀A1 ∈ A , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='11) where it should be understood that this equation holds on the relevant code subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Second, we will show that to leading order at large N, the operator ˆΦ acts like the HKLL operator Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' To see this, consider an arbitrary matrix element on the code subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Two general states of the code subspace can be written as A1|Ψ0⟩, A2|Ψ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Then we have ⟨Ψ0|A† 1 ˆΦA2|Ψ0⟩ = ⟨Ψ0|A† 1A2Φ|Ψ0⟩ = ⟨Ψ0|A† 1ΦA2|Ψ0⟩ + ⟨Ψ0|A† 1[Φ, A2]|Ψ0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='12) In the first equality we used (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Now, the operator A2 is some combination of single-trace operators in the time band, as well as the normalized SO(2, d) generators qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' All of these operators have commutators with Φ which are suppressed by powers of 1/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Hence the last term in the equation above is suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' All in all, we find ⟨Ψ0|A† 1 ˆΦA2|Ψ0⟩ = ⟨Ψ0|A† 1ΦA2|Ψ0⟩ + O(1/N) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='13) which establishes the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This ensures that large N correlators of ˆΦ are the same as Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We emphasize that the operators defined in this section are not exactly the same as the operators (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2) discussed earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For example, unlike (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2) the operators (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2) were defined to act only on the code subspace H0 of |Ψ0⟩ and not on the code subspace HT for T = O(N0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Also, the commutator of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2) with the Hamiltonian is of order e−N2 while it is exactly zero, within the code subspace, for the operators (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 6 Examples In this section we consider various examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Our primary focus will be on examining the validity of equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='18), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='21),(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='22), on which the construction of our operators relies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1 Coherent states In general, we are interested in time-dependent semi-classical geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Many of these states can be thought of as bulk coherent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We will discuss the overlap of these states closely following [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In the CFT, these states are prepared by a Euclidean path integral |Ψ⟩ = Te− � tE<0 dtEdd−1x φb(tE,x)O(tE,x) |0⟩ , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1) where O is a single-trace operator dual to a supergravity field, and the source is scaled appropriately so that it leads to states with non-trivial gravitational backreaction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' the expectation value of the energy and variance of this state will scale like (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In the large N limit the overlap of two such states can be computed by a Euclidean gravitational path integration which in the semi-classical limit can be approximated by a saddle point computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For example, the norm of the state is ⟨Ψ|Ψ⟩ ≈ e−Igrav(λb) , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2) – 41 – where λb is the following boundary condition for the bulk field λb = � φb(tE, x), tE < 0 φ⋆ b(−tE, x), tE > 0 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3) and Igrav(λb) is the on-shell gravitational action in the presence of the sources specified above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Generalizing to two states |Ψ1⟩ and |Ψ2⟩, the normalized inner product between them is R = |⟨Ψ1|Ψ2⟩|2 ⟨Ψ1|Ψ1⟩⟨Ψ2|Ψ2⟩ , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4) which at large N can be computed by a supergravity saddle-point computation R ≈ exp � −2 Re(Igrav(˜λ)) + Igrav(λ1) + Igrav(λ2) � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='5) where the supergravity solutions have the boundary sources ˜λ, λ1 and λ2 which take the following form ˜λ = � φ2(tE, x), tE < 0 φ⋆ 1(−tE, x), tE > 0, λi = � φi(tE, x), tE < 0 φ⋆ i (−tE, x), tE > 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6) where i = 1, 238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Notice that in each of the terms of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='5), the gravitational on-shell action is proportional to 1 GN ∼ N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Since quantum mechanically we need R ≤ 1, we find that the following inequality has to be satisfied 2 Re(Igrav(˜λ)) ≥ Igrav(λ1) + Igrav(λ2) , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='7) for the on-shell value of solutions of the Einstein plus matter equations, for any choice of sources of the form (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' If the two sources are different, we expect a strict inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It would be interesting to explore this inequality directly from the gravitational point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We discuss this further in the discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We now move on to the computation of the return probability for states of the form (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1) after a small (not N-dependent) time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' That is, we take the time-evolved state, |Ψ(T)⟩ = e−iHT |Ψ⟩, and consider the following quantity R(T) = |⟨Ψ(0)|Ψ(T)⟩|2 ⟨Ψ(0)|Ψ(0)⟩⟨Ψ(T)|Ψ(T)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='8) To apply the general formalism described above, we need to analyze how the Euclidean sources φ0 preparing the state |Ψ(0)⟩ need to be modified to φT , in order to prepare |Ψ(T)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' From 38The sources φ2(tE, x) and φ⋆ 1(−tE, x) should decay sufficiently fast at the t = 0 surface such that the states are normalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This also implies that the bra and ket preprations of different states can be smoothly glued to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 42 – a technical point of view computing φT in terms of φ0 is not straightforward, as it requires a solution of the Einstein equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Nevertheless, we can in principle compute the return probability using (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='5) with a modified source ˜λ = � φT (tE, x), tE < 0 φ⋆ 0(−tE, x), tE > 0, λT = � φT (tE, x), tE < 0 φ⋆ T (−tE, x), tE > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='9) Thus we get R(T) = exp � −2 Re(Igrav(˜λ)) + Igrav(λ0) + Igrav(λt) � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='10) and this is exponentially suppressed in the semi-classical limit because of the 1/GN ∼ N2 coefficient in the gravitational action and the condition (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2 Thermofield double state We now consider the thermofield double state |TFD⟩ = 1 � Z(β) � n e− βEn 2 |En⟩L ⊗ |En⟩R , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='11) where the |En⟩’s are the energy eigenstates and Z(β) is the partition function at inverse temperature β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In the strong coupling limit, for temperatures below the Hawking-Page tem- perature, the state is dual to two entangled thermal AdS geometries, while for temperatures higher than the Hawking-Page temperature, it is expected to be dual to the eternal black hole in AdS [101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This geometry has two asymptotically AdS boundaries, on the ”left” and the ”right”, hence the asymptotic symmetry group is SO(2, d)L × SO(2, d)R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The state (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='11) is invariant under certain combinations of the asymptotic charges, for example we have (HR − HL) |TFD⟩ = 0 but (HR + HR) |TFD⟩ ̸= 0 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='12) and similarly for the other charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In this case we can generalize the return probability to include all possible large diffeomorphisms on the two sides R(g1, g2) = |⟨TFD| UL(gL) UR(gR) |TFD⟩|2 , gL/R ∈ SO(2, d)L/R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='13) In this case we expect R(gL, gR) to rapidly decay along certain directions but remain constant along others due to the symmetries of the state (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In what follows we focus on a particular class of deformations, corresponding to evolving with HL + HR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This gives what is usually called the spectral form factor (SFF) defined as R(t) = |⟨TFD|e−i T 2 (HL+HR)|TFD⟩|2 = ���� Z(β + iT) Z(β) ���� 2 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='14) which was introduced in the context of the eternal AdS black hole in [95] and studied in detail in [97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 43 – We are interested in studying (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='14) above the Hawking-Page temperature for small times, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='e, T ∼ O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' One way to proceed is by computing Z(β) and then analytically continuing β → β + iT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' If we are above the Hawking-Page temperature Z(β) can be estimated by the Euclidean AdS-Schwarzschild black hole saddle point Z(β) ≈ e−IBH(β) , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='15) where IBH(β) is the on-shell action on the Euclidean black hole background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For example, we find IBH(β) = − π2 2GNβ (for AdS3) IBH(β) = β GN g(rH) (for AdS5) , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='16) where we have set the AdS radius ℓAdS = 1 and rH is the horizon radius, while g(rH) = V3 8π(−r4 H + r2 H) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='17) where V3 is the dimensionless volume associated with the metric on a unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For the AdS5 case, rH ≈ π/β for small real β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' A detailed discussion of the action can be found in [102].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The central charge of the CFT2 is c = 3/2GN and the rank for the gauge group of the dual four dimensional SU(N) N = 4 super Yang Mills theory is given by N2 = π/2GN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For small T the complexified partition function Z(β + iT) will be given in terms of the analytic continuation of the above actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Thus for T ≪ β, one gets the following for AdS3, R(T) ≈ e − 2π2 β3 c T 2 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='18) which is exponentially small in the large central charge limit39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Similarly for AdS5, we find that Z(β) ∼ e πN2 β3 in the high temperature limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Again for T ≪ β, we have R(T) ≈ e − 12π β5 N2T 2 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='19) As T becomes larger and approaches T ∼ β, the dominant saddle point will no longer be the black hole, as the analytically continued action can start to compete with thermal AdS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In addition the analytically continued black hole saddle point corresponds to a geometry with a complex metric, and as T ∼ O(β) this metric becomes ’unallowable’ according to the criteria of [103], see also [104].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Interestingly, thermal AdS becomes the dominant saddle point before the metric becomes not allowable [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' An exponential decay of R(T) in N is to be expected even when T ∼ β, since in this case the thermal AdS saddle dominates and, |Z(β + iT)|2 ∼ e˜g(T)/β3 where ˜g is O(N0) periodic function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Thus, the numerator of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='14) |Z(β + iT)|2 is N0 while the denominator is O(eN2) leading to an exponentially suppressed R(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 39There will be additional terms suppressed in T 2/β2 which will not affect the exponential decay in the large c limit as long as t is smaller than β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 44 – 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3 Weakly coupled, large N gauge theories It is interesting to consider the behavior of the SFF at small, or even vanishing ’t Hooft coupling λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In this case the bulk dual is stringy and moreover at λ = 0, the spectrum of the dual CFT is (half)-integer-spaced and thus not chaotic at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Nevertheless the decay (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='18) is still valid for a certain time-scale, even in the free theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This was discussed in detail in [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For concreteness, we consider the partition function of free N = 4 SYM on S3 × R, where the sphere has unit radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It has the form [105,106] Z(β) = � DU e � R �∞ m 1 m zR m(β)χR(Um) , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='20) where DU is the invariant Haar measure on the gauge group normalized to one, χR is character in the representation R and zR m(β) = � Ri,B=R e−mβEi + (−1)m+1 � Ri,F =R e−mβEi , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='21) where the first sum is over bosonic states and the sum in the second term is over fermionic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The behavior of the SFF ��� Z(β+iT) Z(β) ��� 2 , as well as of the microcanonical analogue YE,∆E(T), based on the analytic continuation of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='20) was discussed in [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Even at λ = 0 the SFF obeys (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='18), though in this case the Poincare recurrence time is very short, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 4π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='40 While in this limit the bulk theory does not admit a semiclassical gravitational description, we could still apply the procedure (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2) to identify operators with vanishing commutators with the Hamiltonian to all orders in 1/N, though now they do not have a nice bulk interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='41 In doing so, we would need to be careful to take t∗ to be a short O(1) time-scale which is less than 4π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Here we notice that similar results have been derived for the analytically continued super- conformal index [107], which can be thought of as the SFF for 2-sided eternal supersymmetric AdS black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4 Perturbative states around empty AdS We now briefly discuss the return probability for perturbative states around empty AdS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We want to consider states which have a large number of particles, but still small enough so that we can ignore gravitational backreation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We can get some useful estimates by considering a thermal gas of particles in AdSd+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' These are dual to a gas generated by single-trace operators in the CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Suppose we have low-lying single-trace operators with conformal dimension ∆i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For simplicity we consider only scalars and we take the radius of AdSd+1 to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Then 40In our conventions conformal dimensions in the free theory are half-integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 41To start with, the HKLL procedure cannot be implemented at subleading orders in 1/N due to the many stringy fields present in the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Therefore, the issue of non-commutativity with the Hamiltonian does not stand out like it does in the case of Einstein gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 45 – the partition function of single-particle states z(β) and the multi-trace Fock-space partition function are respectively z(β) = � i e−β∆i (1 − e−β)d , Z(β) = exp � ∞ � n=1 1 nz(nβ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='22) It is now straightforward to do the analytic continuation Z(β + iT) = exp � ∞ � n=1 � i e−(nβ+inT)∆i (1 − e−nβ+inT )d � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='23) For scalar BPS operators dual to SUGRA modes, ∆i is integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Then it is obvious that the SFF R(T) = ��� Z(β+iT) Z(β) ��� 2 has periodicity T = T + 2π, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' What we want to estimate is the decay rate of the SFF at early times, and how close to 0 the SFF drops between the recurrences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' First we notice that the partition function factorizes to a product over ∆i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Hence we can study the behavior of a given ∆i and we drop the sum over i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' If we first take the small β limit, before analytically continuing, we find Z(β) ∼ exp � ζ(d + 1) 1 βd � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='24) Using this approximation we find that for early times R(T) ∼ e − d(d+1)ζ(d+1) βd+2 T 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='25) As expected the decay is controlled by the variance of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Of course if we use the high temperature approximation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='24) to perform the analytic continuation, then we do not see the recurrences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' At high temperature the SFF starts decaying quite rapidly, stays close to zero for a while and then goes back to 1 every T = 2π × integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' To find an estimate of how closely it approaches zero it is convenient to evaluate it at T = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Suppose that the conformal dimension is an even integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Then we find R(π) = exp � 2 �∞ n=1 1 n e−nβ∆ (1−(−1)ne−nβ)d � exp � 2 �∞ n=1 1 n e−nβ∆ (1−e−nβ)d � ∼ e −(2−2−d)ζ(d+1) 1 βd − 1 2d log β∆ 2 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='26) So we see significant suppression at small β, though of course, the suppression does not scale like e−N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We expect a similar qualitative behavior for R(T) for generic pure states of similar energy as the states studied above (namely high energy states whose energy scales as O(N0)): they will have recurrences every 2π, but the return probability will quickly decay to small values for 0 < t < 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' If we use (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2) for such states, with t∗ ∼ O(1) < π, then the commutator with H will be suppressed by a factor of the order of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='26) rather than e−N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Note that this is not good enough, since the commutator we are trying to cancel is O(1/N), which in the large N limit is much smaller than the suppression controlled by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 46 – 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='5 LLM geometries An interesting class of semiclassical states with AdS5 × S5 asymptotics in type IIB super- gravity are the LLM geometries [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' These are dual to 1 2-BPS states in N = 4 SYM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' While these geometries do not break all of the asymptotic symmetries, they do provide a useful toy model where we can study in detail the behavior of the return probability as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The 1 2-BPS states in N = 4 SYM on S3 × R are states that preserve 16 of the 32 super- symmetries of the theory in addition to the bosonic symmetries SO(4) × SO(4) × R where R corresponds to the Hamiltonian H − ˆJ where H is the Hamiltonian and ˆJ an R-symmetry generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' These states correspond to operators that lie in the (0, J, 0) representation of the SU(4) ∼ SO(6) R-symmetry and they saturate a unitarity bound for their conformal dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It is illuminating to consider the N = 1 vector and three chiral multiplet decom- position of the N = 4 theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In this case the scalars of the chiral multiplets are organized into Zj = φj + iφj+3, where j = 1, 2, 3, which are in the adjoint representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We will focus on j = 1 from now on without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Then the states we are interested in correspond, via the state-operator map, to single-trace operators of the form Tr(Zni), as well as multi-trace operators of the form Πi(Tr(Zni))ri [81,108,109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Since these operators saturate the unitarity bound ∆ = J, they correspond to the lowest Kaluza-Klein mode of Z on S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This mode has a harmonic oscillator potential due to its conformal coupling to the curvature of S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Thus we are interested in gauge invariant states of the matrix Z in a harmonic potential [109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The ground state, corresponding to empty AdS, is given by a Gaussian wave function Ψvac = Ce− 1 2 N2tr(Z2) , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='27) where C = (π/N)−N2/4 and we introduce the notation trZ = 1 N TrZ = 1 N �N i νii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Fluctua- tions with operators with ∆ = J ≪ N will be small excitations around the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' As discussed in [110] excitations with ∆ = J ∼ N2 will be other coherent states which are given by Ψ = C[J(Z)]1/2e−N2tr( 1 2 φ(Z)2−iψ(Z)) , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='28) parameterized by two functions φ(Z) and ψ(Z) which are monotonically increasing and ar- bitrary functions of Z, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' J[Z] is the Jacobian given by det[∂φ(Z)ij/∂Zkl].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It is well known that one can describe such a system by N fermions in a harmonic potential [111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In the large N limit, states in such a system can be thought of as droplets in a two dimensional phase space, where for example a circular droplet corresponds to the ground state of the system [111–113].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The precise connection between the functions φ(Z) and ψ(Z) and the droplet picture on the phase space will be discussed in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In the bulk, the LLM solutions correspond to 10 dimensional geometries of asymptotically AdS5 × S5 spacetimes, see appendix D, that are completely determined by a function z on a two dimensional surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In particular, specifying whether z takes value 1/2 or −1/2 at each point on this plane completely specifies the full bulk solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This is in parallel with the – 47 – two dimensional fermionic phase space mentioned earlier where the fermion takes occupation number 1 (black) or 0 (white) at each point in the phase space, giving droplet of a given shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For instance, in the fermionic picture the ground state is a circular droplet of a certain radius, say r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It corresponds in the bulk is to the empty AdS5 × S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Fluctuations with operators of ∆ = J ≪ N correspond to having ripples in the edge of the circular droplet and corresponds to having gravitons propagating in the AdS5 × S5 background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' While operators of energy ∆ = J ∼ N correspond to giant gravitons in the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Operators with ∆ = J ∼ N2 correspond to other bulk geometries and different shapes of droplets in the fermionic phase space [81, 114, 115].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The geometries will not be time translation invariant (rotational invariant in the fermionic picture) in general42, but they are invariant under t → t + 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The goal here is to consider a certain geometry that breaks time translation invariance and compute its return probability for short time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In the fermionic picture this corresponds to a droplet that breaks the rotational invariance, an ellipse for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In the matrix quantum mechanics picture it is easy to compute the return probability, evolving (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='28) with the quadratic Hamiltonian and computing the square of the inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' But first, we need to review the dictionary between the two pictures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1 Computation of the return probability The way the matrix quantum mechanics picture and the fermionic picture are related will be obvious once we diagonalize the matrix Z and express it in terms of the eigenvalues (µi), where the Jacobian becomes J(Z) = N � i φ ′(µi) � i̸=j φ(µi) − φ(µj) µi − µj , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='29) which is 1 for the vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In the large N limit, the Gaussian measure dZ exp � −N2tr(Z2) � will reduce to the well known Wigner semi-circle distribution for the density of eigenvalues [116], dϱ(µ) = 1 π(2 − µ2)1/2Θ(2 − µ2)dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='30) Let us now introduce new variables to parameterize the coherent states in the large N limit, w(µ) := dϱ(φ(µ))/dµ which is the density of eigenvalues and v(µ) := ψ(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' These parameters are canonical conjugates of one another43, that is their Poisson bracket is the Dirac delta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In the large N limit, the appropriately renormalized Hamiltonian (hcl) can also be written in terms of w and v ′ = dv/dµ and thus an action can be written for these variables [110,117–119].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In particular, hcl = 1 2 � dµ w(µ)(v ′(µ)2 + π3 3 w(µ)2 + µ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='31) 42There are also static configurations, concentric circles for example [81] 43Note that the two variables are not totally independent and w(µ) has to satisfy a constraint, in particular � dµ w(µ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 48 – Coming back to the two dimensional phase space picture, we consider a blob centered at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We assume the horizontal direction (x-axis) represents the q variable of the phase space, which we take to be the eigenvalues (µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Consider a vertical line crossing the blob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Assuming that the blob has a simple geometry without folds, this vertical line intersects the boundary of the blob twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We parametrize these points by p±(µ) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Then, the density of eigenvalues for any µ is proportional to (p+−p−)(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Computing the kinetic energy of fermions for a given dµ by integrating p2/2 from p = p− to p = p+ and matching this to the kinetic part of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='31), we get p± = ±πw + v ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='32) This has also been mentioned in the context of c = 1 string theory in [120–123].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Note that for the vacuum (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' the empty AdS5 ×S5 geometry), p± = ±(2−µ2)1/2Θ(2−µ2) and v′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Since we are looking for a time dependent geometry, we need a blob in fermion phase space that breaks the rotational symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The simplest non trivial modification of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='32) is to take v to be quadratic44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In this case we have p+(µ) = (2 − µ2)1/2Θ(2 − µ2) + 2µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='33) This can be seen to be half of a tilted ellipse, which combined with an appropriate p− gives the full elliptic blob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This will evolve non trivially under rotation and the corresponding geometry will be a time dependent one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This geometry, together with the five form, can be found using the mapping discussed earlier, by first solving for z(x1, x2, y) then inserting it into (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2), (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3), (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4) and (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Now we proceed with the computation of the return probability for this state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We go back to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='28) and consider a state Ψ(0) with φ = Z and ψ = v = Z2 and after evolving it, compute the overlap ⟨Ψ(0)|Ψ(T)⟩ = � dZ Ψ(Z, 0)∗Ψ(Z, T) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='34) The state we are interested has the form Ψ(Z, 0) = � π N �−N2/4 e− 1 2 N2(1−2i)tr(Z2) = � i,j ϕ(νij) , where ϕ(ν) = � π N �−1/4 e− N 2 (1−2i)ν2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='35) Since we are dealing with matrix quantum mechanics with a quadratic potential, each matrix element evolves independently and governed by the usual harmonic oscillator propagator ϕ(ν, T) = � dν ′K(ν ′, ν, T)ϕ(ν ′) , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='36) where K(ν ′, ν, T) = � N 2πi sinT exp � iN 2sinT ((ν2 + (ν ′)2)cosT − 2νν ′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='37) 44Translated circle blobs will not correspond to physical geometries when the gauge group is SU(N), since the centre of the blob is fixed by imposing the condition Tr(Z)=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 49 – for t < π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We can then compute the overlap ⟨Ψ(0)|Ψ(T)⟩ = [z(T)]N2, z(T) = � dν ϕ⋆(ν, 0)ϕ(ν, T) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='38) Following (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='36) we find ϕ(ν, T) = � N πX �1/4 e−NYν2 and, Ψ(Z, T) = � N πX �−N2/4 e−N2Ytr(Z2) , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='39) where X and Y are periodic functions of time given by, X(T) = (cosT + (2 + i)sinT)2 Y(T) = 1 2 �(1 − 2i) cosT + i sinT (i + 2) sinT + cosT � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='40) Thus, z(T) = � dν ϕ⋆(ν, 0)ϕ(ν, T) = A1/2 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='41) where A = 1 3i sinT + cosT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='42) It can be checked that z(T) is 1 when T = 0 and |z(T)|2 = |A| = � 1 9 sin2T + cos2T �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='43) Since |z(T)|2 ≤ 1, R(T) is an exponentially decaying function in the large N limit, for small times, but a periodic function in T = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' That is R(T) = |⟨Ψ(0)|Ψ(T)⟩|2 = e−N2F(T) , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='44) where the function F(T) = −2log|z(T)| is zero at T = 0, and increases to the local maximum F(T = π/2) = log 9 and goes back to zero at T = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Thus in the time scales we are interested in, in particular T < π/2, the square of the inner product (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='34) which is the return probability of a given LLM semi classical geometry in the large N limit, is exponentially suppressed in N2 as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Note that the return probability is periodic in π, which is due to the symmetry of the particular state considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In general, the period will be 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We can also compute the overlap of states in different code subspaces built upon Ψ(Z, 0) and Ψ(Z, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The simplest is the inner product of the states Ψ(Z, 0) and Tr(Z2n)Ψ(Z, T) which can be written as ⟨Ψ(0)| tr(Z2n) |Ψ(T)⟩ ≡ � dZ Ψ⋆(Z, 0)tr(Z2n)Ψ(Z, T) = � π NX 1/2 �−N2/2 � dZ tr(Z2n) e− S 2 N2 tr(Z2) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='45) – 50 – where S = (1 + 2i) + 2Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Following (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='23), we can rewrite the above integral as ⟨Ψ(0)| tr(Z2n) |Ψ(T)⟩ = ⟨Ψ(0)|Ψ(T)⟩ � dZ tr(Z2n) e− S 2 N2 tr(Z2) � dZ e− S 2 N2 tr(Z2) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='46) The second factor corresponds to an expectation value in a Gaussian matrix model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Keeping only planar diagrams at large N we find ⟨Ψ(0)| tr(Z2n) |Ψ(T)⟩ ≃ ⟨Ψ(0)|Ψ(T)⟩ Cn Sn (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='47) where Cn = 1 n+1 �2n n � are the Catalan numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Similarly, for multi-trace operators the overlap can be computed and using large N fac- torization we get ⟨Ψ(0)| k � i tr(Z2ni) |Ψ(T)⟩ ≃ ⟨Ψ(0)|Ψ(T)⟩ �k i Cni Sn , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='48) where n = n1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' + nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Thus, as long as n does not scale with N the correlator will still be exponentially sup- pressed, otherwise the periodic coefficient can spoil the exponentially decaying behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This is to be expected since in such cases the dimension of the multi-trace operators will be order N and they will not be just small fluctuations of the background and can, in principle, evolve the state back in time to T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In any case, our code subspace is constructed by the action of multitrace operators whose dimension is finite in the large N limit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='e, n is an O(1) number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6 Kourkoulou-Maldacena states in SYK model The SYK model is a quantum mechanical model of N Majorana fermions interacting with random interactions which is given by the Hamiltonian H = � iklm jiklm ψiψkψlψm , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='49) where ψi are the Majorana fermions {ψi, ψj} = δij, and the coupling jiklm has drawn from the distribution P(jiklm) ∼ exp � −N3j2 iklm/12J2� , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='50) leading to disorder average of jiklm = 0, j2 iklm = 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='J2 N3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='51) In a particular realization of the couplings, we consider pure states which are obtained by using the Jordan-Wigner transformation and combining pairs of Majorana fermions into qubit like operators and choosing states with definite eigenvalues for the σ3 components of – 51 – all qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' These states are denoted by |Bs⟩, where s = (s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=', sN/2) with sk = ±1, and they satisfy the relations below Sk |Bs⟩ = sk |Bs⟩ , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='52) where Sk = σk 3/2 ≡ 2i ψ2k−1ψ2k is the spin operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' By choosing all possible combinations of the {sk}’s we get a basis of the Hilbert space whose dimension is 2N/2 (N is an even integer number).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We further evolve these states over some distance l in Euclidean time in order to get low energy states |Bs,l⟩ = e−lH |Bs⟩ which we will refer to as Kourkoulou-Maldacena (KM) states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' To stay in the low-energy regime where the SYK model exhibits conformal invariance we take 1 ≪ lJ ≪ N [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' As discussed in [76] the KM states can be thought of as a toy model of pure black hole microstates which are out of equilibrium and which contain excitations behind the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Hence they are states which exhibit time-dependence and our general formalism should be applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We start by discussing the behavior of the return probability for these states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1 Analytical computation of the return probability at large N We start with the normalization of the KM states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In the large N limit, due to the approxi- mate O(N) symmetry of the theory it can be shown [76] that ⟨Bs,l |Bs,l⟩ = ⟨Bs| e−2lH |Bs⟩ = 2−N/2Z(β) , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='53) where β = 2l [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The return probability then in the large N limit is given by R(T) = ���⟨Bs,l|e−iHT |Bs,l⟩ ⟨Bs,l |Bs,l⟩ ��� 2 = ���Z(β + iT) Z(β) ��� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='54) In a low temperature expansion, the partition function can be estimated [124] using the Schwarzian approximation to be Z(β) ∝ e2 √ 2π2αS N βJ (βJ)3/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='55) Using (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='55) we find for the return probability R(T) = 1 (1 + T 2 β2 )3/2 e −(4 √ 2π2αS N Jβ3 )T 2 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='56) which is compatible with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='18), after we take into account the different N-dependence in the SYK model vs N = 4 SYM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We can now try to test the more general decay of the inner product between states in time-shifted code subspaces (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Let us denote the unit-normalized KM states as | �Bs,l⟩ = |Bs,l⟩ � ⟨Bs,l|Bs,l⟩ , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='57) – 52 – and denote their time-dependence as | ˆBs,l(T)⟩ = e−iHT | �Bs,l⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We consider an operator A(t) which is a simple combination of the fermions, so that the state A(t)| ˆBs,l⟩ is in the code subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Then we write ⟨ �Bs,l(0)|A(t)| �Bs,l(T)⟩ = ⟨ �Bs,l(0)| �Bs,l(T)⟩ × ⟨Bs,l(0)|A(t)|Bs,l(T)⟩ ⟨Bs,l(0)|Bs,l(T)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='58) Let us focus on the last ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We can rewrite it as ⟨Bs,l(0)|A(t)|Bs,l(T)⟩ ⟨Bs,l(0)|Bs,l(T)⟩ = ⟨Bs|e−(l+i T 2 )HA(t − T 2 )e−(l+i T 2 )H|Bs⟩ ⟨Bs|e−(l+i T 2 )He−(l+i T 2 )H|Bs⟩ , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='59) which depends holomorphically on l + i T 2 , so we can evaluate if by analytic continuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' All in all we find ⟨ �Bs,l(0)|A(t)| �Bs,l(T)⟩ = ⟨ �Bs,l(0)| �Bs,l(T)⟩ × � ⟨ ˆBs,l(0)|A(t − T 2 )| ˆBs,l(0)⟩ � l→l+i T 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='60) At large N and for flip-invariant operators [76] we can also write this as ⟨ �Bs,l(0)|A(t)| �Bs,l(T)⟩ = ⟨ �Bs,l(0)| �Bs,l(T)⟩ × ⟨A(t − T 2 )⟩β|β→β+iT , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='61) where in the last term we first compute the thermal 1-point function ⟨A(t− T 2 )⟩β as a function of β and then analytically continue β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' As an example, we consider the case where A = ψk(t)ψk(t′) (no summation over k implied).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Following [76] we have for real time and large N ⟨ �Bs,l(0)|ψk(t)ψk(t′) | �Bs,l(0)⟩ = Gβ(t − t′) , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='62) where, for t > t′, we have Gβ(t − t′) = π1/4 √2βJ e−iπ/4 � sinh[π(t − iϵ)/β] , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='63) Therefore, using (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='60) we get ⟨ �Bs,l(0)|ψk(t)ψk(t′)| �Bs,l(T)⟩ = ⟨ �Bs,l(0)| �Bs,l(T)⟩ Gβ+iT (t − t′) , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='64) where the last term can be computed as the analytic continuation of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='63).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Similarly for A = ψ2k−1(t)ψ2k(t′)Sk we have [76] ⟨ �Bs,l(0)|ψ2k−1(t)ψ2k(t′)Sk| �Bs,l(0)⟩ = −2iskGβ(t)Gβ(t′) + O(1/N), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='65) hence ⟨ �Bs,l(0)|ψ2k−1(t)ψ2k(t′)Sk| �Bs,l(T)⟩ = ⟨ �Bs,l(0)| �Bs,l(T)⟩× (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='66) × � −2iskGβ+iT (t − T 2 )Gβ+iT (t′ − T 2 ) + O(1/N) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='67) The examples (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='64) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='66) are consistent with our general expectations, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='21) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 53 – β 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='20 (a) N = 14 β 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='25 (b) N = 20 β 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='35 (c) N = 24 Figure 2: The blue lines are the numerical results for the variance of Hamiltonian as a function of β while the yellow ones are the Schwarzian approximation ∆H2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='396N/β3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2 Some numerical checks In this subsection we perform some simple numerical checks of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='21) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='24), as well as the behavior of the operators (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2) for KM states in the SYK model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The first step is to select an appropriate value for the inverse temperature β = 2l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The early time decay of the return probability is R(T) = e−∆H2T 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='68) Earlier we used the Schwarzian approximation to compute the partition function (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='55) from which we can also get the variance ∆H2 = 4 √ 2π2αS N β3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='396 N β3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='69) We compare this result with a numerical computation of the variance ∆H2 for a KM state constructed from |Bs⟩ = |+ − −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='−⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In Figure 3, we show the value of the plateau for the KM state, as defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='19) for various values of N and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For the range of values of N we are interested in, we can take the inverse temperature to be β = 5, which is the value we will use in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' l R 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6 N=14 N=16 N=18 N=20 N=20 N=24 Figure 3: The plateau height ¯R as a function of l = β/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In Figure 4 we can see the return probability as a function of t for different values of N for the corresponding KM state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' As discussed in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6, we expect that the overlap – 54 – JT R(T) 20 40 60 80 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='0 N=14 N=16 N=18 N=20 N=20 N=24 Figure 4: Return probability as a function of T for different values of N between any state in the code subspace at t = 0 will and the one at t = T will also decay exponentially fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We can encode the overlap between all such pairs of states by Rcode(T) = 1 dcode Tr[PT P0] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='70) For the numerical computation we need to make some choice about the code subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' One condition is that the dimension dcode of the code subspace should satisfy dcode ≪ 2N/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' As an example, and for the purpose of the numerical computation, we can define the code subspace as Hcode = span{Oi1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='Oik k |Bs⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' ij = 0, 1} , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='71) for some choice of the operators Oi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Here Dcode = 2k the value of k should be such that D ≪ 2N/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Note that the states in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='71) are generally not orthonormal but it is easy to write a projector on the code subspace in terms of elements of this basis, see [125] for a related discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 5, we see plots of the behavior of Rcode(T) as a function of time for some specific choices of such a code subspace: a : the dimension of the code subspace is D = 8 and the operators are chosen to be O1 = ψ1(t = 0), O2 = ψ1(t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1), O3 = ψ1(t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' b : the dimension of the code subspace is D = 8 and the operators are chosen to b O1 = ψ1(t = 0), O2 = ψ1(t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1), O3 = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' c : the dimension of the code subspace is D = 16 and the operators are chosen to be O1 = ψ1(t = 0), O2 = ψ1(t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1), O3 = ψ1(t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='5), O4 = ψ1(t = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' where in case (b) the operator h is the normalized Hamiltonian h = 1 √ N (H − ⟨H⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='72) – 55 – 0 20 40 60 80 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='0 R (T) code JT 0 20 40 60 80 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='0 N=14 N=16 N=18 N=20 (a) 0 20 40 60 80 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='0 N=14 N=16 N=18 N=20 0 20 40 60 80 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='8 R (T) code JT (b) R (T) code JT 0 20 40 60 80 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='0 N=14 N=16 N=18 N=20 (c) Figure 5: Rcode(T) as a function of T for three different examples of codesubspaces in the form of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='71).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 0 20 40 60 80 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='0 N=14 N=16 N=18 N=20 R (T) code JT (a) ■ ■ ■ ■ ■ 2 4 6 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='5 ■ boundary-dressed state-dressed Jt (b) Figure 6: Results for the code subspace (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='73).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (a) Rcode(T) as a function of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (b) The blue line is ⟨ψ3(0)ψ3(t)⟩ as a function of t, while in the case of the yellow line, ψ3(0) is replaced by the dressed operator obtained from our proposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Here N=20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We finally check that the operator (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2) has similar correlators as the boundary-dressed operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We take the code subspace as Hcode = span{|Bs⟩, O1|Bs⟩, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='Ok|Bs⟩, h|Bs⟩, hO1|Bs⟩, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='hOk|Bs⟩}, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='73) where the dimension of the code subspace is dcode = 2(k + 1) ≪ 2N/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 6, we plot the result for the case of k = 5 and where the operators chosen to be O1 = ψ1(t = 0), O2 = ψ1(t = 2), O3 = ψ1(t = 4) O4 = ψ1(t = 6), O5 = ψ1(t = 8) for N = 20 (dcode = 12 ≪ 210) are plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' One can see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6b that the state-dressed operator for ψ3 has approximately the same correlation function as the original one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='7 Holographic boundary states The KM states discussed in the previous section can be thought of as certain a-typical black hole microstates in the context of SYK/AdS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Interesting analogs in higher dimensional examples of AdS/CFT can be found by considering boundary states in CFTs [126–128].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' A boundary state characterizes boundary conditions which can be imposed on a boundary of space-time on which the CFT lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For each allowed boundary condition, we can evolve the – 56 – state along the Euclidean time to suppress the high-energy contributions and obtain a state of finite energy which is called a regularized boundary state of the CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For holographic theories, the CFT path integral maps onto the gravity path integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Therefore, we will be able to make use of the AdS/CFT correspondence to deduce the cor- responding geometries if we can choose a state for which we can understand a gravity pre- scription for dealing with the boundary condition at the initial Euclidean time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' As discussed in [129], we can describe boundary states by starting with the TFD state of two CFTs labeled by L and R |TFD(β/2)⟩ = 1 Z � i e−βEi/4 |Ei⟩L ⊗ |Ei⟩R , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='74) and then project the TFD state onto some particular pure state |B⟩ of the left CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' As a result we obtain a pure state of the right CFT given by |ΨB,β⟩ = 1 Z � i e−βEi/4⟨B |Ei⟩ |Ei⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='75) If the temperature is high enough, the TFD state is dual to the maximally extended AdS-Schwarzschild black hole in the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The geometry which is dual to these regularized boundary states is expected to contain a significant portion of the left asymptotic region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Therefore, in a holographic CFT, this class of regularized boundary states can be regarded as microstates of a single-sided black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' These black hole microstates can be thought of as black holes with end of the world (EOW) branes on the left side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='45 Generally the EOW brane configuration is time-dependent at the macroscopic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Hence these are states with energy and energy variance compatible with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3), so we expect to be able to apply our construction and define operators (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' As we will discuss in the next section, one way to think of them is that the gravitational dressing has been moved over to the EOW brane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1 Computation of the return probability and correlators First we define unit-normalized boundary states | �Ba(0)⟩ = e− βH 4 |Ba⟩ � ⟨Ba| e− βH 2 |Ba⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='76) Then we want to show that return probability of a boundary state R(T) = |⟨ �Ba(0) | �Ba(T)⟩ |2 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='77) decays exponentially fast at early time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For boundary states in holographic 2d CFTs we have (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='11) G(β) = ⟨Ba|e− βH 2 |Ba⟩ ≃ e π2c 6β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='78) 45Proving from first principles that boundary states dual to EOW branes exist is far from trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It has been investigated from a bootstrap perspective in [130], where it was suggested that such boundary states must be extremely fine-tuned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In [131], the full classification of boundary states in large N symmetric orbifolds was carried out, and typical boundary states are not of this form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 57 – where we have taken the CFT to be defined on a spatial circle of length 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For small T we have R(T) = |G(β + 2iT)|2 |G(β)|2 ≃ e − 4π2c 3β3 T 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='79) The energy variance of the boundary state can be easily computed from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='78) and we find ∆H2 = ⟨H2⟩ − ⟨H⟩2 = 4π2c 3β3 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='80) so the initial decay (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='79) is, not surprisingly, consistent with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='16), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='18) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='80).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In higher dimensional cases we can read from (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='13) G(β) = e αd βd−1 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='81) thus R(T) = |G(β + 2iT)|2 |G(β)|2 ≃ exp � − αd βd+1 4d(d − 1)T 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='82) We can again check that ∆H2 = ⟨H2⟩ − ⟨H⟩2 = αd βd+1 4d(d − 1), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='83) which is compatible with (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='82).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We now proceed with checking that the other states in the code subspace around a boundary state are orthogonal to the time evolved code subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Consider for example the state O(t, x)| �Ba⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Following similar reasoning as in subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6 we can show that |⟨ �Ba(0)|O(t, x)| �Ba(T)⟩|2 = ⟨ �Ba(0)|O(t, x)| �Ba(T)⟩ ⟨O(t − T 2 , x)⟩β→β+2iT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='84) where ⟨ �Ba(0)|O(t, x)| �Ba(T)⟩ = Ga(I,β+2iT) Ga(I,β) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' More generally ⟨ �Ba(0)|O(t1, x1)O(t2, x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='O(tn, xn)| �Ba(T)⟩ = ⟨ �Ba(0)|O(t, x)| �Ba(T)⟩⟨O(t1 − T 2 , x1)O(t2 − T 2 , x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='O(tn − T 2 , xn)⟩β→β+2iT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='85) Thus, as long as the analytical continuation of the correlation function in β does not introduce any surprising N-dependent factors we will get the expected behavior (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We now check this condition for low-point functions in 2d boundary states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Here we assume that for a holographic CFT, and if we are working in the large N limit, the 1-point function of light conformal primaries can be computed by a method of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Then for a 1-point function of a scalar primary O with dimension ∆ on a boundary state we have ⟨ �Ba(0)|O(t, x)| �Ba(0)⟩ = AO ( β π cosh[ 2π β t])∆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='86) – 58 – for some constant AO which depends on the boundary state a and the operator O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' After the analytic continuation necessary for (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='84) we find ⟨O(t − T 2 , x)⟩β→β+2iT = AO ( (β+2iT) π cosh[ 2π (β+2iT)(t − T 2 )])∆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='87) Hence we notice that the results (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='84),(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='87) are consistent with our general expectations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='21),(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We can also check 2-point functions, which we can compute in the large N limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' First we compute the 2-point function on the boundary state, using the method of images ⟨ �Ba(0)|O(t1, x1)O(t2, x2)| �Ba(0)⟩ = +∞ � n=−∞ 1 �� β π sinh � π β[(x1 − x2 + 2πn) − (t1 − t2)] ���2∆ ± 1 �� β π cosh � π β[(x1 − x2 + 2πn) − (t1 + t2)] ���2∆ , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='88) After the analytic continuation necessary for (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='85) we find from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='88) that we do not notice any unexpected behavior of this part of the correlator as T increases, so the result (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='85) is dominated by the decay of the return probability, and is consistent with our expectations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='21),(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 7 Black Hole microstates One question which is particularly interesting is whether we can apply our construction to black hole microstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We have already mentioned in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1 that there are various classes of black hole microstates, some of which have macroscopic time dependence and some of which do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We will now discuss these various cases in more detail and interpret our operators for these types of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1 States with macroscopic time-dependence We will start with the simplest situation: states with macroscopic time-dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This can be visible outside the horizon, for example black holes in the presence of infalling matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Alternatively it can be that the geometry appears to be static outside the horizon but there is no corresponding Killing isometry in the interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' As the first case is more straightforward, we focus on the second case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Two examples of such states are boundary states of the CFT, corresponding to end-of-the-world branes inside the horizon, which have already been dis- cussed in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' A second example is states prepared by the Euclidean path integral on some surface of higher topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The dual geometries have topology behind the horizon, and are often referred to as geons [80, 132, 133].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It is worth re-emphasizing that both of these states are usually prepared by the Euclidean path integral and are in fact very a-typical states, even if the CFT 1-point functions are very close to those in a thermal state – 59 – (or said differently, even if the classical geometry is exactly that of a black hole outside the horizon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Both of these examples involve pure states |Ψ0⟩ that have a large energy variance, of order N2, such that the return probability will decay as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We can thus apply our construction to build local operators that are not dressed to the boundary CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The interpretation is that the operators are dressed with respect to the time-dependence of the interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Consider for example the genus-2 geon in d = 2, which is prepared by the Euclidean path integral on half of a genus-2 surface [80,134].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Microscopically, the state can be described by |Ψ0⟩ ∼ � i,j Ciije−Eiβi/2−Ejβj |Ej⟩ , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1) where ∼ indicates that we have not been careful about the parametrization of the genus- 2 surface, but βi,j are related to the moduli of the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The un-normalized overlap of this state corresponds to a genus-2 partition function in the dumbbell channel, where βj parametrizes the length of the two handles, and βi parametrizes the length of the neck between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It is not straightforward to write down a metric that covers the entire space-time of such states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Outside the horizon whose size is controlled by βj, they look exactly like the BTZ geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Inside the horizon, they have macroscopic time-dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' A nice coordinate patch that covers the Wheeler-de Witt patch of the t = 0 slice of the geometry can be written down in a very simple form ds2 = −dt2 + cos2 t dΣ2 2 , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2) where dΣ2 2 is the constant negative curvature metric on half of a genus-2 surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This coordinate patch covers the entire t = 0 slice of the geometry, which is precisely half of a genus-2 surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The neck corresponds to the horizon, and there is topology (one handle) behind the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' From this metric, we explicitly see the time dependence of the geometry, even if a metric for the full spacetime is hard to write down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The interpretation of our operator is that the dressing is to the time-dependence of the geometry that sits inside the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For end-of-the-world brane geometries, the situation is similar and the operator is dressed to the end-of-the-world brane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2 Typical states The question we would now like to ask is whether our prescription works in typical black hole microstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Contrary to states with end-of-the-world branes or topology behind the horizon, it seems reasonable to expect that typical states should also look like the thermal state a finite distance inside the black hole (see for example [135,136]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Whether or not our prescription works depends on the definition of a typical black hole microstate, and in particular on the energy spread we are choosing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' One possibility is to define typical states using an ensemble of energy eigenstates with spread O(N0) in energy (recall that there are still eS with S ∼ O(N2) states in this energy band).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In that situation, – 60 – our prescription does not work, as the variance of energy is O(N0) and the return probability will not decay fast enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Another possibility is to consider typical states with an energy spread similar to that of the canonical ensemble, that is (∆E)2 ∼ O(N2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3) For such states, the return probability will decay following the behaviour (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Therefore, we can follow our prescription and define the operators in the same way and they will satisfy the two properties of commuting with the Hamiltonian to all orders in 1/N and acting like HKLL operators to leading order at large N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' While these operators are certainly diff-invariant, since they are operators defined in the CFT, the bulk interpretation of their gravitational dressing on typical black hole microstates is not entirely clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' When the gravitational configurations are macroscopically time-dependent, our operators are dressed with respect to the features of the geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The typical states are still time-dependent, but only microscopically, as it seems plausible to assume that macro- scopically they are featureless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In some sense our operators are dressed to the microscopic time-dependence of the state (the phases of the ci in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2)), but it is unclear exactly what that means in the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Notice however, that if we start with a particular typical pure state |Ψ0⟩ and act with a unitary made out of the operator (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2), associated to that state, then the predictions for what an infalling observer jumping into the black hole will see are unambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For example, the operators (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2) will generally create an excitation in the bulk and the location in time relative to that of the infalling observer who jumps from the boundary at a particular boundary time, can be unambiguously computed for each state |Ψ0⟩ and corresponding operators (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We emphasize that for this interpretation it is important to remember that the operators (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2) are state-dependent and cannot generally be promoted to a single operator which acts in a specific way globally on most typical states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We briefly comment on black hole interior reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Suppose we start with a typ- ical black hole microstate with energy spread of order (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' If we assume that the interior geometry contains part of the left asymptotic region, then the possibility of removing the dressing of the operators implies that we can deform the state behind the horizon by creat- ing some particles there, in such a way that these excitations cannot be detected from the boundary CFT by the measurement of single-trace correlators, including the Hamiltonian, in the 1/N expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This was also discussed in [137, 138].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We emphasize that this does not contradict the statements made in [25, 135, 136] that for typical states with microcanonical energy spread, it is impossible to add excitations without affecting single-trace correlators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3 Two entangled CFTs Similar considerations apply to geometries with two asymptotically AdS regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Consider two non-interacting CFTs with total Hamiltonian H = HL+HR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We take the full system to be in a pure state |Ψ0⟩ which may be entangled, but we will assume the pattern of entanglement – 61 – is generic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In particular, we do not consider states like the thermofield-double which have a very fine-tuned structure of entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We can imagine the state |Ψ0⟩ to be, for example, UL |TFD⟩, where UL is a complicated random unitary acting on the left CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In this case we can consider the following generalization of our construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Let us consider the 2-parameter family of time-shifted states e−i(TLHL+TRHR)|Ψ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We start with an HKLL operator Φ dressed with respect the to left system, which commutes with HR but not HL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We now consider the following generalization of the operators (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2) �Φ = c � dTLdTRe−i(TLHL+TRHR)P0ΦP0ei(TLHL+TRHR) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4) using P0 = P L 0 ⊗ P R 0 and [Φ, P R 0 ] = 0 then �Φ = c � dTLe−iTLHLP L 0 ΦP L 0 eiTLHL ⊗ � dTRP R TR (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='5) The resulting operator commutes with both HL and HR on the relevant code subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In this case, the operator is not dressed with respect to the overall time-dependence of the full system, but rather to the time dependence of the “left” subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' There are states with special entanglement pattern such as the TFD state, which was already discussed in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The generalized return amplitude ⟨Ψ0|e−i(HLTL+HrTR)|Ψ0⟩ which is a function of TL and TR does not decay in all directions for these special states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For example, in the TFD state it is constant along the line TL = −TR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In those cases we cannot set both commutators with HL, HR to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' So we can move the dressing from one side to another if we wish to, but there it is always dressed to one of the boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This happens because the TFD state has a symmetry, it is annihilated by HL − HR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4 Island discussion Our prescription is also useful to resolve some paradoxes in the context of black hole evapo- ration and islands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Consider a setup where a holographic CFT is coupled to a bath such that the bulk description is given by an evaporating black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' After the Page time, a non-trivial quantum extremal surface appears in the bulk delimiting an island, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' a part of the interior of the black hole that is encoded in the bath degrees of freedom rather than in those of the CFT [11,139].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' There is an apparent tension in this context related to gravitational dressing [140].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' If we create an excitation in the island by acting with a local operator φisland, where does the gravitational dressing go?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It appears that the only place for the dressing to go is the boundary CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' But this implies that the local operator will have the property [φisland, HCFT] ̸= 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6) But this seems to be inconsistent, because since the operator is in the island, it should be reconstructable from the bath degrees of freedom, and commute with the CFT degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 62 – Our operators provide a way out of this paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We can apply our prescription above in terms of two entangled systems with a generic pattern of entanglement (there is a subtlety here since the bath and CFT are actually coupled rather than non-interacting, but we can treat this interaction as weak).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In that case, even if we did start with an operator that had a non-trivial commutator (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6), we would engineer a new operator which commutes with HCFT up to exponentially small corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This new operator is now dressed with respect to the radiation, rather than the boundary CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The interpretation of the dressing is similar to that of the typical states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' While it would be tempting to imagine dressing the operator to the quantum extremal surface, the bulk geometry only has extremely slow time-dependence so it is unclear if time-dependent features of the geometry are sharp enough to dress with respect to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It appears that that the dressing is towards the microscopic time-dependence of the radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The story becomes less subtle if we consider a doubly holographic model (see for example [10,141]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In that case, the dressing to the bath can be directly geometrized in the higher-dimensional geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Our operators can perhaps be thought as a counter-part of the operators in the doubly-holographic setup, but in cases where the dressing cannot be so easily geometrized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Finally, we would like to clarify the distinction between reconstruction and dressing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' To make things simple, let us consider the TFD state and consider an HKLL operator on the left φL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This operator is dressed to the left CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Now we run our protocol, and as explained above, we can move the dressing to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The operator ˆφL now commutes with HL but no longer with HR [95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This does not mean that it can be reconstructed from the right degrees of freedom, but that it can be detected from the right CFT via the Gauss law tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It is still mostly built from the left CFT degrees of freedom, only its dressing has been pushed to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 8 Discussion In this paper, we have investigated whether information can be localized in perturbative quantum gravity, in the context of the AdS/CFT correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The challenge at hand is to construct local diff-invariant operators that are not dressed to the boundary where the CFT lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We have presented evidence that such operators exist, at least around high energy states with a large energy variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Such states include semi-classical geometries with features that break the symmetries of the dual CFT and for such states, local operators can be dressed to the features of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We have argued that there exist CFT operators that commute with all single-trace operators in a narrow time-band to all orders in the 1/N expansion, including the Hamiltonian and other charges that generate conformal transformations, while at the same time act like standard HKLL operators to leading order at large N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We have presented an explicit construction of such operators, and checked that they commute with the Hamiltonian to all orders in the 1/N expansion, and act like HKLL oper- ators to leading order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Technically the construction of such operators is made possible due to the fact that different semi-classical states have exponentially small overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We have also – 63 – discussed a generalization of our operators that would commute with all boundary charges of the conformal group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Moreover, we presented a definition of operators that commutes with all single-trace operators, not just conserved charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The construction of these operators is slightly less explicit, and we define them by specifying their action on the code subspace around a semi-classical geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We argue that such operators commute with all single- trace operators in a narrow CFT time-band, while also acting like HKLL operators to leading order at large N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Acting with such operators creates excitations that are completely invisible to CFT correlation functions in a narrow time-band, even if they become accessible at later times when a lightray from the location where the bulk excitation was created reaches the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This suggests that information can be lozalized in perturbative quantum gravity, to all orders in GN perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We conclude with some open questions that we raised along the way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1 The variance of the energy from semi-classical gravity A quantity that played a primordial role throughout the paper is the variance of energy, which controls the early time decay of the return probability through (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' One question that would be interesting to understand better is how we can compute the variance ⟨Ψ0|∆H2|Ψ0⟩ from semi-classical gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In appendix A we give an example that we can change the O(N2) coefficient of the variance of the Hamiltonian without changing the semi-classical geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This implies that the variance of the energy is not just a property of the geometry, but also of the quantum state of the fields on top of that geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Of course, if the metric changes as a function of time, this puts a bound on the variance through (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This suggests that if we start with some time-dependent semi-classical geometry with a matter QFT state with large variance, it should not be possible to change the state in a way to make the variance decrease to O(1) without changing the metric towards a time-independent solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The mechanism by which this would happen is unclear, and it would be interesting to pursue it further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' On a related note, we can ask how we can quantize the bulk mode associated to the Hamil- tonian directly in gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The expectation value of the Hamiltonian is extracted through the fall-off of the metric near the AdS boundary, as is standard in AdS/CFT, but this does not capture its quantum 2-point function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' If one computes the stress-tensor connected 2-point function on the geometry, takes the relevant components and performs the spatial integrals, one should obtain the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It would be desirable to have a more direct representation of the variance in terms of the the bulk wavefunction of the non-propagating s-wave mode of the graviton and also understand from this point of view the lower bound on the variance for time-dependent geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2 Gravitational proof for the decay of the return probability A central part of this paper was played by the decay of the return probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The physical interpretation of this decay for a semi-classical time-dependent geometry is that it computes (the square) of an overlap between two distinct geometries, namely the original one and the – 64 – time-evolved one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The general expectation is that the overlap of two distinct coherent states should be given by ⟨λ1|λ2⟩ ∼ e−N2f(λ1,λ2) , (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1) where f is some O(1) function whose real part is positive (we have assumed that the states |λ1,2⟩ are normalized).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The intuition is that N2 plays the role of 1/ℏ which controls the overlap of coherent states, and from a gravitational stand-point, the on-shell action of any geometry will be proportional to 1/GN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' However, this gravitational argument does not necessarily imply that the real part of f is positive, which is required by reflection positivity of the CFT dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' As we have seen in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='7), interpreting geometries as quantum states implies constraints on various on-shell actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It would be interesting to understand this problem directly in gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Can reflection positivity be proven directly at the level of the gravitational path integral?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This requires proving (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='7) directly in gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' A possible way to prove this is the following: we consider two states λ1 and λ2 with fixed sources, and their associated geometries contributing to the overlaps ⟨λ1,2|λ1,2⟩, with geometries g1 and g2 and on-shell actions I1 and I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We start by considering a gravitational configuration which is half of g1 (say the northern hemi-ball) and half of g2 (the southern hemi-ball).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This configuration has action Itot = I1 + I2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2) Note that the geometry is off-shell at the gluing surface between g1 and g2, and there could be another contribution Ijunction to the action coming from the gluing, which we will not include for now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' To find the smooth saddle-point geometry, we need to let this geometry relax by modifying its configuration near the junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' One may be able to prove that this smoothing of the glued geometry comes with a definite sign in the action, therefore proving (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It would be interesting to pursue this idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3 Microscopically time-dependent states We have seen that for any state with large energy and large energy variance, we can find bulk local operators who commute with the time-band algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The interpretation of these operators is that they are dressed with respect to features of the state (in particular the time-depdence of the state), rather than to the boundary CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This intuitive picture is clear when the state describes a semi-classical geometry that is macroscopically time-dependent, as the time-dependence can be seen directly from the background metric which has features with respect to which we can attach a gravitational dressing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' As we have discussed, our prescription also works for typical states with energy variance of O(N2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In that context, the interpretation of the dressing is less clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The dual geometry is not macroscopically time-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We can declare that the operator is dressed with respect to the microscopic time-dependence, but it is unclear what that means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It would be interesting to have a better physical understanding of the dressing for such type of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We hope to return to this question in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 65 – It is also important to note that our operators are state-dependent, even outside the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For a given typical state, we can use our construction to find the state-dressed local operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' However, if we now pick a different typical state then the operator will not act in the desired fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In this sense, our operators are similar to mirror operators [6], but they can live outside the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Nevertheless, we wish to emphasize again that independently of questions surrounding the interpretation of these operators, an important message of this paper is that these operators exist and that states created by acting on the corresponding typical state with unitaries built from these state-dressed operators have identical correlators of single-trace operators in a narrow time-band in the 1/N expansion as the original state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Moreover, this can be done around any typical state once the state has been fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4 Microcanonical states and small energy variance There are also typical states with a small energy variance, of O(N0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For example, when one refers to the microcanonical ensemble, one often has in mind picking a state with spread in energy which is O(N0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For such states, the return probability does not decay to values which are exponentially small in N2 after an order one time, which means we cannot use our construction to define state-dressed operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The variance of the energy is a very coarse way to define how time-dependent a state is, and for states with energy variance of size O(N0), the state is not time-dependent enough to dress operators to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Of course, all these states look macroscopically time independent, and all the information is in the microscopic phases of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It would be interesting to study this further, and have a better physical picture of whether one can find state-dressed operators to these small variance states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It is worthing mentioning that if the variance is O(Nc) for any 0 < c < 2, our pre- scription does work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For typical states, this is some kind of intermediate regime between canonical states and microcanonical states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For coherent states that are macroscopically time-dependent, this situation would occur if the profile of the fields are not O(1), but rather scale with some positive power of GN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In that case, backreaction is small, but the return probability still decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It would be interesting to understand these regimes better, they interpolate between coherent states of the bulk quantum fields propagating on a frozen AdS background, and semi-classical geometries with a non-trivial metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='5 The AdS vacuum and low-energy states For low-energy states like the AdS vacuum or states with an O(N0) energy above it, our construction does not work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Therefore, the results of this paper do not contradict the claims of [46], that for perturbative excitations on top of the AdS vacuum one can reconstruct the state directly from the time-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Technically, this happens because the return probability does not decay to exponentially small values for such states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Physically, states like the AdS vacuum have no features to which we can dress operators, so the only possible diff-invariant way to specify a point is with relation to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Even classically, there are no diff- invariant local observables in classical general relativity for the case of vacuum AdS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It thus appears that the failure of constructing approximately local diff-invariant operators around – 66 – the AdS vacuum happens because of the special nature of the state, rather than a fundamental obstruction due to the non-locality of quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For excited states on top of the AdS vacuum, it is less obvious why local diff-invariant states cannot be constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' One may imagine that if the VEV of a scalar field has a quantum lump in some region of space-time, we could dress an operator to the location of this lump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Technically, we see that at least our operators cannot achieve this goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It would be interesting to have a more physical understanding of why it is not possible to dress operators to quantum profiles, rather than semi-classical ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' As we have seen in the previous subsection, it is not completely related to backreaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' If we consider a coherent state on top of vacuum AdS corresponding to a source which scales as N1/4, the return probability would decay fast enough for our construction to work, even if backreaction can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Note however that such a state is not really part of the low-energy EFT on top of vacuum AdS, since it has energy that scales with some fractional power of the Planck scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It would be interesting to understand this better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Acknowledgments We are happy to thank Micha Berkooz, Jan de Boer, Monica Guica, Elias Kiritsis, Shota Komatsu, Hong Liu, Olga Papadoulaki, Suvrat Raju, Erik Verlinde, Spenta Wadia, and Sasha Zhiboedov for stimulating discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' EB and NV would like to thank CERN-TH for their hospitality during the preparation of this work and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Bertolini for his invaluable support during this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The work of EB and NV is partially supported by INFN Iniziativa Specifica - String Theory and Fundamental Interactions project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' A Changing the variance of H We would like to understand whether the variance of the energy is accessible within semi- classical gravity, simply from the geometry, or whether it requires more knowledge and in particular, the knowledge of the bulk quantum state for the fields propagating on the back- ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' As we will see, knowledge of the quantum state seems to be required to extract the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The quantity we would like to compute is ⟨Ψ0|H2|Ψ0⟩ − ⟨Ψ0|H|Ψ0⟩2 ≡ ⟨Ψ0|H2|Ψ0⟩c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1) This is a connected correlation function in holography, which usually would be compute from the 2-point function of the associated propagating fields on the relevant background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This 2-point function is sensitive both to the geometry and to the bulk quantum state of the propagating fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' However, here the situation is more subtle, because we are not studying the local correlation function of an operator, but rather the 2-point function of the spatial integral of a local operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In this particular case, the situation is a lot more confusing – 67 – because the dual bulk field would be the s-wave graviton, which is not a propagating degree of freedom in gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' So what computes this variance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We will not be able to answer this question, and we believe it to be an interesting open problem which we hope to return to in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Nevertheless, we will study some particular states that should be interpreted as adding an s- wave graviton in the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Even though this mode doesn’t propagate, we will see that adding it can affect the CFT variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We will consider two type of deformations of the thermofield double (TFD) state, both of which are related to adding an integrated stress-tensor operator on the cylinder that prepares the TFD state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Let us start with some basics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We consider the TFD state |TFD⟩ = 1 √ Z � i e−βEi/2 |Ei⟩ |Ei⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2) We assume that the partition function has the usual large N behavior Z(β) = exp � N2 � F0(β) + 1 N2 F1(β) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' �� , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3) from which we can compute ⟨Hn⟩β = (−1)n 1 Z dn dβn Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4) where H is HL or HR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We have ⟨TFD| H |TFD⟩ = ⟨H⟩β = −N2F ′ 0 − F ′ 1 , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='5) ⟨TFD| H2 |TFD⟩ − ⟨TFD| H |TFD⟩2 = ⟨H2⟩β,c ≡ ⟨H2⟩β − ⟨H⟩2 β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6) We have ⟨H2⟩β,c = N2F ′′ 0 + F ′′ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='7) Now, consider the following state |ψ⟩ = H |TFD⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='8) We now have ⟨ψ|ψ⟩ = ⟨H2⟩β (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='9) Let us now see how the energy and variance of the state have evolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We have ⟨ψ| H |ψ⟩ ⟨ψ|ψ⟩ = ⟨TFD| H3 |TFD⟩ ⟨TFD| H2 |TFD⟩ = ⟨H⟩3 β + 3 ⟨H2⟩β,c ⟨H⟩β + ⟨H3⟩β,c ⟨H⟩2 β + ⟨H2⟩β,c , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='10) where we defined ⟨H3⟩β,c ≡ ⟨H3⟩β − 3 ⟨H2⟩β,c ⟨H⟩β − ⟨H⟩3 β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='11) Large N factorization implies that we can expand this answer and we find ⟨ψ| H |ψ⟩ ⟨ψ|ψ⟩ = ⟨H⟩β + 2 ⟨H2⟩β,c ⟨H⟩β + · · · = −N2F ′ 0 − F ′ 1 − 2F ′′ 0 F ′ 0 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='12) – 68 – We see that we obtain the TFD answer, up to a correction term, which is of size N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This means we have not changed the geometry classically, but only added a quantum particle on top of the TFD state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Similarly, one can compute ⟨ψ| H2 |ψ⟩ ⟨ψ|ψ⟩ − �⟨ψ| H |ψ⟩ ⟨ψ|ψ⟩ �2 = ⟨H⟩4 β + 6 ⟨H2⟩β,c ⟨H⟩2 β + · · · ⟨H⟩2 β + ⟨H2⟩β,c − � ⟨H⟩2 β + 4 ⟨H2⟩β,c + · · · � = ⟨H2⟩β,c + · · · = N2F ′′ 0 + · · · (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='13) We see that that the energy has changed at N0, but the variance has not changed at order N2, only at order N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' So this state modifies both the variance and the energy at subleading order compared to the TFD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We will now build a state that modifies the energy at subleading order, but the variance at leading order compared to the TFD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Consider the state |φ⟩ = (H − ⟨H⟩β) |TFD⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='14) We now have ⟨φ|φ⟩ = ⟨H2⟩β,c , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='15) and we can now compute the energy in this state: ⟨φ| H |φ⟩ ⟨φ|φ⟩ = ⟨H3⟩β − 2 ⟨H2⟩β ⟨H⟩β + ⟨H⟩3 β ⟨H2⟩β,c = ⟨H⟩β + ⟨H3⟩β,c ⟨H2⟩β,c = −N2F ′ 0 − F ′ 1 − 2F ′′′ 0 F ′′ 0 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='16) We see that this state modifies again the energy only at order N0, and in a slightly different way than the previous state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In a similar way, we compute the variance and find ⟨φ| H2 |φ⟩ ⟨φ|φ⟩ − �⟨φ| H |φ⟩ ⟨φ|φ⟩ �2 = ⟨H⟩2 β + 3 ⟨H⟩2 β,c + 2 ⟨H3⟩β,c ⟨H⟩β + ⟨H4⟩β,c ⟨H2⟩β,c − � ⟨H⟩β + ⟨H3⟩β,c ⟨H2⟩β,c �2 = 3 ⟨H2⟩β,c + ⟨H4⟩β,c ⟨H2⟩β,c − � ⟨H3⟩β,c ⟨H2⟩β,c �2 = 3N2F ′′ 0 + 3(F ′′ 0 )2F ′′ 1 − (F ′′′ 0 )2 + F ′′ 0 F ′′′′ 0 (F ′′ 0 )2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='17) One can see that the change in the variance is order N2 (it is three times the variance of the TFD state), so this is a modification of the variance at the order we were looking for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' From this, we can conclude that the semi-classical geometry is not enough to extract the variance of the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The quantum state of the bulk fields is equally important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For the state |φ⟩, we have the same leading large N properties, but a different quantum state for the graviton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The fact that it is the s-wave of the graviton that enters is still puzzling, and it would be interesting how to propertly quantize this non-propagating degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We leave this for the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 69 – B Boosts in global AdS As we have discussed in section 3, the conformal generators on the d-dimensional cylinder R × Sd−1 organize themselves as time-translations, rotations, and 2d remaining generators which correspond to boosts in the dual AdS geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The goal of this section is to discuss whether there exist states that can preserve the boost symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' As we have seen throughout the paper, symmetries that are broken by semi-classical states allow us to specify bulk points by dressing the location of a bulk point to the feature of the state that breaks the symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It is important to understand which symmetries are broken, and which symmetries can be preserved by semi-classical states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For time translations and rotations, this is straightforward, but it is somewhat more subtle for boosts, which is the purpose of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The 2d boost generators can be realized as d non-independent copies of SL(2, R) [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For simplicity, we will study the case of AdS3, but the higher dimensional versions follow in a straight forward manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In d = 2, the two copies of SL(2, R) are well-known and correspond to the left and right moving sectors of conformal transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The generators are given by L−1, L0, L1 and ¯L−1, ¯L0, ¯L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Time-translations and rotations are obtained by the combinations H = L0 + ¯L0 , J = L0 − ¯L0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1) The four residual generators correspond to boosts in AdS3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For explicit expressions, see [142].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We would now like to analyze whether non-trivial states can be annihilated by these boosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' As a starting point, notice that there are obviously CFT states which are annilitated by L−1 and ¯L−1: primary states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' However, we would like to consider generators that can be exponentiated to norm-preserving group elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This means the generators should be Hermitian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The generators L−1 and ¯L−1 do not satisfy this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' However, we can assemble them into the combinations L+ = L−1 + L1 , L− = i(L−1 − L1) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2) Using that L† −1 = L1, we see that L± are hermitian operators and can thus be exponentiated to form unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The question we would like to ask is whether there are states in the Hilbert space that are eigenstates of L±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We will see that the only finite energy eigenstates of these operators are those where the left-moving part of the CFT is in the vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' To see this, we consider the commutator [L+, L−] = 4iL0 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3) Suppose now that |ψ⟩ is a normalizable eigenstate of —say— L+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Computing the expectation value of this equation we find ⟨ψ|L0|ψ⟩ = 0 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4) From the positivity of the energy spectrum this is possible only if L0|ψ⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The only states with this property are states where the left moving sector of the CFT is in the vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 70 – Non-trivial states will thus break boost invariance, which can be use to specify the radial location of an operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For the construction of operators presented in this paper, this would require considering the states obtained by acting with the unitary operators on semi-classical states |ψ0⟩ as e−iγL± |ψ0⟩ , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='5) and studying the generalized return probability R(γ) ≡ ��⟨ψ0| e−iγL± |ψ0⟩ ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6) These return probabilities have not been studied but for semi-classical states, it is natural to expect them to be exponentially small for γ ∼ O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' C Early time decay of the return probability We wish to estimate the early time decay of the return probability (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We will see that at very early times, namely t ∼ 1 N , we can find the decay purely from large N factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We will first recall a general property of coherent state overlaps which follows from large N factorization, and then adapt the situation slightly to the return probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1 Overlap of coherent states and large N factorization Coherent states of quantum gravity in AdS/CFT can be described by states prepared by a Euclidean path integral with sources turned on for single-trace operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' These states are thus given by |λ⟩ = e � x0<0 dxdλ(x)O(x) |0⟩ , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1) where we have not written the appropriate time-ordering which is left implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We will now show that the overlap is given by ⟨λ1|λ2⟩ = e � Rd λ∗ 1(y)λ2(x)⟨O(y)O(x)⟩ + O(1/N) , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2) where it should be understood that y is integrated over the upper half plane while x is integrated over the lower half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We can explicitly expand out the integrals of the bra and the ket states, and use large N factorization: this implies that the operators should be paired up and contracted using Wick’s theorem, up to 1/N corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' At a given power in the source, we will have a term of the form �� dxdy �k 1 (k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' )2 λ∗ 1(y)kλ2(x)k ⟨0| Ok(y)Ok(x) |0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3) We can now apply Wick’s theorem and find �� dxdy �k 1 (k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' )2 λ∗ 1(y)kλ2(x)k ⟨0| Ok(y)Ok(x) |0⟩ = 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' �� dxdyλ∗ 1(y)λ2(x) ⟨0| O(y)O(x) |0⟩) �k , – 71 – which we can re-exponentiate to find (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Note that we have not written the normalization of the states, which takes care of the Wick contraction between any two operators living both in the lower half plane, or upper half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Similarly, terms which have a different powers of upper and lower operators do not give contributions to leading order at large N because we cannot pair the operators and use Wick’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For this to work, we have implicitly assumed that λ ∼ O(N0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' To see this, note that the connected correlation functions of higher-point operators are suppressed by 1/N, but also have more sources than lower-point functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' If we scale the sources as λ ∼ N1/2, which is the correct scaling to induce O(1) back-reaction on the dual spacetime46, we have to be more careful, as some of the terms we dropped involving connected correlators will be the same size as the Wick contractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For example, we have λ∗ 1(y)λ2(x) ⟨O(y)O(x)⟩ ∼ N2 (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4) (λ∗ 1(y)λ2(x))2 ⟨O(y)O(y)O(x)O(x)⟩c ∼ N2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='5) This means that we cannot truncate to the sector of Wick contraction, and we must resum the entire expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Note however that the contributions corresponding to loop diagrams in AdS are still suppressed by 1/N, so we are resumming tree-level diagrams to build the backreacted geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The upshot of this analysis is that we can use large-N factorization to easily compute the overlap of coherent states, but only if the sources are O(1), in which case the exponent in the exponential is also O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' If we try to make the sources scale with N, the exponent will be of order N2 and then infinitely many contributions must be resummed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We will now apply this logic to the return probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2 The return probability We can now apply the same logic as above, taking the operator e−iHT to be seen as an imaginary Euclidean source for the Hamiltonian (which is the integral of the stress-tensor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We want to compute R(T) = ⟨Ψ0|e−iHT |Ψ0⟩⟨Ψ0|eiHT |Ψ0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6) Applying the logic above, we would find that to leading order we have R(T) = e−iT⟨Ψ0|H0|Ψ0⟩eiT⟨Ψ0|H0|Ψ0⟩ = 1 + O(1/N) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='7) So we see that the candidate leading term vanishes, and we must go to the next order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This is due to the nature of the return probability, which is a square of overlaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' A quick expansion of the exponentials shows that at order T 2, we have T 2� − ⟨Ψ0|H2|Ψ0⟩ + � ⟨Ψ0|H|Ψ0⟩ �2� = −T 2∆H2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='8) 46For operators that have unit 2-point function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 72 – For reasons similar to those explained above, this term can be exponentiated such that we find R(T) = e−T 2∆H2 + O(1/N) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='9) As in the previous section, we can only trust this approximation if the exponent is O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Be- cause we are considering states that have ∆H ∼ N2, we see that we can trust this exponential decay of the return probability for time-scales up to t ∼ 1/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For larger time-scales, it may still hold, but it cannot be justified based solely on large N factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It is instructive to consider the case of the thermofield double state and the spectral form factor, as we already discussed in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For simplicity, we set d = 2 where we have Z(β) = e c 12 4π2 β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='10) The spectral form factor then gives R(T) = e π2c 3 � 1 β+IT + 1 β−iT � = e 2π2c 3 β β2+T 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='11) We can expand this expression in T, as long as T ≪ β, to find R(T) ≈ Z(β)2e − 2π2c 3 T 2 β3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='12) We find the exponential decay that goes like T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' What is important is that even though T must be much smaller than β, it is allowed to scale as N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' This cannot be justified solely from large N factorization, but still holds in this particular context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We expect the return probability to satisfy this property for holographic states more generally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' D LLM solutions in the bulk The LLM geometries correspond to solutions of type IIB supergravity with symmetry SO(4)× SO(4)×R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We assume the axion and dilaton are constant and the IIB three forms are vanish- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We introduce coordinates xµ = (t, y, x1, x2) and Ω3, ˜Ω3 for two 3-spheres corresponding to the SO(4) isometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We parametrize the five form as F5 = Fµνdxµ ∧ dxν ∧ dΩ3 + ˜Fµνdxµ ∧ dxν ∧ d˜Ω3 , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1) where the self duality of the five form implies that the two forms F and �F are dual to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' After demanding that the geometry preserves the Killing spinor in the presence of the five form, we arrive at the following solution for the 1 2-BPS bulk states [81] ds2 = −(dt + Vidxi)2 h2 + h2(dy2 + dxidxi) + yeGdΩ2 3 + y eG d˜Ω2 3 , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2) – 73 – where every function in the metric is expressed in terms of a function z(x1, x2, y) and we defined z = 1 2tanh G, h−2 = 2y cosh G, and y∂yVi = ϵij∂jz, y(∂iVj − ∂jVi) = ϵij∂yz .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3) For the forms F, ˜F we have F = dBt ∧ (dt + V ) + BtdV + d ˆB, ˜F = d ˜Bt ∧ (dt + V ) + ˜BtdV + d ˆ˜B , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4) where Bt = − 1 4y2e2G and ˜Bt = − 1 4y2e−2G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' On the other hand, d ˆB = −1 4y3 ⋆3 d(z + 2 y2 ), d ˆ˜B = −1 4y3 ⋆3 d(z − 2 y2 ) , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='5) where ⋆3 is the epislon symbol in the flat three dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The only free function, z, is constrained to solve the equation, ∂i∂jz + y∂y(∂yz y ) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6) We focus our attention on the plane y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Since the product of the radii of the two 3-spheres is y, there will be a conical singularity at y = 0 unless the function z has a special behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Let’s consider the case where R1 is kept finite, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='e, e−G → 0 as y → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Thus, one has, z ∼ 1/2 − e−2G + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' If one assumes that z = 1/2 at y = 0, then one gets the expansion, z ∼ 1/2 − y2f(x1, x2) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' for some positive function f, with our boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Thus, e−G ∼ yc(x1, x2) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' and h2 ∼ c(x1, x2) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Therefore, close to y = 0, the part of the metric involving R2 will look like, h2dy2 + R2d˜Ω2 3 ≈ c(dy2 + y2d˜Ω2 3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='7) Thus the conical singularity is resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In the case where R2 is kept fixed, the same argument goes through but now with the condition that z = −1/2 at y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' With these boundary values of z at y = 0 as a source, one can solve the Laplace equation47 (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6) and compute z(x1, x2, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In addition, Vi can also be expressed in terms of an integral of z(x1, x2, 0) over the two dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' E Notes on boundary states Some useful references for this section are [79,143–145].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 47More precisely, it is a Laplace equation for z/y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 74 – E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1 Boundary states in 2D CFT Boundary states in a 2d CFT need to satisfy [143] (Ln − ˜Ln) |B⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1) In any Verma module, one can find a simple solution to these conditions as |Ih⟩ = � ⃗k |⃗k, h⟩L ⊗ |⃗k, h⟩R , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2) where |⃗k, h⟩L is a linear combination of Virasoro descendants of the primary state |h⟩ char- acterized by an infinite dimensional vector ⃗k = (k1, k2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=') with non-negative integer compo- nents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We identify these states by starting with descendants of the form .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='LKn −n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='LK1 −1 |h⟩L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3) and forming an orthonormal basis selected such that L⟨⃗k, h|⃗k′, h⟩L = δ⃗k,⃗k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The state |Ih⟩ is called the Ishibashi state for the primary state |h⟩L, where the states |⃗k, h⟩ are the descendant on top of the primary labeled by h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' It can be seen easily that Ln|Ih⟩ = ˜Ln|Ih⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='4) It is clear that the Ishibashi states have maximal entanglement between the left-moving and right-moving sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Linear combinations of the Ishibashi states satisfy the constraint (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='1) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Physical boundary sates are given by special linear combinations of Ishibashi states which are called Cardy states |Ba⟩ = � h Ca,h |Ih⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='5) Physical boundary states should satisfy a consistency condition of the partition function on a finite cylinder related to open-closed duality [143].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The Cardy states are singular because the norm of the Ishibashi states is divergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' One can define regularized boundary states by evolving in Euclidean time as |Ba,β⟩ = e− β 4 Hc |Ba⟩ , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6) where β is a positive constant and Hc = L0 + ˜L0 − c 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Since [L0 − ˜L0, Hc] = 0, the state (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='6) is still space-translational invariant on the circle, but it is time-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Ishibashi states are orthogonal to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The amplitude of Euclidean time evolution by β/2 between two such states is computed as ⟨Ik|e−βHc/2|Il⟩ = δklχk(e−β/2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='7) – 75 – χk is the character for the primary k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' On the other hand, the Cardy states are not orthogonal to each other but satisfy the open-closed duality relation as follows ⟨Ba|e− β 2 Hc|Bb⟩ = � k N(k) a,b Trk[e− 4π2 β Ho] (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='8) where Ho = Lo− c 24 denotes the Hamiltonian in the dual channel, characterized by the bound- ary conditions a, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' On the right hand side, Trk[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='] denotes a trace in the sector associated to a primary k as well as its descendants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Moreover, N(k) a,b counts the degeneracy of sectors which belong to the primary k with boundary conditions a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In the high temperature limit β → 0, we find that ⟨Ba|e− β 2 Hc|Bb⟩ ≃ N(km) a,b e− 4π2 β (h(min) a,b − c 24 ) , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='9) where km is the lightest primary among those satisfy N(km) a,b ̸= 0, whose conformal dimension is denoted as h(min) a,b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We can estimate the inner products between two normalized boundary states in this limit as ⟨ψa|e− β 2 Hc|ψb⟩ = ⟨Ba|e− β 2 Hc|Bb⟩ � ⟨Ba|e− β 2 Hc|Ba⟩⟨Bb|e− β 2 Hc|Bb⟩ ≃ δa,b + N(km) a,b e− 4π2 β h(min) a,b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='10) Note that N(0) a,a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In this way, a large gap in the open string channel leads to a large exponential suppression of off-diagonal elements of inner products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In holographic BCFT, the inner product between two boundary states can be computed by evaluating the gravity action on the dual background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' When we consider the gravity dual of a cylinder, there are two candidates of classical gravity solutions depending on whether the end of the word brane is connected or disconnected which are called connected and disconnected solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' When we consider the overlap for an identical boundary condition a, then both the connected and disconnected solution are allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In the limit β → 0, the connected solution is favored and one can find that ⟨Ba|e− β 2 Hc|Ba⟩ ≃ e π2c 6β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='11) We will use it later to calculate the return probability for boundary states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In addition to it, one can find the inner product between two boundary states with different boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In this case, only the disconnected solutions are allowed and ⟨Ba|e− β 2 Hc|Bb⟩ ≃ e cβ 12 +S(a) bdy+S(b) bdy, (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='12) where S(i) bdy, i = a, b are the boundary entropies [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' – 76 – E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='2 Boundary states in higher dimensions One can generalize to higher dimensions and define a boundary state |Ba⟩ as a state associated to a (d − 1)-dimensional boundary in d-dimensional CFT [79,146].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Taking the boundary to be a torus Td−1, the inner product between two boundary states in a holographic BCFT can be computed as a partition function on a d-dimensional open manifold Iβ/2 ×Td−1 where Iβ/2 is a length β/2 interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' As in the 2d case, there are two bulk solutions, a connected and a disconnected one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In the β → 0 limit the connected solution is dominant and one can find the inner product between two identical boundary states using the gravity solution as ⟨Ba|e− β 2 Hc|Ba⟩con ≃ eαd/βd−1 , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='13) where αd = (4ζ(T))d Rd−1 16GN Ld−1 , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='14) where R is the AdS radius, L is the length of the compactified spatial directions and ζ(T) is a function of tension which is defined when T < 0 as ζ(T) ≡ Γ(1/d)Γ(1/2) Γ(1/d + 1/2) R|T| d(d − 1)(1− R2T 2 (d − 1)2 )1/d−1/2F(1, 1/d, 1/2+1/d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' 1− R2T 2 (d − 1)2 ) , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='15) and when T > 0, ζ(T) = 2π d − ζ(−T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The tension takes values in the range |T| < d−1 R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' For d > 2, ζ(T) non-trivially depends on T and there is an upper bound of the tension T < T∗ which T∗ > 0 and ζ(T∗) = 0 [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='3 Correlation functions in BCFTs Let us first start with the simplest case where the CFT is defined on the upper half plane and the boundary state |B⟩ is placed along the real axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We consider the 1-point function of a local operator placed at z in the upper half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In the case of a CFT on the plane, the 1-point function of a primary operator in the vacuum is required to vanish by the symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' These are partly broken in a BCFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The remaining symmetries constraint the 1-point function to have the form ⟨O(z)⟩UHP = AO (2 Im(z))∆ , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='16) where AO is determined by the details of the theory and the precise boundary state in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' One could think of this as the boundary providing a source for the operator O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The 2-point function of a primary operator in a BCFT is more complicated than the case with no boundaries where it is exactly fixed by the symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Non-trivial information about the operator content and OPE coefficients is necessary to compute the 2-point function exactly in a BCFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We assume that for large N holographic CFTs the large N 2-point function takes the form ⟨O(z1)O(z2)⟩UHP = ⟨O(z1)⟩UHP ⟨O(z2)⟩UHP + ⟨O(z1)O(z2)⟩ ± ⟨O(z1)O(z∗ 2)⟩ , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='17) – 77 – where ⟨O(z1)O(z2)⟩ = 1 |z1 − z2|2∆ , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='18) where the contribution from an image insertion placed at z∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' The sign of the last term is governed by the boundary conditions, being either Dirichlet (−) or Neumann (+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Mapping the z coordinate to a new coordinate w by w → z = exp(2πw/β + i2π/4) , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='19) we can map the upper half plane to the a strip of width β/2, where the positive (negative) real axis is mapped to the lower (upper) edge of the strip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Since primary operators continue to transform in the usual way, the correlation functions now transform to ⟨O(w)⟩strip = AO ( β π cos[ 2π β τ])∆ ⟨O(w1)O(w2)⟩connected strip = 1 | β π sinh[ π β(w1 − w2)]|2∆ ± 1 | β π cosh[ π β(w1 − ¯w2)]|2∆ , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='20) where the second line is only the connected piece of the large N 2-point function [126].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Higher order correlation function can be found through large N factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' Correlation functions on a state defined on a circle by |Bβ⟩ = e−βH/4 |B⟩ , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='21) can be thought of as correlation function on a cylinder of width β/2 where the boundary state is placed on both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We can instead consider a strip of width β/2, from τ = −β/4 to τ = β/4 with periodicity x ∼ x + R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' We choose R = 2π for simplicity from now on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' In large N holographic CFTs correlation functions on the cylinder can be found from the correlation function on the strip using the method of images ⟨O(w1)O(w2)⟩connected cylinder = ∞ � n=0 ⟨O(w1 + 2πn)O(w2)⟩connected strip .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content='22) References [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9FAT4oBgHgl3EQf8x72/content/2301.08753v1.pdf'} +page_content=' M.' metadata={'source': 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100644 index 0000000000000000000000000000000000000000..a8a1045226376a301395cc79b4f7e4b84f956eea --- /dev/null +++ b/jdAzT4oBgHgl3EQfpf0M/content/tmp_files/2301.01612v1.pdf.txt @@ -0,0 +1,1883 @@ +arXiv:2301.01612v1 [cond-mat.stat-mech] 4 Jan 2023 +A range three elliptic deformation of the Hubbard model +Marius de Leeuw1, Chiara Paletta1, Balázs Pozsgay2 +January 5, 2023 +1School of Mathematics & Hamilton Mathematics Institute, Trinity College Dublin, Ireland +2MTA-ELTE “Momentum” Integrable Quantum Dynamics Research Group, Department of +Theoretical Physics, Eötvös Loránd University, Budapest, Hungary +mdeleeuw@maths.tcd.ie, palettac@maths.tcd.ie, pozsgay.balazs@gmail.com +Abstract +In this paper we present a new integrable deformation of the Hubbard model. Our de- +formation gives rise to a range 3 interaction term in the Hamiltonian which does not pre- +serve spin or particle number. This is the first non-trivial medium range deformation of the +Hubbard model that is integrable. Our model can be mapped to a new integrable nearest- +neighbour model via a duality transformation. The resulting nearest-neighbour model also +breaks spin conservation. We compute the R-matrices for our models, and find that there is +a very unusual dependence on the spectral parameters in terms of the elliptic amplitude. +1 +Introduction +The Hubbard model describes the physics of interacting spin-1/2 fermions on the lattice, +and it is one of the most important models in the condensed matter literature. In one space +dimension it is exactly solvable by the Bethe Ansatz [1, 2], enabling the exact computation of +interesting phenomena such as spin-charge separation. The model is integrable and it can be +embedded into the standard framework of the Yang-Baxter equation; this is achieved using the +R-matrix of Shastry [3]. The transport properties of the model have been an object of interest +for many decades (see for example [4]), and research in this direction is still ongoing [5, 6, 7, 8, 9]. +Recently so-called integrable quantum quenches have also been considered in the 1D Hubbard +model [10], using information also about exact overlaps [11, 12]. +The Hubbard model is also important for research on the AdS/CFT correspondence [13]. +It turns out that the R-matrix, which is relevant for the AdS/CFT correspondence, is related +to Shastry’s R-matrix [14, 15]. This remarkable relation shed some new light on the symmetry +algebra of the Hubbard model. It was known for a long time that the Hubbard model exhibits +SU(2) × SU(2) symmetry [16, 17]. By using the map to string theory these could be seen as +coming from a centrally extended superalgebra from which the Hubbard model can be obtained +in a certain limit [18]. Moreover, this observation recently lead to the formulation of the so-called +quantum spectral curve for the Hubbard model [19]. +Over the years, many extensions and generalizations of the Hubbard model appeared, and +many of the models were found to be integrable. Examples include the models found from the +R-matrix of Shastry [20, 21], the models of Bariev and Alcaraz [22] (see also [23]), the Essler- +Korepin-Schoutens model, [24], and multi-component generalizations [25, 26, 27]. +In this paper we consider a new extension of the Hubbard model. Our model belongs to +the class of medium range spin chains: it has next-to-nearest-neighbour interactions and it is +1 + +still integrable. The model depends on two parameters: the Hubbard interaction strength and a +deformation parameter. If both parameters are real, the model is Hermitian. The deformation +violates both the spin and charge conservation, therefore our model is reminiscent of the XYZ +spin chain. Accordingly, we find that the R-matrix is elliptic. However the dependence of the +R-matrix on the spectral parameter in very unusual. +We furthermore find that the new model can be transformed into a spin chain with nearest- +neighbour interactions after applying a certain duality (or bond-site) transformation. This model +is also characterized by two parameters, whose reality determines the Hermiticity of the model. +However, after the transformation there is no direct connection to the Hubbard model. +The paper is structured as follows. In Section 2, we will first briefly discuss the Hubbard +model, with both the fermionic and the bosonic formulations, and the symmetries. After this, in +Section 3 we introduce the 3-site extension of the Hubbard model and show that it is integrable. +In Section 4, we introduce the bond-site transformation and show that our model becomes a new +integrable model with nearest-neighbour interactions. In Section 5 we prove the integrability +properties of our model; the explicit form of the R-matrix is presented in the Appendix A. +Finally, in Section 6 we discuss the large coupling limit of the models. +2 +The Hubbard model +In this section, we give the basic definition of the Hubbard model. We also discuss several +transformations and reformulations to bring it into a form, which is more convenient for our later +purposes. We also briefly discuss the symmetries of the Hubbard model. +Definition +Let us consider a fermionic Hilbert space, with two species of particles which can +be identified with electrons with spin up and down. We use the standard fermionic creation and +annihilation operators (c↑,↓ +j )†, c↑,↓ +j , which satisfy the canonical anti-commutation relations +{cα +j , cβ +k} = 0, +α, β =↑, ↓ +{cα +j , (cβ +k)†} = δα,βδj,k, +(2.1) +where j, k refer to the local Hilbert spaces. +We will also use the local particle number operators nα +j = cα† +j cα +j . The local Hilbert space is +spanned by the four vectors +|∅⟩, +|↑⟩ = (c↑)†|∅⟩, +|↓⟩ = (c↓)†|∅⟩, +|↕⟩ = (c↓)†(c↑)†|∅⟩. +(2.2) +The Hubbard model [2, 28] is defined by the Hamiltonian +H = +� +j +� +(c↑ +j)†c↑ +j+1 + (c↑ +j+1)†c↑ +j + (c↓ +j)†c↓ +j+1 + (c↓ +j+1)†c↓ +j + Un↑ +jn↓ +j +� +, +(2.3) +where U ∈ R is the coupling constant of the model. We will consider the model with both periodic +and free boundary conditions. In the periodic case it is understood that the sum over j runs from +1 to L with the identification L + 1 ≡ 1, whereas in the case of free boundary conditions j runs +from 1 to L − 1. +The model has particle number conservation for both species separately. Hence the Hamilto- +nian commutes with the “total particle number” N and the “total spin” Sz defined as +N = +� +j +n↑ +j + n↓ +j, +Sz = +� +j +n↑ +j − n↓ +j. +(2.4) +2 + +Therefore, it is possible to add two magnetic fields. A convenient choice is to add magnetic +fields so that the interaction term becomes particle/hole symmetric. This choice preserves the +integrability of the model and its explicit form is +H′ = +� +j +� +(c↑ +j)†c↑ +j+1 + (c↑ +j+1)†c↑ +j + (c↓ +j)†c↓ +j+1 + (c↓ +j+1)†c↓ +j + U +4 (1 − 2n↑ +j)(1 − 2n↓ +j) +� +. +(2.5) +This Hamiltonian enjoys SU(2) × SU(2) symmetry; the symmetry properties will be discussed +in more detail below. +Spin chain formulation +For our purposes it is convenient to work with the “bosonic” version +of the model. In order to do this, we perform an (inverse) Jordan-Wigner transformation to +commuting spin chain operators. The operation can be performed in the case of open boundary +conditions. The local Hilbert space is the tensor product +Vj = C2 ⊗ C2 +(2.6) +with the full Hilbert space being the tensor product +V = ⊗L +j=1Vj, +(2.7) +with L the length of the spin chain. Using a standard notation in the literature, we introduce +two sets of Pauli matrices σa and τ a, a = x, y, z that act respectively in the first or in the second +copy of C2. The connection between the operators is +σ− +j = +�j−1 +� +k=1 +(−1)n↑ +k +� +c↑ +j, +τ − +j = +�j−1 +� +k=1 +(−1)n↓ +k +� +c↓ +j, +(2.8) +σ+ +j = (c↑ +j)† +�j−1 +� +k=1 +(−1)n↑ +k +� +, +τ + +j = (c↓ +j)† +�j−1 +� +k=1 +(−1)n↓ +k +� +, +(2.9) +σz +j = 1 − 2n↑ +j, +τ z +j = 1 − 2n↓ +j. +(2.10) +This transforms the Hubbard model Hamiltonian (2.5) to its bosonic formulation +H′′ = +� +j +� +σ+ +j σ− +j+1 + σ− +j σ+ +j+1 + τ + +j τ − +j+1 + τ − +j τ + +j+1 + U +4 σz +j τ z +j +� +, +(2.11) +where U is still the coupling constant of the model. At U = 0 the model describes two independent +XX spin chains which do not interact with each other. +Let us now consider the model with periodic boundary conditions and volume L = 4k, k ∈ N. +In this case, we can perform a similarity transformation by the diagonal operator +D = DσDτ, +(2.12) +with +Dσ = ⊗L +j=1 +� �ij +0 +0 +1 +� +⊗ 12 +� += ik exp +� � +j +iπj +4 σz +j +� +, +(2.13) +Dτ = ⊗L +j=1 +� +12 ⊗ +�ij +0 +0 +1 +� � += ik exp +� � +j +iπj +4 τ z +j +� +, +(2.14) +3 + +12 is the 2 × 2 Identity matrix. +Then we obtain +H1 ≡ D−1H′′D = +� +j +� +hσ +j,j+1 + hτ +j,j+1 + U +4 σz +j τ z +j +� +, +(2.15) +where +hσ +j,j+1 ≡ i +� +σ+ +j σ− +j+1 − σ− +j σ+ +j+1 +� +, +hτ +j,j+1 ≡ i +� +τ + +j τ − +j+1 − τ − +j τ + +j+1 +� +. +(2.16) +The notation H1 for the Hamiltonian signals that the interaction term is a one-site operator. +Later we will also introduce Hamiltonians Hk with k = 2, 3. Our convention will be the same: Hk +is a Hamiltonian where the kinetic term is a standard two-site hopping term, but the interaction +term spans k sites. +The kinetic terms above are known as “Dzyaloshinskii–Moriya interaction” terms [29], which +becomes apparent after the rewriting +hσ +j,j+1 = 1 +2 +� +σx +j σy +j+1 − σy +j σx +j+1 +� +, +(2.17) +and similarly for hτ +j,j+1. These hopping terms are antisymmetric with respect to space reflection. +If the volume L is divisible by 4, then model Hamiltonians (2.11) and (2.15) are completely +equivalent, despite the apparent spatial asymmetry. However, the Hamiltonian (2.15) defines an +integrable model in itself, and we take this model as the starting point of our discussion. +Symmetries +Now we discuss the symmetries of the Hubbard model in more detail, focusing on +the Hamiltonian (2.15). The Hubbard model has both continuous as well as discrete symmetries. +For what follows we would like to introduce the so-called Shiba transformation [2]. It is +defined on a chain of even length L by +Sσ = σy +Lσx +L−1 . . . σy +2σx +1, +Sτ = τ y +Lτ x +L−1 . . . τ y +2 τ x +1 . +(2.18) +A similarity transformation with either Sσ or Sτ preserves the kinetic term of H1, while changing +the sign of the interaction term. Explicitly, +SσH1Sσ = SτH1Sτ = +� +j +� +hσ +j,j+1 + hτ +j,j+1 − U +4 σz +j τ z +j +� +. +(2.19) +As a result, the combination of the two Shiba transformations is a discrete symmetry: +SτSσH1SσSτ = H1. +(2.20) +The Hubbard model Hamiltonian also enjoys invariance under the continuous group SU(2)× +SU(2) [16, 17]. For future reference, let us explicitly work out these symmetries for the Hamil- +tonian (2.15). +The first SU(2) corresponds to rotations in spin space, which can be interpreted also as a +mixing of the σ and τ operators. The generators are local in space if we express them using the +original fermionic variables. However, when we work with the spin variables, the Jordan-Wigner +strings appear. Formally we have +Az = +� +j +σz − τ z +2 +(2.21) +4 + +and +A+ = +� +j +�� � +k 100τ). +Insets: +δ = 0 and δ = 1 on the left and δ = 0.5 on the right. +Solid lines are the theory. +mixture), the thickness is approximately half the +value for δ = 0 as passive disks cannot wet. The +layer thickness exhibits a transient overshoot be- +fore reaching stationarity. +The gap between the +peak of ⟨h⟩(t) and ⟨h⟩(t → ∞) depends on δ. Such +overshoot will be elucidated below. +The evolution of the layer composition is shown +in Figure 2 (see Movie at bit.ly/3GgIuBN). The +layer is always richer in fast particles than the over- +all system. In phase-separation problems, this is +known as “fractionation” [16, 18]. The fractiona- +tion degree, however, is not constant. There is a +first stage where the ratio of fast and slow particles +remains constant, depending on δ. A second slower +stage then starts, in which the composition is finely +adjusted towards the SS. For δ > 0.7, excess slow +particles are eliminated, while for δ < 0.7, addi- +tional slow particles are incorporated. This slow +dynamics occurs simultaneously with changes in + +100 +00 +80 +0 +C +60 +80 +40 +O +8 +20 +0 0 +0 +160 +180 +200 +220 +2403 +⟨h⟩, with both processes being non-monotonic. +FIG. 2. Top: Wetting layer composition evolution for δ +from 0 to 1 every 0.1 and φ = 0.18 (6000 particles in to- +tal), measured by the amount of fast and slow particles +in it. The diagonal corresponds to the same amount of +fast and slow particles. Star symbols show results for +the SS. The colorbar shows time (in a nonlinear scale, +to focus on the final stage). Bottom: Theory. +To model the accumulation dynamics, we first +analyze the spatial distribution of orientations by +computing α(x) ≡ ⟨ˆn · ˆν⟩, shown in Fig. 3, where +ˆn is the inwards normal to the walls and the aver- +age is performed over all particles within a y-axis +stripe of width d0, centered at position x. +Ini- +tially, all particles are randomly oriented, imply- +ing α(x) = 0 everywhere. Later, particles pointing +away from the walls abandon them, leaving regions +close to the walls with particles mostly moving to- +wards them. This manifests as regions of α(x) > 0 +which grow linearly in time; see red and blue lines. +This “cleaning signal” has the mean velocity in the +x direction at which a randomly oriented particle +joins a wall, ⟨v(f/s) +x +⟩ = +� π/2 +−π/2 vf/s cos θdθ/π = +2vf/s +π , +considering only particles moving towards the wall. +At the interface, particles must point towards the +wall as otherwise they escape. Consequently, the +maximum of α(x) independently locates the inter- +face [see Fig. 3 and compare with the thickness +from Fig. 1(b)]. +This maximum in polarization +near the interface, pointing toward the layer, is +consistent with simulations and theoretical predic- +tions for ABPs [38–41]. +0 +250 500 750 1000 +t +0 +20 +40 +60 +80 +100 +120 +distance to wall +0 +250 500 750 1000 +t +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +α +FIG. 3. Spatiotemporal diagram of the self-propulsion +orientation parameter α(x) = ⟨ˆn · ˆν⟩, for fast (left) +and slow (right) particles, with δ = 0.5 and φ = 0.18. +The black dashed lines show the mean thickness data +shown in Fig. 1(b). The dashed red and blue lines are +the “cleaning signals” with velocities ⟨v(f/s) +x +⟩ (see main +text). +Finally, in Fig. 4, we present the stationary con- +centration profiles for slow, fast, and all particles +for various values of speed diversity degree δ. Small +oscillations arise from the fact that, similarly to +the case of molecular fluids, near the walls, par- +ticles accumulate in a series of stable one-particle +layers. +IV. +KINETIC THEORY +To understand the above results, we develop a +simple kinetic theory that estimates the emission +and absorption rates of fast and slow particles, +k(f/s) +out +and k(f/s) +in +, and thus the layer thickness and +composition versus time. For that, we generalize +a previous theory originally developed for systems +without walls [30] to include mixtures (beyond the +simpler approximation for mixtures in Ref. [19]). +Since the global density is low, we use an ideal +gas approximation in the gas (see Fig. 4), i.e., +particles there do not interact. +The rate of ab- +sorption of particles by the layer, i.e., the incom- +ing flux per unit length, is written as k(f/s) +in += +ρ(f/s) +g +2π +� π/2 +−π/2 v(f/s) cos θdθ = +ρ(f/s) +g +v(f/s) +π +, where we in- +tegrate vx over random orientations leading to the +particle entering the layer (on the right without +loss of generality) weighted by the distribution +ρ(f/s) +g +/2π where ρ(f/s) +g +are the gas number densities +in contact with the layer. For a time t∗ +(f/s), ρ(f/s) +g +is approximately constant and equal to the initial + +4 +0 +10 +20 +30 +40 +50 +60 +70 +80 +x distance to wall +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +¡f +± = 0.0 +± = 0.1 +± = 0.2 +± = 0.3 +± = 0.4 +± = 0.5 +± = 0.6 +± = 0.7 +± = 0.8 +± = 0.9 +± = 1.0 +0 +20 +40 +60 +80 +100 +x distance to wall +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +¡f + ¡s +± = 0.0 +± = 0.1 +± = 0.2 +± = 0.3 +± = 0.4 +± = 0.5 +± = 0.6 +± = 0.7 +± = 0.8 +± = 0.9 +± = 1.0 +0 +20 +40 +60 +80 +100 +x distance to wall +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +¡s +± = 0.0 +± = 0.1 +± = 0.2 +± = 0.3 +± = 0.4 +± = 0.5 +± = 0.6 +± = 0.7 +± = 0.8 +± = 0.9 +± = 1.0 +FIG. 4. +Stationary concentration profiles for slow +[φs(x)] (top), fast [φs(x)] (medium), and all [φf(x) + +φs(x)] (bottom) particles for various values of speed +diversity degree δ. The global area fraction is φ = 0.18. +density as the front of non-interacting gas particles +arrives at the layer. Only later, once the “clean- +ing signal” mentioned before has overcome the en- +tire system, ρ(f/s) +g +evolves into the current bulk gas +density obtained from the absorption-evaporation +balance. We estimate t∗ +(f/s) = Lx/⟨v(f/s) +x +⟩. Finally, +ρ(f/s) +g +are assumed to change abruptly at t∗ +(f/s) be- +tween their two values. This approximation, which +leads to abrupt changes in layer growth rate at two +instants, is more accurate for fast particles (see +Fig. 3-left), as the crossover time is smaller and +the rotational diffusion has not significantly acted +yet; for slow particles, the transition is smoother +(see Fig. 3-right). +The SS kout is calculated by solving the diffusion +equation in angular space for P, the distribution +of orientations at the interface, i.e., ∂tP(θ, t) = +η∂2 +θP(θ, t) with absorbing boundaries at ±π/2 and +initial condition given by the distribution of inci- +dent particles, i.e., P(±π/2, t) = 0 and P(θ, 0) = +cos θ/2 (as particles with |θ| ≥ π/2 cannot reach +the wall and those with adequate θ will hit it +with probability proportional to the x-axis veloc- +ity, normalized by integrating between ±π/2). The +solution is P(θ, t) = e−ηt cos θ/2. +For average +diameter σ and identical speeds, one can write +kout ≡ − +˙Ninterface +σNinterface = +η +σ → κη +σ where Ninterface = +� π/2 +−π/2 P(θ, t) is the number of particles at the inter- +face and the dot is the time derivative. The result +is corrected by a factor κ: when a particle escapes, +some inner particles pointing towards the gas fol- +low it in an avalanche-like effect (see Fig. 5 for +an example of such phenomenon). In the SS, the +average number of particles leaving the layer per +escape event is denoted κ = 1 + κexcess. The value +of κexcess is treated as a fitting parameter (Ref. [30] +found that κexcess ≈ 3.5 works well for all studied +v and φ in one-component systems without walls; +in 1D, κexcess = 1 [42]). However, since at early +stages particles in the layer are highly oriented to- +wards the wall (see Fig. 3), avalanche effects be- +come strong only after τ. Before that, once a par- +ticle escapes, other particles are likely to be still +pointing towards the wall and therefore will not +escape. This is incorporated by considering that +κ is time-dependent: κ(t) = 1 + κexcess(1 − e−ηt), +meaning that avalanche events occur with proba- +bility (1 − e−ηt) as particles start to rotate away +from the wall. Finally, with speed diversity, one +has +k(f/s) +out = +N (f/s) +ℓ +N (f) +ℓ ++ N (s) +ℓ +κ(t)η +σ +, +(2) +where +N (f/s) +ℓ +is +the +number +of +particles +of +each type in the layer. +Crucially, the factor +N (f/s) +ℓ +/ +� +N (f) +ℓ ++ N (s) +ℓ +� +, which states that particle +emission is taken as proportional to the fraction +of particles of each type in the layer, nonlinearly +couples the occupations of both types. +The evolution of the parameters involved in +the absorption and emission rates provides the +layer thickness and composition at any time via +dN (f/s) +ℓ +/dt = +� +k(f/s) +in +− k(f/s) +out +� +Ly. Assuming par- +ticle conservation and that the layer is rectan- + +5 +FIG. 5. Sequence of snapshots (from left to right and +top to bottom), showing an avalanche event for δ = 0. +The first escaping particle is shown in red, followed by +a small avalanche of one particle (in light blue). +gular and close-packed with the hard-disk occu- +pied area fraction φcp = π/(2 +√ +3) (as observed +in simulations; see Fig. 4), we obtained a theory +for ⟨h(t)⟩. +A good agreement occurs for δ near +0 and near 1—see left side of inset of Fig. 1(b). +For intermediate δ, the value of t∗ +s grows larger +than the diffusion time and the theory becomes +less good for intermediate times—see right side +of inset of Fig. 1(b). +Also, the theory predicts +that the initial deposition rate is independent of +δ, dNℓ/dt = Ly(ρ0v0/π − η/σ), in agreement with +the simulations [Fig. 1(b)]. Notably, the overshoot +in ⟨h(t)⟩ is well captured, which is not the case if +either the effect of t∗ or the relaxation of κ are not +included in the model. The layer composition evo- +lution is also well captured (Fig. 2-bottom), show- +ing also the two stages found in the simulations. +For δ = 1, the theory predicts no slow particles +are in the layer as they are nonmotile; however, +in simulations the transient concentration of slow +particles is finite, with an ulterior elimination of +them. This difference, also present for δ = 0.9, is +due to an induced accumulation of slow particles +pushed by fast ones, an effect that is neglected by +the ideal gas assumption in the gas. Figure 6 com- +pares theory and simulation for the fraction of slow +particles in the layer. In the inset, this comparison +is shown for the SS layer thickness, with very good +agreement. +V. +CONCLUSIONS +Self-propelled Brownian particles under repul- +sive interactions spontaneously exhibit complete +wetting layer formation in the presence of a flat +wall due to persistent motion. With simulations +and a theory with speed diversity, we calculate and +explain the wetting layer composition and thick- +ness. We reveal a two-stage evolution for the layer +composition and a transient overshoot for the layer +0 +0.2 +0.4 +0.6 +0.8 +1.0 +δ +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +N (s) +ℓ /Nℓ +Transient +Stationary +0 +0.2 +0.4 +0.6 +0.8 +1.0 +δ +0 +5 +10 +15 +20 +25 +⟨h⟩ +FIG. 6. +Ratio between the number of slow parti- +cles and the total number of particles in the layer for +φ = 0.18 as a function of δ. Dashed and solid lines +show the theory for the transient and for the SS, re- +spectively. Triangles and stars show simulation results +for the transient and for the SS, respectively. Inset: +Stationary mean layer thickness. Stars are simulation +and the solid line is the theory. +thickness, explained only when the theory consid- +ers delayed avalanche-like emissions outwards and +a transient front of particles moving towards the +walls. +An implicit assumption of the theory is that no +segregation develops inside the layers [mean-field +approximation in Eq. (2)]. +However, Figs. 1(a) +and 4 indicate that spatial segregation does ex- +ist, with fast particles closer to the wall; a more +detailed analysis is beyond our scope here. Note +that in Ref. 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Sollich, Predicting phase equilibria in poly- +disperse systems, Journal of Physics: Condensed +Matter 14, R79 (2001). + diff --git a/n9AzT4oBgHgl3EQf5P5T/content/tmp_files/load_file.txt b/n9AzT4oBgHgl3EQf5P5T/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0d26381df12a41125bbfcfb3322536740245e11e --- /dev/null +++ b/n9AzT4oBgHgl3EQf5P5T/content/tmp_files/load_file.txt @@ -0,0 +1,542 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf,len=541 +page_content='Wetting dynamics by mixtures of fast and slow self-propelled particles Mauricio Rojas-Vega,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' 2 Pablo de Castro,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='3 and Rodrigo Soto1 1Departamento de F´ısica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' FCFM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Universidad de Chile,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Santiago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Chile 2Institute of Science and Technology Austria,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Klosterneuburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Austria 3ICTP South American Institute for Fundamental Research & Instituto de F´ısica Te´orica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Universidade Estadual Paulista - UNESP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' S˜ao Paulo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Brazil (Dated: January 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' 2023) We study active surface wetting using a minimal model of bacteria that takes into account the intrinsic motility diversity of living matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' A mixture of “fast” and “slow” self-propelled Brownian particles is considered in the presence of a wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' The evolution of the wetting layer thickness shows an overshoot before stationarity and its composition evolves in two stages, equilibrating after a slow elimination of excess particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Non-monotonic evolutions are shown to arise from delayed avalanches towards the dilute phase combined with the emergence of a transient particle front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' INTRODUCTION Natural active matter, such as collections of organisms, is not composed of identical self- propelling agents [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Instead, a wide distribution of motility properties exists due to different ages, reproduction stages, shapes, and sizes [2–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' More- over, active particles typically interact with “sur- faces”, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=', bacteria swimming near boundaries of their host body or of contaminated medical instru- ments [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' For simplicity, models usually ignore at least one of these two ingredients, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=', diversity and surface effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' A persistent particle has a self-propulsion direc- tion that fluctuates stochastically and, typically, slowly [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Consequently, active matter accumu- lates on surfaces to an extent dependent on per- sistence and density [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' For bacteria, this mech- anism, together with other factors, contributes to initiate biofilm formation [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Surface accumula- tion by persistence is called active wetting [10–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Three phases are possible [8]: complete wetting, where the wetting layer covers the wall completely;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' incomplete wetting, where only a fraction of the wall becomes covered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' and “unwetting” or “dry- ing”, where no dense phase exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Active wetting was studied mostly for identical particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' How- ever, passive and active phase behaviors can de- pend strongly on “diversity” in some particle at- tribute [14–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' In this Letter, we study a mixture of “fast” and “slow” active Brownian disks moving in 2D, in the presence of a flat impenetrable wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Each type has its own self-propulsion speed, defining a de- gree of speed diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Besides simulations, a dy- namical kinetic theory is developed by extending the approach of Redner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' [30] in three fronts: to mixtures, to include walls, and to incorporate time-dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' This approach calculates the absorption and emission rates for the agglomer- ate directly from microscopic considerations and is therefore different than free-energy-like approxi- mations [31] or phenomenological theories [32] that can be harder to connect with microscopic proper- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Our theory relies on one fitting parameter only (similarly to Redner’s original theory [30]), which assumes a single value across all parame- ters, somewhat like a “universal constant”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' To iso- late surface effects, we choose a range of densities that allows for significant complete wetting while bulk motility-induced phase separation (MIPS) re- mains absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Instead of focusing on “equilibrium” wetting-drying transitions [8], we study the wet- ting dynamics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=', the mechanisms involved in set- ting the composition and thickness of the wetting layer versus time and how motility diversity af- fects those.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' A two-stage evolution is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' More- over, we identify a transient overshoot of the layer thickness, which occurs even without diversity but whose intensity depends non-monotonically on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' MODEL AND SIMULATION METHOD We consider a binary mixture in 2D composed of N active Brownian disks (labeled by i) where N/2 of them are “fast” particles, with self-propulsion speed vi = vf ≡ v0(1 + δ), and the other N/2 are “slow” particles, with vi = vs ≡ v0(1 − δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Thus δ ∈ [0, 1] is the degree of speed diversity [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Here- after “f” and “s” denote “fast” and “slow” particles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Their dynamics obeys ∂tri = vi ˆνi + µF i + ξi, ∂tθi = ηi(t), (1) where ˆνi = (cos θi, sin θi) is the self-propulsion di- rection, µ is the mobility and F i = � j F ij + F wall i is the net force on particle i due to in- teractions with other particles and with the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' The noise terms ξi(t) and ηi(t) are Gaussian and white, with zero mean and correlations ⟨ξiλ(t)ξjβ(t′)⟩ = 2ξδijδλβδ(t − t′) (the Greek letters denote coordinates) and arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='01856v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='soft] 5 Jan 2023 2 ⟨ηi(t)ηj(t′)⟩ = 2ηδijδ(t − t′), where ξ and η are the translational [34] and rotational diffusion coeffi- cients, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' We model particle interactions by a soft repul- sive WCA-like potential [35], U = 23/2(σij/rij)3 − 3(σij/rij)6 + (σij/rij)12 − 3/4 for rij ≤ 2 1 6 σij and U = 0 otherwise [36], with rij the interparticle distance and σij ≡ 1 2(di + dj), where di is the di- ameter of particle i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' To avoid crystallization [37], each particle is randomly assigned one of two di- ameters, dsmall = d0 or dlarge = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='4d0, uncorrelated with self-propulsion speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' We focus on speed di- versity effects and thus the system is said to be just binary (the observed size segregation is weak).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' We choose v0 = 1, d0 = 1, µ = 1, ξ = 5 × 10−4, η = 5 × 10−3 and the forward Euler method with time step ∆t = 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Initially, positions and ve- locity directions are randomly distributed indepen- dent of types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Figure 1(a) shows the system in the steady state (SS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' The simulation box—which has total dimen- sions Lx = 400 and Ly = 100 and periodic bound- ary conditions in the y direction—is shown only partially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' An impenetrable flat wall (with sides at x = 195 and x = 205) is placed at the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' For particle-wall interactions, the same potential is used with dj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' The occupied area fraction φ is the total area occupied by particles divided by the area of the simulation box minus the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' In all simulations, φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='18, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=', 6000 particles, leading to complete wetting during the whole dy- namics without bulk MIPS, and we focus on vary- ing δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' The average free-particle persistence length is ℓ ≡ v0/η = 200, which is comparable to the system size but sufficiently small to avoid ballis- tic motion between wall sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Thus, each wall is treated independently and we average data from both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' For complete wetting, increasing φ or ℓ trivially increases the layer thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' For SS averages, only configurations after t = 100τ were used, where τ ≡ η−1 = 200 is the rotational diffu- sion time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' WETTING To characterize accumulation, two particles were considered “connected” if rij < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='1σij, allowing us to identify the cluster of connected particles in con- tact with each wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' The mean wetting layer thick- ness ⟨h⟩ is obtained by averaging the position of the outermost particle in each of the 128 bins in which Ly is divided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Figure 1(b) shows ⟨h⟩(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' The initial growth rate is constant and independent of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' However, at long times, the higher the δ, the thinner the SS layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' For δ = 1 (active-passive FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' (a) Snapshot for speed diversity δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='5 and φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='18 in the SS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Fast (slow) particles are in red (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' (b) Temporal evolution of the mean wetting layer thickness for φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='18 and δ from 0 to 1 every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Stars indicate SS averages (t > 100τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Insets: δ = 0 and δ = 1 on the left and δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='5 on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Solid lines are the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' mixture), the thickness is approximately half the value for δ = 0 as passive disks cannot wet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' The layer thickness exhibits a transient overshoot be- fore reaching stationarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' The gap between the peak of ⟨h⟩(t) and ⟨h⟩(t → ∞) depends on δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Such overshoot will be elucidated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' The evolution of the layer composition is shown in Figure 2 (see Movie at bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='ly/3GgIuBN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' The layer is always richer in fast particles than the over- all system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' In phase-separation problems, this is known as “fractionation” [16, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' The fractiona- tion degree, however, is not constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' There is a first stage where the ratio of fast and slow particles remains constant, depending on δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' A second slower stage then starts, in which the composition is finely adjusted towards the SS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' For δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='7, excess slow particles are eliminated, while for δ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='7, addi- tional slow particles are incorporated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' This slow dynamics occurs simultaneously with changes in 100 00 80 0 C 60 80 40 O 8 20 0 0 0 160 180 200 220 2403 ⟨h⟩, with both processes being non-monotonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Top: Wetting layer composition evolution for δ from 0 to 1 every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='1 and φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='18 (6000 particles in to- tal), measured by the amount of fast and slow particles in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' The diagonal corresponds to the same amount of fast and slow particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Star symbols show results for the SS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' The colorbar shows time (in a nonlinear scale, to focus on the final stage).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Bottom: Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' To model the accumulation dynamics, we first analyze the spatial distribution of orientations by computing α(x) ≡ ⟨ˆn · ˆν⟩, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' 3, where ˆn is the inwards normal to the walls and the aver- age is performed over all particles within a y-axis stripe of width d0, centered at position x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Ini- tially, all particles are randomly oriented, imply- ing α(x) = 0 everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Later, particles pointing away from the walls abandon them, leaving regions close to the walls with particles mostly moving to- wards them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' This manifests as regions of α(x) > 0 which grow linearly in time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' see red and blue lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' This “cleaning signal” has the mean velocity in the x direction at which a randomly oriented particle joins a wall, ⟨v(f/s) x ⟩ = � π/2 −π/2 vf/s cos θdθ/π = 2vf/s π , considering only particles moving towards the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' At the interface, particles must point towards the wall as otherwise they escape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Consequently, the maximum of α(x) independently locates the inter- face [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' 3 and compare with the thickness from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' 1(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' This maximum in polarization near the interface, pointing toward the layer, is consistent with simulations and theoretical predic- tions for ABPs [38–41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' 0 250 500 750 1000 t 0 20 40 60 80 100 120 distance to wall 0 250 500 750 1000 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='0 α FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Spatiotemporal diagram of the self-propulsion orientation parameter α(x) = ⟨ˆn · ˆν⟩, for fast (left) and slow (right) particles, with δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='5 and φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' The black dashed lines show the mean thickness data shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' The dashed red and blue lines are the “cleaning signals” with velocities ⟨v(f/s) x ⟩ (see main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Finally, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' 4, we present the stationary con- centration profiles for slow, fast, and all particles for various values of speed diversity degree δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Small oscillations arise from the fact that, similarly to the case of molecular fluids, near the walls, par- ticles accumulate in a series of stable one-particle layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' KINETIC THEORY To understand the above results, we develop a simple kinetic theory that estimates the emission and absorption rates of fast and slow particles, k(f/s) out and k(f/s) in , and thus the layer thickness and composition versus time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' For that, we generalize a previous theory originally developed for systems without walls [30] to include mixtures (beyond the simpler approximation for mixtures in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' [19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Since the global density is low, we use an ideal gas approximation in the gas (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' 4), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=', particles there do not interact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' The rate of ab- sorption of particles by the layer, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=', the incom- ing flux per unit length, is written as k(f/s) in = ρ(f/s) g 2π � π/2 −π/2 v(f/s) cos θdθ = ρ(f/s) g v(f/s) π , where we in- tegrate vx over random orientations leading to the particle entering the layer (on the right without loss of generality) weighted by the distribution ρ(f/s) g /2π where ρ(f/s) g are the gas number densities in contact with the layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' For a time t∗ (f/s), ρ(f/s) g is approximately constant and equal to the initial 4 0 10 20 30 40 50 60 70 80 x distance to wall 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='0 ¡f ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='0 ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='1 ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='2 ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='3 ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='4 ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='5 ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='6 ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='7 ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='8 ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='9 ± = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='0 0 20 40 60 80 100 x distance to wall 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='0 ¡f + ¡s ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='0 ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='1 ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='2 ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='3 ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='4 ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='5 ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='6 ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='7 ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='8 ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='9 ± = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='0 0 20 40 60 80 100 x distance to wall 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='0 ¡s ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='0 ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='1 ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='2 ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='3 ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='4 ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='5 ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='6 ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='7 ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='8 ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='9 ± = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Stationary concentration profiles for slow [φs(x)] (top), fast [φs(x)] (medium), and all [φf(x) + φs(x)] (bottom) particles for various values of speed diversity degree δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' The global area fraction is φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' density as the front of non-interacting gas particles arrives at the layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Only later, once the “clean- ing signal” mentioned before has overcome the en- tire system, ρ(f/s) g evolves into the current bulk gas density obtained from the absorption-evaporation balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' We estimate t∗ (f/s) = Lx/⟨v(f/s) x ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Finally, ρ(f/s) g are assumed to change abruptly at t∗ (f/s) be- tween their two values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' This approximation, which leads to abrupt changes in layer growth rate at two instants, is more accurate for fast particles (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' 3-left), as the crossover time is smaller and the rotational diffusion has not significantly acted yet;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' for slow particles, the transition is smoother (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' 3-right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' The SS kout is calculated by solving the diffusion equation in angular space for P, the distribution of orientations at the interface, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=', ∂tP(θ, t) = η∂2 θP(θ, t) with absorbing boundaries at ±π/2 and initial condition given by the distribution of inci- dent particles, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=', P(±π/2, t) = 0 and P(θ, 0) = cos θ/2 (as particles with |θ| ≥ π/2 cannot reach the wall and those with adequate θ will hit it with probability proportional to the x-axis veloc- ity, normalized by integrating between ±π/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' The solution is P(θ, t) = e−ηt cos θ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' For average diameter σ and identical speeds, one can write kout ≡ − ˙Ninterface σNinterface = η σ → κη σ where Ninterface = � π/2 −π/2 P(θ, t) is the number of particles at the inter- face and the dot is the time derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' The result is corrected by a factor κ: when a particle escapes, some inner particles pointing towards the gas fol- low it in an avalanche-like effect (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' 5 for an example of such phenomenon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' In the SS, the average number of particles leaving the layer per escape event is denoted κ = 1 + κexcess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' The value of κexcess is treated as a fitting parameter (Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' [30] found that κexcess ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='5 works well for all studied v and φ in one-component systems without walls;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' in 1D, κexcess = 1 [42]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' However, since at early stages particles in the layer are highly oriented to- wards the wall (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' 3), avalanche effects be- come strong only after τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Before that, once a par- ticle escapes, other particles are likely to be still pointing towards the wall and therefore will not escape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' This is incorporated by considering that κ is time-dependent: κ(t) = 1 + κexcess(1 − e−ηt), meaning that avalanche events occur with proba- bility (1 − e−ηt) as particles start to rotate away from the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Finally, with speed diversity, one has k(f/s) out = N (f/s) ℓ N (f) ℓ + N (s) ℓ κ(t)η σ , (2) where N (f/s) ℓ is the number of particles of each type in the layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Crucially, the factor N (f/s) ℓ / � N (f) ℓ + N (s) ℓ � , which states that particle emission is taken as proportional to the fraction of particles of each type in the layer, nonlinearly couples the occupations of both types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' The evolution of the parameters involved in the absorption and emission rates provides the layer thickness and composition at any time via dN (f/s) ℓ /dt = � k(f/s) in − k(f/s) out � Ly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Assuming par- ticle conservation and that the layer is rectan- 5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Sequence of snapshots (from left to right and top to bottom), showing an avalanche event for δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' The first escaping particle is shown in red, followed by a small avalanche of one particle (in light blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' gular and close-packed with the hard-disk occu- pied area fraction φcp = π/(2 √ 3) (as observed in simulations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' 4), we obtained a theory for ⟨h(t)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' A good agreement occurs for δ near 0 and near 1—see left side of inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' For intermediate δ, the value of t∗ s grows larger than the diffusion time and the theory becomes less good for intermediate times—see right side of inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Also, the theory predicts that the initial deposition rate is independent of δ, dNℓ/dt = Ly(ρ0v0/π − η/σ), in agreement with the simulations [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' 1(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Notably, the overshoot in ⟨h(t)⟩ is well captured, which is not the case if either the effect of t∗ or the relaxation of κ are not included in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' The layer composition evo- lution is also well captured (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' 2-bottom), show- ing also the two stages found in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' For δ = 1, the theory predicts no slow particles are in the layer as they are nonmotile;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' however, in simulations the transient concentration of slow particles is finite, with an ulterior elimination of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' This difference, also present for δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='9, is due to an induced accumulation of slow particles pushed by fast ones, an effect that is neglected by the ideal gas assumption in the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Figure 6 com- pares theory and simulation for the fraction of slow particles in the layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' In the inset, this comparison is shown for the SS layer thickness, with very good agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' CONCLUSIONS Self-propelled Brownian particles under repul- sive interactions spontaneously exhibit complete wetting layer formation in the presence of a flat wall due to persistent motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' With simulations and a theory with speed diversity, we calculate and explain the wetting layer composition and thick- ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' We reveal a two-stage evolution for the layer composition and a transient overshoot for the layer 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='0 δ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='5 N (s) ℓ /Nℓ Transient Stationary 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='0 δ 0 5 10 15 20 25 ⟨h⟩ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Ratio between the number of slow parti- cles and the total number of particles in the layer for φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='18 as a function of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Dashed and solid lines show the theory for the transient and for the SS, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Triangles and stars show simulation results for the transient and for the SS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Inset: Stationary mean layer thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Stars are simulation and the solid line is the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' thickness, explained only when the theory consid- ers delayed avalanche-like emissions outwards and a transient front of particles moving towards the walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' An implicit assumption of the theory is that no segregation develops inside the layers [mean-field approximation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' (2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' However, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' 1(a) and 4 indicate that spatial segregation does ex- ist, with fast particles closer to the wall;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' a more detailed analysis is beyond our scope here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Note that in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' [19], for a much denser case showing bulk MIPS (φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content='6), the opposite is seen: faster particles accumulate at cluster boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Since bacteria located inside thick layers may be protected, our work shows how biological variabil- ity of motility properties can play a central role in determining the survivability of microorganisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' More broadly, our results provide important in- sights into the behavior of active matter such as the origin of swim pressure overshoots previously seen in confined systems [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' Furthermore, our frame- work can be adapted to study bacterial types com- peting to colonize niches in confined systems [44] as well as the puzzling formation of multi-cellular ag- gregates such as ameboid slime mold, where slower cells hijack the motion of faster cells to move fur- ther and spread their spores at low energy cost [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' C 800 806 ACKNOWLEDGMENTS MR-V and RS are supported by Fondecyt Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} +page_content=' 1220536 and ANID – Millennium Science Ini- tiative Program – NCN19 170D, Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQf5P5T/content/2301.01856v1.pdf'} 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MOHAN, AND RENHAI WANG +Abstract. This article is concerned with the asymptotically autonomous robustness (almost surely and in proba- +bility) of non-autonomous random attractors for two stochastic versions of convective Brinkman-Forchheimer (CBF) +equations defined on the whole space Rd: +∂v +∂t − µ∆v + (v · ∇)v + αv + β|v|r−1v + ∇p = f(t) + “stochastic terms”, +∇ · v = 0, +with initial and boundary vanishing conditions, where d = 2, 3, µ, α, β > 0, r ≥ 1 and f(t) is a given time- +dependent external force field. By the asymptotically autonomous robustness of a non-autonomous random attractor +A = {A (τ, ω) : τ ∈ R, ω ∈ Ω} we mean its time-section A (τ, ω) is robust to a time-independent random set as +time τ tends to negative infinity according to the Hausdorff semi-distance of the underlying space. Our goal is to +study this topic, almost surely and in probability, for the non-autonomous CBF equations when the stochastic term +is a linear multiplicative or additive noise, and the time-dependent forcing converges towards a time-independent +function. Our main results contain three cases: i) d = 2 and r ∈ {1} ∪ [2, ∞); ii) d = 3 and r ∈ (3, ∞); iii) +d = 3, r = 3 and 2βµ ≥ 1. The main procedure to achieve our goal is how to justify that the usual pullback +asymptotic compactness of the solution operators is uniform on some uniformly tempered universes over an infinite +time-interval (−∞, τ]. +This can be done by a method based on Kuratowski’s measure of noncompactness by +showing the backward uniform “tail-smallness” and “flattening-property” of the solutions over (−∞, τ] in order to +overcome the lack of compact Sobolev embeddings on unbounded domains. Several rigorous calculations dealing +the pressure term p and the fast growing term β|v|r−1v play key role in the whole analysis. When α = β = 0, +the present result can be viewed as a generation of the authors’s recent work [73] for the standard Navier-Stokes +equations on unbounded Poincar´e domains. +1. Introduction +1.1. The model. In this article, we consider a stochastic fluid dynamic model concerning the convective +Brinkman-Forchheimer (CBF) equation driven by stochastic and non-autonomous forcing simultaneously +defined on the whole space Rd (d = 2, 3): +(1.1) + + + + + + + + + + + + + + + +∂v +∂t − µ∆v + (v · ∇)v + αv + β|v|r−1v + ∇p = f + S(v) ◦ dW(t) +dt +, in Rd × (τ, ∞), +∇ · v = 0, +in Rd × (τ, ∞), +v(x)|t=τ = v0(x), +x ∈ Rd and τ ∈ R, +v(x)|t=τ → 0, +as |x| → ∞, +2020 Mathematics Subject Classification. Primary 37L55; Secondary 37B55, 35B41, 35B40. +Key words and phrases. Asymptotically autonomous robustness, pullback random attractor, stochastic convective Brinkman- +Forchheimer equations, backward uniform-tail estimate, backward flattening-property. +1 + +2 +KUSH KINRA, MANIL T. MOHAN, AND RENHAI WANG +where v(x, t) ∈ Rd, p(x, t) ∈ R and f(x, t) ∈ Rd represent the velocity field, pressure field and external +forcing, respectively. The constants µ, α, β > 0 stand for the Brinkman (effective viscosity), Darcy (perme- +ability of the porous medium) and Forchheimer coefficients, respectively. Note that r ∈ [1, ∞) is called the +absorption exponent and r = 3 is called the critical absorption exponent. The one-dimensional two-sided +Wiener W(t) is defined on a probability space (Ω, F, P) (see Subsection 2.2). The diffusion coefficient of +the noise is S(v) = v (multiplicative noise) or independent of v (additive noise). The symbol ◦ means that +the stochastic integral should be understood in the sense of Stratonovich. +The CBF equations are also referred to as the tamed Navier-Stokes equations with a modified damping +term αv +β|v|r−1v. If α = β = 0, then the system (1.1) is reduced to the standard Navier-Stokes equations. +It has been proved by Hajduk and Robinson [31, Proposition 1.1] that the CBF equations and the Navier- +Stokes equations have the same scaling only when r = 3 and α = 0, but have no scaling invariance for other +values of α and r. From the physical point of views, system (1.1) is applied to the flows when the velocities +are sufficiently high and porosities are not too small, that is, when the Darcy law for a porous medium does +not apply, see [47]. In this case, system (1.1) is also referred as the non-Darcy model. +1.2. Literature results for CBF equations. For deterministic 2D/3D CBF equations, the existence and +uniqueness of weak/strong solutions on bounded, periodic and unbounded domains has been investigated +in [1, 26, 31, 47, 48]; the existence, uniqueness, regularity, stability and Hausdorff/fractal dimension of +global/pullback/exponential attractors have studied in [?, 37, 50, 51] and the references therein. For sto- +chastic 2D/3D CBF equations, the existence of a global in time pathwise mild solution was justified in [52] +when the equations are defined on the whole space and driven by fractional Brownian noise; the existence +and uniqueness of strong solutions (in probabilistic sense) was established in [34] when the equations are +defined on unbounded Poincar´e domains and forced by Gaussian noise. +However, similar to the 3D Navier-Stokes equations, the existence of the unique global weak solution and +unique pathwise strong solution of 3D deterministic and stochastic CBF equations are still open problems +for r ∈ [1, 3) with β, µ > 0, and r = 3 with 2βµ < 1. +1.3. Literature survey for random attractors. The theory of various types of attractors, such as +global,pullback,exponential and trajectory attractors, of deterministic dynamical systems has been exten- +sively studied in [3, 4, 10, 11, 16, 17, 18, 56, 57, 60] and many others. For the long term dynamics of +stochastic ordinary/partial differential equations which generates a random dynamical system (RDS) [2], +the extension of global attractors to the random attractors was introduced in [5, 20, 21, 58] and success- +fully applied to 2D stochastic Navier-Stokes equations and other stochastic equations. Since the evolution +equations arriving from physics and other fields of science are often driven by non-autonomous and sto- +chastic forcing simultaneously, random attractors of autonomous RDS are generalized to pullback random +attractors in [65] under the framework of non-autonomous RDS. In view of these abstract results, there +is a large number of literature on the random attractors for autonomous and non-autonomous stochastic + +ASYMPTOTIC AUTONOMY OF RANDOM ATTRACTORS FOR SCBF EQUATIONS +3 +equations, see [6, 14, 15, 29, 37, 38, 39, 64, 73, 74, 75]. As per the existing literature, the existence of ran- +dom attractors for stochastic systems is based on some transformation which converts the stochastic system +into a pathwise deterministic system. This transformation is available in the literature only when the noise +is either linear multiplicative or additive one, see [6, 27, 37, 43, 64]. In order to deal with the nonlinear +diffusion term of the noise, the concept of mean random attractors was introduced in [66] and applied to +stochastic Navier-Stokes and CBF equations (Itˆo sense) with Lipschitz nonlinear diffusion term in [67] and +[36], respectively, see [68, 70] for other physically relevant stochastic models. Another different approach in +the direction of random attractors, when the diffusion term is nonlinear, is the Wong-Zakai approximation +of pathwise random attractors, see [29, 30, 38, 75] and the references therein. +1.4. Motivation, assumptions and main results. In general, a non-autonomous random attractor car- +ries the form Aς = {Aς(τ, ω) : τ ∈ R, ω ∈ Ω}, where ς stands for some external perturbation parameter. +In the literature, the robustness of pullback random attractors of stochastic CBF equations have been +established in [37, 38] with respect to the external parameter ς. For the robustness with respect to the ex- +ternal/internal parameters of pullback random attractors of 2D stochastic Navier-Stokes, we refer interested +readers to the works [22, 29, 30, 39, 73] and the references therein. Currently, the questions of robustness +of pullback random attractors of stochastic CBF equations defined on unbounded domains with respect to +the internal parameter τ, however, is still unsolved. +As per our expectations, if the time-dependent forcing term f(x, t) converges to some time-independent +forcing term f ∞(x) in some sense, the non-autonomous random dynamics of the system (1.1) becomes more +and more autonomous. Our main motivation is to examine the asymptotically autonomous robustness of +pullback random attractors of (1.1) when the parameters in (1.1) are discussed in the following three cases. +Cases +d +r +conditions on µ & β +I +d = 2 +r ∈ {1} ∪ [2, ∞) +for any µ > 0 and β > 0 +II +d = 3 +r ∈ (3, ∞) +for any µ > 0 and β > 0 +III +d = 3 +r = 3 +for µ > 0 and β > 0 with 2βµ ≥ 1 +Table 1. Values of µ, β and r for d = 2, 3. +Assumption 1.1. f ∈ L2 +loc(R; L2(Rd)) converges to f∞ ∈ L2(Rd) : +lim +τ→−∞ +� τ +−∞ +∥f(t) − f ∞∥2 +L2(Rd)dt = 0. +(1.2) +Moreover, f(·, ·) satisfies +sup +s≤τ +� s +−∞ +eκ(t−s)∥f(t)∥2 +L1(Rd)dt < +∞, +∀ κ > 0, τ ∈ R. +(1.3) +Theorem 1.2 (Multiplicative noise case). Let Assumption 1.1 be satisfied. Then, for all the cases +given in Table 1, the non-autonomous RDS Φ generated by (1.1) with S(v) = v has a unique pullback + +4 +KUSH KINRA, MANIL T. MOHAN, AND RENHAI WANG +random attractor A = {A (τ, ω) : τ ∈ R, ω ∈ Ω} such that +� +s∈(−∞,τ] +A (s, ω) is precompact in H (the +definition of H is given below, see Section 2.1) and +lim +t→+∞ e−γt +sup +s∈(−∞,τ] +∥A (s−t, ϑ−tω)∥H = 0, for any γ > 0, +τ ∈ R and ω ∈ Ω. In addition, the time-section A (τ, ω) is asymptotically autonomous robust in H, and the +limiting set of A (τ, ω) as τ → −∞ is just determined by the random attractor A∞ = {A (ω) : ω ∈ Ω} of +stochastic CBF equations (1.1) with the autonomous forcing f ∞, that is, +lim +τ→−∞ distH(A (τ, ω), A∞(ω)) = 0, P-a.s. ω ∈ Ω. +(1.4) +Moreover, the asymptotically autonomous robustness in probability is also justified: +lim +τ→−∞ P +� +ω ∈ Ω : distH(A (τ, ω), A∞(ω)) ≥ δ +� += 0, +∀ δ > 0. +(1.5) +In addition, for any ε > 0 and sequence τn → −∞, there exists Ωε ∈ F with P(Ωε) > 1 − ε such that +lim +n→∞ sup +ω∈Ωε +distH(A (τn, ω), A∞(ω)) = 0. +(1.6) +Theorem 1.3 (Additive noise case). Under the Assumption 1.1 and for all the cases given in Table 1 +(excluding d = 2 with r = 1), all results in Theorem 1.2 hold for the non-autonomous RDS generated by +(1.1) with S(v) = g with g ∈ D(A), where D(A) is the domain of the Stokes operator A defined in (2.1). +Remark 1.4. (i) An example of Assumption 1.1 is f(x, t) = f∞(x)et +f ∞(x) with f ∞ ∈ L2(Rd)∩L1(Rd). +(ii) Assumption 1.1 implies the following conditions (cf. [9]): +Uniform integrability: +sup +s≤τ +� s +−∞ +eκ(ξ−s)∥f(ξ)∥2 +L2(Rd)dξ < +∞, ∀ κ > 0, τ ∈ R, +(1.7) +Uniform tails-smallness: +lim +k→∞ sup +s≤τ +� s +−∞ +eκ(ξ−s) +� +|x|≥k +|f(x, ξ)|2dxdξ = 0, ∀ κ > 0, τ ∈ R. +(1.8) +(iii) We only use Assumption 1.1 for f in the whole paper. +(iv) In Poincar´e domains (bounded or unbounded), we can relax the condition (1.3) (see [73]). +(iv) Due to technical difficulties, we are not able to establish the present results for d = 2 and r ∈ (1, 2). +(v) In the additive noise case, we do not need to assume, as in [73, Hypothesis 1.3], that there exists a +constant ℵ > 0 such that g ∈ D(A) satisfies +���� +d +� +i,j=1 +� +Rd vi(x)∂gj(x) +∂xi +vj(x)dx +���� ≤ ℵ∥v∥2 +L2(Rd), +∀ v ∈ L2(Rd). +(1.9) +1.5. Novelties, difficulties and approaches. In order to prove Theorems 1.2 and 1.3, the uniform pre- +compactness of +� +s∈(−∞,τ] +A (s, ω) in H is a pivotal point. The well-known abstract theory of pullback random +attractors from [65] tells us that the pullback asymptotic compactness of Φ gives the compactness of A (τ, ω) +for each τ ∈ R, but it cannot provide the precompactness of +� +s∈(−∞,τ] +A (s, ω) in H, since (−∞, τ] is an in- +finite interval. However, motivated by the ideas of [65], this can be done if one is able to show that the + +ASYMPTOTIC AUTONOMY OF RANDOM ATTRACTORS FOR SCBF EQUATIONS +5 +usual pullback asymptotic compactness of Φ is uniform with respect to a uniformly tempered universe (see +(2.13)) over (−∞, τ]. +Note that in the bounded domain case, one can obtain the uniform pullback asymptotic compactness of +Φ over (−∞, τ] via a compact uniform pullback absorbing set by using compact Sobolev embeddings (see +[39, Theorem 3.10]). Moreover, the same idea is used for several stochastic Navier-Stokes, g-Navier-Stokes, +magneto-hydrodynamics, Brinkman-Forchheimer equations on bounded domains, see [39, 44, 72, 76]. Due +to the lack of compact Sobolev embeddings in unbounded domains as considered in the present work, to +demonstrate such backward uniform pullback asymptotic compactness is therefore harder than that in the +bounded domain case. We mention that the criteria of Kuratowski’s measure of noncompactness ([41, 55]) is +useful to resolve the difficulty created by the noncompactness of Sobolev embeddings on unbounded domains +(cf. [42, Lemma 2.7]). In order to apply such criteria, we use the idea of uniform tail-estimates introduced +by Wang [62] and flattening-properties introduced by Ma et. al. [46](deterministic case) and Kloeden and +Langa [33](random case). Using the cut-off technique, we show that the solutions of (1.1) are sufficiently +small in L2(Oc +k) uniformly over (−∞, τ], when k is large enough, where Ok = {x ∈ Rd : |x| ≤ k} and +Oc +k = Rd \ Ok, that is, we obtain the backward uniform tail-estimates for the solutions. Furthermore, using +the same cut-off function, we can also establish the backward flattening-properties of the solutions. +Note that parabolic and hyperbolic stochastic models as considered in the works [12, 13, 9, 19, 42, 59, 62, +69] etc., do not contain pressure term p. But some physically relevant models such as Navier-Stokes (cf. [73]), +Brinkman-Forchheimer equations (cf. [76]) and many others, contain the pressure term p. While proving +the backward uniform tail-estimates as well as backward flattening-properties of the solutions, when we take +a suitable inner product, the pressure term p does not vanish with the help of divergence free condition (or +incompressibility condition) of the solutions of (1.1). However, by taking the divergence in (1.1) formally +and using the divergence free condition, we end up with the rigorous expression of the pressure term +p = (−∆)−1 +� +d +� +i,j=1 +∂2 +∂xi∂xj +(vivj) + ∇ · {|v|r−1v} − ∇ · f +� +, +(1.10) +in the weak sense, which is the most difficult term to handle in an appropriate way. Then it is possible +to obtain these backward uniform tail-estimates as well as backward flattening-property with the help of +Gagliardo-Nirenberg ([54, Theorem 1]. In this paper it has been used very carefully in each case to get +appropriate estimates by using H¨older, interpolation and Young inequalities (cf. Lemmas 3.8-3.9 and 4.7- +4.8). +It is worth mentioning here that we are able to prove the backward uniform tail-estimate as well as the +backward flattening-property for d = 2 and d = 3 with r ∈ {1} ∪ [2, ∞) and r ∈ [3, ∞), respectively. But +establishing the backward uniform tail-estimate as well as the backward flattening-properties for d = 2 with +1 < r < 2 on the whole space is still not yet resolved (that is, the difficulty in estimating the pressure term +(1.10) on the whole space is not resolved for d = 2 with 1 < r < 2). Note that one can estimate the pressure +term (1.10) for d = 2 with 1 < r < 2 on unbounded Poincar´e domains O by using the elliptic regularity (cf. +[35, Lemma 6.5]). As a result of these backward uniform tail-estimates and backward flattening-property + +6 +KUSH KINRA, MANIL T. MOHAN, AND RENHAI WANG +of the solutions to (1.1), the backward uniform pullback asymptotic compactness of Φ in H follows. The +wide-spread idea of energy equations introduced in [3] can be used to overcome the noncompactness of +Sobolev embeddings on unbounded domains, see the works [6, 7, 29, 38, 63, 64, 71], etc. and many others. +A remark is that we are currently unable to use the idea of energy equations to prove the backward uniform +pullback asymptotic compactness of Φ in H since (−∞, τ] is an infinite time-interval. +Since we have to consider the uniformly tempered universe to prove the backward uniform pullback +asymptotic compactness of Φ, we shall establish the measurability of the uniformly compact attractor. This +is not straightforward compared with the usual case since the radii of the uniform pullback absorbing set +is taken as the supremum over an uncountable set (−∞, τ] (see Proposition 3.7). In order to overcome the +difficulty, we first observe that the measurability of the usual random attractor is known in the literature, +see for example, [6, 7, 29, 65], etc., and then prove that such a uniformly compact attractor is just equal +to the usual random attractor. This idea has been successfully used by the authors in [9, 72, 73] etc., for +different stochastic models. +1.6. Advantages of the damping term. CBF equations are also known as damped Navier-Stokes equa- +tions (cf. [32]). The damping arises from the resistance to the motion of the flow or by friction effects. +Due to the presence of the damping term αv + β|v|r−1v, we are able to establish better results than which +are available for the Navier-Stokes equations. The existence of global as well as random attractors for the +Navier-Stokes equations on the whole space or general unbounded domains is an interesting and challeng- +ing open problem. In the literature, for Navier-Stokes equations, these types of results are available on +unbounded Poincar´e domains only (cf. [39, 73]). For 2D Navier-Stokes equations forced by a linear mul- +tiplicative noise, we refer to [40]. For stochastic CBF equations (1.1), we are considering the whole space, +where the linear damping term αv plays a crucial role to establish the required results on the whole space. +This is different from the 2D Navier-Stokes equations on unbounded Poincar´e domains, see [73]. +1.7. Outline of the article. In the next section, we provide the necessary function spaces and abstract +formulation of (1.1), and discuss the Ornstein-Uhlenbeck process with its properties. In Section 3, we prove +Theorem 1.2 for the system (1.1) driven by multiplicative noise. In the final section, we prove Theorem 1.3 +for the problem (1.1) driven by additive noise. +2. Mathematical formulation +We start this section with some necessary function spaces whose elements satisfy the divergence free +conditions, that is, ∇ · v = 0. Next, in order to obtain the abstract formulation of the system (1.1), we +define linear, bilinear and nonlinear operators along with their properties. Finally, we discuss the Ornstein- +Uhlenbeck process with some of its properties and the backward tempered random sets. +2.1. Function spaces and operators. Let C∞ +0 (Rd; Rd) denote the space of all Rd-valued, infinitely dif- +ferentiable functions with compact support in Rd. Let Ls(Rd) := Ls(Rd; Rd) and Hk(Rd) := Hk(Rd; Rd) for + +ASYMPTOTIC AUTONOMY OF RANDOM ATTRACTORS FOR SCBF EQUATIONS +7 +s ∈ [2, ∞) and k ∈ N. Define the spaces +H := {v ∈ C∞ +0 (Rd; Rd) : ∇ · v = 0} +L2(Rd), +V := {v ∈ C∞ +0 (Rd; Rd) : ∇ · v = 0} +H1(Rd), +�Lp := {v ∈ C∞ +0 (Rd; Rd) : ∇ · v = 0} +Lp(Rd), +p > 2. +The spaces H, V and �Lp are endowed with the norms +∥v∥2 +H := +� +Rd |v(x)|2dx, ∥v∥2 +V = +� +Rd |v(x)|2dx + +� +Rd |∇v(x)|2dx and ∥v∥p +�Lp := +� +Rd |v(x)|pdx, +for p ∈ (2, ∞), respectively. The inner product in the Hilbert space H is represented by (·, ·). The duality +pairing between the spaces V and V′, and �Lp and its dual �L +p +p−1 is denoted by ⟨·, ·⟩. Also, the space H can +be identified with its own dual H′. We endow the space V ∩ �Lp with the norm ∥v∥V + ∥v∥�Lp, for v ∈ V ∩ �Lp +and its dual V′ + �Lp′ with the norm (cf. [24, Subsection 2.1]) +inf +� +∥u1∥V′ + ∥u2∥�Lp′ : u = u1 + u2, u1 ∈ V′, u2 ∈ �Lp′� +. +2.1.1. Linear operator. Let P : L2(Rd) → H be the Helmholtz-Hodge (or Leray) projection. Note that the +projection operator P can be expressed in terms of the Riesz transform (cf. [53]). We define the Stokes +operator +(2.1) +Av := −P∆v, v ∈ D(A) := V ∩ H2(Rd). +Moreover, P and ∆ commutes in Rd, that is, P∆ = ∆P. +2.1.2. Bilinear operator. Let us define the trilinear form b(·, ·, ·) : V × V × V → R by +b(v1, v2, v3) = +� +Rd(v1(x) · ∇)v2(x) · v3(x)dx = +d +� +i,j=1 +� +Rd v1,i(x)∂v2,j(x) +∂xi +v3,j(x)dx. +If v1, v2 are such that the linear map b(v1, v2, ·) is continuous on V, the corresponding element of V′ is +denoted by B(v1, v2). We also denote B(v) = B(v, v) = P[(v · ∇)v]. An integration by parts yields +(2.2) +�b(v1, v2, v2) = 0, +for all v1, v2 ∈ V, +b(v1, v2, v3) = −b(v1, v3, v2), +for all v1, v2, v3 ∈ V. +Remark 2.1 ([61, Chapter 2, Section 2.3]). For all v1, v2, v3 ∈ V, +|b(v1, v2, v3)| ≤ C × +� +∥v1∥1/2 +H ∥∇v1∥1/2 +H ∥∇v2∥H∥v3∥1/2 +H ∥∇v3∥1/2 +H , +for d = 2, +∥v1∥1/4 +H ∥∇v1∥3/4 +H ∥∇v2∥H∥v3∥1/4 +H ∥∇v3∥3/4 +H , +for d = 3. +(2.3) +Remark 2.2. Note that ⟨B(v1, v1 − v2), v1 − v2⟩ = 0 (for all v1, v2 ∈ V) gives us +(2.4) +⟨B(v1) − B(v2), v1 − v2⟩ = ⟨B(v1 − v2, v2), v1 − v2⟩ = −⟨B(v1 − v2, v1 − v2), v2⟩. + +8 +KUSH KINRA, MANIL T. MOHAN, AND RENHAI WANG +2.1.3. Nonlinear operator. Let us consider the nonlinear operator C(v) := P(|v|r−1v), for v ∈ V ∩ �Lr+1. +The map C(·) : V ∩ �Lr+1 → V′ + �L +r+1 +r +and ⟨C(v), v⟩ = ∥v∥r+1 +�Lr+1. +Remark 2.3. For any v1, v2 ∈ V ∩ �Lr+1, we have (cf. [49, Subsection 2.4]) +⟨C(v1) − C(v2), v1 − v2⟩ ≥ 1 +2∥|v1| +r−1 +2 (v1 − v2)∥2 +H + 1 +2∥|v2| +r−1 +2 (v1 − v2)∥2 +H ≥ 0, +for all r ≥ 1. +(2.5) +2.2. Abstract formulation and Ornstein-Uhlenbeck process. By taking the projection P on the +SCBF equations (1.1), we obtain the following abstract formulation by linear, bilinear and nonlinear oper- +ators: +(2.6) + + + +dv +dt + µAv + B(v) + αv + βC(v) = Pf + S(v) ◦ dW +dt , +t > τ, +v(x)|t=τ = vτ(x), +x ∈ Rd, +where S(v) = v (multiplicative noise) or S(v) is independent of v (additive noise). Here, the symbol ◦ +represents that the stochastic integral is understood in the sense of Stratonovich and W(t, ω) is the standard +scalar Wiener process on the probability space (Ω, F, P), where Ω = {ω ∈ C(R; R) : ω(0) = 0}, endowed +with the compact-open topology given by the metric +dΩ(ω, ω′) := +∞ +� +m=1 +1 +2m +∥ω − ω′∥m +1 + ∥ω − ω′∥m +, where ∥ω − ω′∥m := +sup +−m≤t≤m +|ω(t) − ω′(t)|, +F is the Borel sigma-algebra induced by the compact-open topology of (Ω, dΩ) and P is the two-sided +Wiener measure on (Ω, F). From [28], it is clear that the measure P is ergodic and invariant under the +translation-operator group {ϑt}t∈R on Ω defined by +ϑtω(·) = ω(· + t) − ω(t), +for all t ∈ R, ω ∈ Ω. +The operator ϑ(·) is known as Wiener shift operator. +2.2.1. Ornstein-Uhlenbeck process. Consider for some σ > 0 +y(ϑtω) = +� t +−∞ +e−σ(t−ξ)dW(ξ), +ω ∈ Ω, +(2.7) +which is the stationary solution of the one-dimensional Ornstein-Uhlenbeck equation +dy(ϑtω) + σy(ϑtω)dt = dW(t). +(2.8) +It is known from [25] that there exists a ϑ-invariant subset �Ω ⊂ Ω of full measure such that y(ϑtω) is +continuous in t for every ω ∈ �Ω, and +lim +t→±∞ +|y(ϑtω)| +|t| += +lim +t→±∞ +1 +t +� t +0 +y(ϑξω)dξ = lim +t→∞ e−δt|y(ϑ−tω)| = 0, +(2.9) +for all δ > 0. For further analysis of this work, we do not distinguish between �Ω and Ω. +Since, ω(·) has sub-exponential growth (cf. [8, Lemma 11]), Ω can be written as Ω = � +N∈N +ΩN, where +ΩN := {ω ∈ Ω : |ω(t)| ≤ Ne|t|, for all t ∈ R}, for all N ∈ N. +Moreover, for each N ∈ N, (ΩN, dΩN ) is a polish space (cf. [8, Lemma 17]). + +ASYMPTOTIC AUTONOMY OF RANDOM ATTRACTORS FOR SCBF EQUATIONS +9 +Lemma 2.4. For each N ∈ N, suppose ωk, ω0 ∈ ΩN are such that dΩ(ωk, ω0) → 0 as k → ∞. Then, for +each τ ∈ R, T ∈ R+ and a ∈ R, +sup +t∈[τ,τ+T] +� +|y(ϑtωk) − y(ϑtω0)| + |eay(ϑtωk) − eay(ϑtω0)| +� +→ 0 as k → ∞, +(2.10) +sup +k∈N +sup +t∈[τ,τ+T] +|y(ϑtωk)| ≤ C(τ, T, ω0). +(2.11) +Proof. See the proofs of [23, Corollary 22] and [45, Lemma 2.5]. +□ +2.2.2. Backward-uniformly tempered random set. A bi-parametric set D = {D(τ, ω)} in a Banach space X +is said to be backward-uniformly tempered if +lim +t→+∞ e−ct sup +s≤τ +∥D(s − t, ϑ−tω)∥2 +X = 0 ∀ (τ, ω, c) ∈ R × Ω × R+, +where ∥D∥X = sup +x∈D +∥x∥X. +(2.12) +2.2.3. Class of random sets. +• Let D be the collection of subsets of H defined as: +D = +� +D = {D(τ, ω) : (τ, ω) ∈ R × Ω} : +lim +t→+∞ e−ct sup +s≤τ +∥D(s − t, ϑ−tω)∥2 +H = 0, ∀ c > 0 +� +. +(2.13) +• Let B be the collection of subsets of H defined as: +B = +� +B = {B(τ, ω) : (τ, ω) ∈ R × Ω} : +lim +t→+∞ e−ct∥B(τ − t, ϑ−tω)∥2 +H = 0, ∀ c > 0 +� +. +• Let D∞ be the collection of subsets of H defined as: +D∞ = +� +�D = {�D(ω) : ω ∈ Ω} : +lim +t→+∞ e−ct∥�D(ϑ−tω)∥2 +H = 0, ∀ c > 0 +� +. +3. 2D and 3D SCBF equations: Multiplicative noise +In this section, we consider 2D and 3D SCBF equations driven by a linear multiplicative white noise, that +is, S(v) = v and establish the asymptotic autonomy of pullback random attractors. Let us define +u(t, τ, ω, uτ) := e−y(ϑtω)v(t, τ, ω, vτ) with uτ = e−y(ϑτ ω)vτ, +where y satisfies (2.8) and v(·) := v(·, τ, ω, vτ) is the solution of (1.1) with S(v) = v. +Then u(·) := +u(·, τ, ω, uτ) satisfies: +(3.1) + + + + + + + + + + + + + + + + + + + + + +du(t) +dt +− µ∆u(t) + ey(ϑtω)(u(t) · ∇)u(t) + αu(t) + βe(r−1)y(ϑtω)|u(t)|r−1u(t) += −e−y(ϑtω)∇p(t) + f(t)e−y(ϑtω) + σy(ϑtω)u(t), +in Rd × (τ, ∞), +∇ · u = 0, +in Rd × (τ, ∞), +u(x)|t=τ = u0(x) = e−y(ϑτ ω)v0(x), +x ∈ Rd and τ ∈ R, +u(x)|t=τ → 0 +as |x| → ∞, + +10 +KUSH KINRA, MANIL T. MOHAN, AND RENHAI WANG +as well as (projected form) +(3.2) + + + + + + + + + +du(t) +dt ++ µAu(t) + ey(ϑtω)B +� +u(t) +� ++ αu(t) + βe(r−1)y(ϑtω)C +� +u(t) +� += e−y(ϑtω)Pf(t) + σy(ϑtω)u(t), +t > τ, τ ∈ R, +u(x)|t=τ = u0(x) = e−y(ϑτ ω)v0(x), +x ∈ Rd, +in V′ + �L +r+1 +r , where r ≥ 1. Due to some technical difficulties, we restrict ourselves to all the cases given in +Table 1 (see Lemmas 3.8 and 3.9). +3.1. Non-autonomous random dynamical system (NRDS). Lusin continuity helps us to define the +NRDS. The following lemma (energy inequality) will be frequently used. +Lemma 3.1. For all the cases given in Table 1, assume that f ∈ L2 +loc(R; L2(Rd)). Then, the solution of +(3.2) satisfies the following energy inequality: +d +dt∥u∥2 +H + +� +α − 2σy(ϑtω) + α +2 +� +∥u∥2 +H + 2µ∥∇u∥2 +H + 2βe(r−1)y(ϑtω)∥u∥r+1 +�Lr+1 ≤ 2e2|y(ϑtω)| +α +∥f∥2 +L2(Rd). +(3.3) +Proof. From the first equation of the system (3.2), using (2.2) and the Cauchy-Schwarz inequality, one can +obtain (3.3) immediately. +□ +Lemma 3.2. For all the cases given in Table 1, let f ∈ L2 +loc(R; L2(Rd)). For each (τ, ω, uτ) ∈ R×Ω×H, the +system (3.2) has a unique weak solution u(·, τ, ω, uτ) ∈ C([τ, +∞); H) ∩ L2 +loc(τ, +∞; V) ∩ Lr+1 +loc (τ, +∞; �Lr+1) +such that u is continuous with respect to the initial data. +Proof. One can prove the existence and uniqueness of solutions by a standard Faedo-Galerkin approximation +method, cf. [31, 34, 47], etc. For continuity with respect to initial data uτ, see [38, Lemma 3.5]. +□ +Next result shows the Lusin continuity of the mapping of solution to the system (3.2) in sample points. +Proposition 3.3. For all the cases given in Table 1, suppose that f ∈ L2 +loc(R; L2(Rd)). For each N ∈ N, the +mapping ω �→ u(t, τ, ω, uτ) (solution of (3.2)) is continuous from (ΩN, dΩN ) to H, uniformly in t ∈ [τ, τ +T] +with T > 0. +Proof. Assume that ωk, ω0 ∈ ΩN, N ∈ N such that dΩN (ωk, ω0) → 0 as k → ∞. Let U k(·) := uk(·) − u0(·), +where uk(·) := u(·, τ, ωk, uτ) and u0 := u(·, τ, ω0, uτ). Then, U k(·) satisfies: +dU k +dt += −µAU k − (α − σy(ϑtωk))U k − ey(ϑtωk)� +B +� +uk� +− B +� +u0�� +− +� +ey(ϑtωk) − ey(ϑtω0)� +B +� +u0� +− βe(r−1)y(ϑtωk)� +C +� +uk� +− C +� +u0�� +− β +� +e(r−1)y(ϑtωk) − e(r−1)y(ϑtω0)� +C +� +u0� ++ f +� +e−y(ϑtωk) − e−y(ϑtω0)� ++ σ[y(ϑtωk) − y(ϑtω0)]u0, +(3.4) +in V′ + �L +r+1 +r . Taking the inner product with U k(·) in (3.4), and using (2.2) and (2.4), we obtain +1 +2 +d +dt∥U k∥2 +H = −µ∥∇U k∥2 +H − (α − σy(ϑtωk))∥U k∥2 +H − βe(r−1)y(ϑtωk)� +C +� +uk� +− C +� +u0� +, uk − u0� + +ASYMPTOTIC AUTONOMY OF RANDOM ATTRACTORS FOR SCBF EQUATIONS +11 ++ ey(ϑtωk)b(U k, U k, u0) + +� +ey(ϑtωk) − ey(ϑtω0)� +b(u0, U k, u0) +− β +� +e(r−1)y(ϑtωk) − e(r−1)y(ϑtω0)�� +C +� +u0� +, U k� ++ +� +e−y(ϑtωk) − e−y(ϑtω0)� +(f, U k) ++ σ[y(ϑtωk) − y(ϑtω0)](u0, U k). +(3.5) +We know by (2.5) that +− +� +C +� +uk� +− C +� +u0� +, uk − u0� +≤ −1 +2∥|uk| +r−1 +2 (uk − u0)∥2 +H − 1 +2∥|u0| +r−1 +2 (uk − u0)∥2 +H. +(3.6) +Using H¨older’s and Young’s inequalities, we obtain +��� +� +e−y(ϑtωk) − e−y(ϑtω0)� +(f, U k) +��� ≤ C +���e−y(ϑtωk) − e−y(ϑtω0)��� +2 +∥f∥2 +L2(Rd) + α +4 ∥U k∥2 +H, +(3.7) +���σ[y(ϑtωk) − y(ϑtω0)](u0, U k) +��� ≤ C|y(ϑtωk) − y(ϑtω0)|2∥u0∥2 +H + α +4 ∥U k∥2 +H, +(3.8) +��� +� +e(r−1)y(ϑtωk) − e(r−1)y(ϑtω0)�� +C +� +u0� +, U k���� ≤ C +���e(r−1)y(ϑtωk) − e(r−1)y(ϑtω0)��� +� +∥u0∥r+1 +�Lr+1 + ∥uk∥r+1 +�Lr+1 +� +. +(3.9) +Next, we estimate the remaining terms of (3.5) separately. +Case I: d = 2 and r ≥ 1. Applying (2.2), (2.3), H¨older’s and Young’s inequalities, we estimate +���ey(ϑtωk)b(U k, U k, u0) +��� ≤ Cey(ϑtωk)∥U k∥H∥∇U k∥H∥∇u0∥H +≤ Ce2y(ϑtωk)∥∇u0∥2 +H∥U k∥2 +H + µ +4 ∥∇U k∥2 +H, +(3.10) +and +��� +� +ey(ϑtωk) − ey(ϑtω0)� +b(u0, U k, u0) +��� ≤ C +���ey(ϑtωk) − ey(ϑtω0)��� +2 +∥u0∥2 +H∥∇u0∥2 +H + µ +4 ∥∇U k∥2 +H. +(3.11) +Case II: d = 3 and r > 3. Using H¨older’s and Young’s inequalities, we infer +���ey(ϑtωk)b(U k, U k, u0) +��� ≤ µ +4 ∥∇U k∥2 +H + β +4 e(r−1)y(ϑtωk)∥|U k||u0| +r−1 +2 ∥2 +H + C∥U k∥2 +H, +(3.12) +and +��� +� +ey(ϑtωk) − ey(ϑtω0)� +b(u0, U k, u0) +��� +≤ +���1 − ey(ϑtω0)−y(ϑtωk)���ey(ϑtωk)∥∇u0∥H∥|U k||u0|∥H +≤ β +4 e(r−1)y(ϑtωk)∥|U k||u0| +r−1 +2 ∥2 +H + C +���1 − ey(ϑtω0)−y(ϑtωk)��� +2 +∥∇u0∥2 +H + C∥U k∥2 +H. +(3.13) +Case III: When d = r = 3 with 2βµ ≥ 1. Applying (2.2), H¨older’s and Young’s inequalities, we obtain +���ey(ϑtωk)b(U k, U k, u0) +��� = +���ey(ϑtωk)b(U k, U k, uk) +��� ≤ 1 +2β ∥∇U k∥2 +H + β +2 e2y(ϑtωk)∥|U k||uk|∥2 +H, +(3.14) +and +���ey(ϑtωk) − ey(ϑtω0)||b(u0, U k, u0) +��� ≤ C +���1 − ey(ϑtω0)−y(ϑtωk)��� +2 +∥∇u0∥2 +H + β +2 e2y(ϑtωk)∥|U k||u0|∥2 +H. +(3.15) +Combining (3.5)-(3.15), we arrive at +d +dt∥U k(t)∥2 +H ≤ P(t)∥U k(t)∥2 +H + Q(t), for a.e. t ∈ [τ, τ + T] with T > 0, +(3.16) + +12 +KUSH KINRA, MANIL T. MOHAN, AND RENHAI WANG +where +P = + + + + + +y(ϑtωk) + Ce2y(ϑtωk)∥∇u0∥2 +H, +for d = 2 with r ≥ 1, +y(ϑtωk) + C, +for d = 3 with r > 3, +y(ϑtωk), +for d = r = 3 with 2βµ ≥ 1, +Q = C +���e−y(ϑtωk) − e−y(ϑtω0)��� +2 +∥f∥2 +L2(Rd) + C|y(ϑtωk) − y(ϑtω0)|2∥u0∥2 +H + C +���e(r−1)y(ϑtωk) − e(r−1)y(ϑtω0)��� +× +� +∥u0∥r+1 +�Lr+1 + ∥uk∥r+1 +�Lr+1 +� ++ C × + + + + + +��ey(ϑtωk) − ey(ϑtω0)��2∥u0∥2 +H∥∇u0∥2 +H, +for d = 2 with r ≥ 1, +��1 − ey(ϑtω0)−y(ϑtωk)��2∥∇u0∥2 +H, +for d = 3 with r > 3, +��1 − ey(ϑtω0)−y(ϑtωk)��2∥∇u0∥2 +H, +for d = r = 3 with 2βµ ≥ 1. +From (3.3), we deduce +� τ+T +τ +2βe(r−1)y(ϑtωk)∥uk(t)∥r+1 +�Lr+1dt ≤ ∥uτ∥2 +H + 2 +α +� τ+T +τ +e−2y(ϑtωk)∥f(t)∥2 +L2(Rd)dt +≤ ∥uτ∥2 +H + 2 +α +sup +t∈[τ,τ+T] +� +e−2y(ϑtωk)� � τ+T +τ +∥f(t)∥2 +L2(Rd)dt, +which gives +sup +k∈N +� τ+T +τ +e(r−1)y(ϑtωk)∥uk(t)∥r+1 +�Lr+1dt ≤ C(τ, T, ω0, uτ, f), +(3.17) +where we have used (2.11) and the fact f ∈ L2 +loc(R; L2(Rd)). Using (2.11) and u0 ∈ L2 +loc(τ, +∞; V), we +deduce +� τ+T +τ +P(t)dt ≤ C(τ, T, ω0). +(3.18) +Now, from (3.17), f ∈ L2 +loc(R; H), u0 ∈ C([τ, +∞); H)∩L2 +loc(τ, +∞; V)∩Lr+1 +loc (τ, +∞; �Lr+1) and Lemma 2.4, +we conclude +lim +k→+∞ +� τ+T +τ +Q(t)dt = 0. +(3.19) +Making use of the Gronwall inequality in (3.16), we get +∥U k(t)∥2 +H ≤ e +� τ+T +τ +P (t)dt +�� τ+T +τ +Q(t)dt +� +, +for all t ∈ [τ, τ + T]. +(3.20) +In view of (3.18)-(3.20), we complete the proof. +□ +Lemma 3.2 ensures that we can define a mapping Φ : R+ × R × Ω × H → H by +Φ(t, τ, ω, vτ) := v(t + τ, τ, ϑ−τω, vτ) = ey(ϑtω)u(t + τ, τ, ϑ−τω, uτ). +(3.21) +The Lusin continuity in Proposition 3.3 provides the F-measurability of Φ. Consequently, in view of Lemma +3.2 and Proposition 3.3, we have the following result for NRDS. +Proposition 3.4. The mapping Φ defined by (3.21) is an NRDS on H, that is, Φ has the following properties: +(i) Φ is (B(R+) × B(R) × F × B(H); B(H))-measurable, + +ASYMPTOTIC AUTONOMY OF RANDOM ATTRACTORS FOR SCBF EQUATIONS +13 +(ii) Φ satisfies the cocycle property: Φ(0, τ, ω, ·) = I, and +Φ(t + s, τ, ω, vτ) = Φ(t, τ + s, ϑsω, Φ(s, τ, ω, vτ)), +t, s ≥ 0. +3.2. Backward convergence of NRDS. Consider the autonomous SCBF equations driven by linear +multiplicative white noise: +(3.22) + + + +d�v(t) +dt ++ µA�v(t) + B(�v(t)) + α�v(t) + βC(�v(t)) = Pf ∞ + �v(t) ◦ dW(t) +dt +, +t > 0, +�v(x, 0) = �v0(x), +x ∈ Rd. +Let �u(t, ω) = e−y(ϑtω)�v(t, ω). Then, �u(·) satisfies +(3.23) + + + + + + + + + +d�u(t) +dt ++ µA�u(t) + ey(ϑtω)B +��u(t) +� ++ α�u(t) + βe(r−1)y(ϑtω)C +��u(t) +� += Pf ∞e−y(ϑtω) + σy(ϑtω)�u(t), +t > 0, +�u(x, 0) = �u0(x) = e−y(ω)�v0(x), +x ∈ Rd, +in V′ + �L +r+1 +r . +Proposition 3.5. For all the cases given in Table 1, suppose that Assumption 1.1 is satisfied. +Then, +lim +τ→−∞ ∥uτ − �u0∥H = 0 implies that the solution u of the system (3.2) backward converges to the solution �u +of the system (3.23) , that is, +lim +τ→−∞ ∥u(T + τ, τ, ϑ−τω, uτ) − �u(t, ω, �u0)∥H = 0, +for all T > 0 and ω ∈ Ω. +(3.24) +Proof. Let U τ(·) := u(· + τ, τ, ϑ−τω, uτ) − �u(·, ω, �u0). From (3.2) and (3.23), we obtain +dU τ +dt += −µAU τ − αU τ − ey(ϑtω)� +B +� +u +� +− B +��u +�� +− βe(r−1)y(ϑtω)� +C +� +u +� +− C +��u +�� ++ e−y(ϑtω)[Pf(t + τ) − Pf ∞] + σy(ϑtω)U τ, +(3.25) +in V′ + �L +r+1 +r . Taking the inner product with U τ(·) in (3.25), and using (2.2) and (2.4), we get +1 +2 +d +dt∥U τ∥2 +H = −µ∥∇U τ∥2 +H − (α − σy(ϑtω))∥U τ∥2 +H − βe(r−1)y(ϑtω)� +C +� +u +� +− C +��u +� +, u − �u +� ++ ey(ϑtω)b(U τ, U τ, �u) + e−y(ϑtω)(f(t + τ) − f∞, U τ). +(3.26) +From (2.5), one can write +− +� +C +� +u +� +− C +��u +� +, u − �u +� +≤ −1 +2∥|u| +r−1 +2 (u − �u)∥2 +H − 1 +2∥|�u| +r−1 +2 (u − �u)∥2 +H. +(3.27) +Applying H¨older’s and Young’s inequalities, we infer +���e−y(ϑtω)(f(t + τ) − f ∞, U τ) +��� ≤ ∥f(t + τ) − f ∞∥2 +L2(Rd) + Ce−2y(ϑtω)∥U τ∥2 +H, +(3.28) +and +ey(ϑtω)b(U τ, U τ, �u) +≤ + + + + + +Ce2y(ϑtω)∥∇�u∥2 +H∥U τ∥2 +H + µ +2 ∥∇U τ∥2 +H, +for d = 2 and r ≥ 1, +µ +2∥∇U τ∥2 +H + β +4 e(r−1)y(ϑtω)∥|U τ||�u| +r−1 +2 ∥2 +H + C∥U τ∥2 +H, +for d = 3 and r > 3, +1 +2β∥∇U τ∥2 +H + β +2 e2y(ϑtω)∥|U τ||�u|∥2 +H, +for d = r = 3 and 2βµ ≥ 1. +(3.29) + +14 +KUSH KINRA, MANIL T. MOHAN, AND RENHAI WANG +Combining (3.26)-(3.29), we achieve +d +dt∥U τ(t)∥2 +H ≤ S(t)∥U τ(t)∥2 +H + ∥f(t + τ) − f∞∥2 +L2(Rd), +(3.30) +where +S(t) = C × + + + + + +e2y(ϑtω)∥∇�u(t)∥2 +H + e−2y(ϑtω) + |y(ϑtω)|, +for d = 2 and r ≥ 1, +e−2y(ϑtω) + |y(ϑtω)| + 1, +for d = 3 and r > 3, +e−2y(ϑtω) + |y(ϑtω)|, +for d = r = 3 and 2βµ ≥ 1, +for a.e. t ∈ [τ, τ + T]. Making use of Gronwall’s inequality in (3.30) over (0, T), we obtain +∥U τ(T)∥2 +H ≤ +� +∥U τ(0)∥2 +H + +� T +0 +∥f(t + τ) − f ∞∥2 +L2(Rd)dt +� +e +� T +0 S(t)dt. +Since y is continuous and �u ∈ L2(0, T; V), it implies that +� T +0 S(t)dt is bounded. From Assumption 1.1 +(particularly, (1.2)), we deduce +� T +0 +∥f(t + τ) − f ∞∥2 +L2(Rd)dt ≤ +� τ+T +−∞ +∥f(t) − f ∞∥2 +L2(Rd)dt → 0 as τ → −∞. +(3.31) +Using the fact that +� T +0 S(t)dt is bounded, (3.31) and lim +τ→∞ ∥U τ(0)∥2 +H = 0, we conclude the proof. +□ +3.3. Increasing random absorbing sets. In this subsection, we prove the existence of a pullback D- +random absorbing set for the system (2.6) with S(v) = v. +Lemma 3.6. For all the cases given in Table 1, suppose that f ∈ L2 +loc(R; L2(Rd)). Then, for each (τ, ω, D) ∈ +R × Ω × D, there exists a time T := T(τ, ω, D) > 0 such that +sup +s≤τ +sup +t≥T +sup +u0∈D(s−t,ϑ−tω) +� +∥u(s, s − t, ϑ−sω, u0)∥2 +H ++ α +2 +� s +s−t +eα(ζ−s)−2σ � ζ +s y(ϑη−sω)dη∥u(ζ, s − t, ϑ−sω, u0)∥2 +Hdζ ++ 2µ +� s +s−t +eα(ζ−s)−2σ +� ζ +s y(ϑη−sω)dη∥∇u(ζ, s − t, ϑ−sω, u0)∥2 +Hdζ ++ 2β +� s +s−t +e(r−1)y(ϑζ−sω)+α(ζ−s)−2σ � ζ +s y(ϑη−sω)dη∥u(ζ, s − t, ϑ−sω, u0)∥r+1 +�Lr+1dζ +� +≤ 4 +α sup +s≤τ +� 0 +−∞ +eαζ+2|y(ϑζω)|+2σ +� 0 +ζ y(ϑηω)dη∥f(ζ + s)∥2 +L2(Rd)dζ =: 4 +α sup +s≤τ +K(s, ω). +(3.32) +Furthermore, for 2 < k1 < ∞ and k2 > 0, there exists a time T∗ := T∗(τ, ω, D, k1) > 0 such that +sup +s≤τ +sup +t≥T∗ +sup +u0∈D(s−t,ϑ−tω) +� s +s−t +ek2|y(ϑζ−sω)|+α(ζ−s)−2σ � ζ +s y(ϑη−sω)dη∥u(ζ, s − t, ϑ−sω, u0)∥k1 +H dζ +≤ C +0 +� +−∞ +ek2|y(ϑζω)|+ α +k1 ζ−(k1−2)σ +� 0 +ζ y(ϑηω)dηdζ +× +� +0 +� +−∞ +e +2(k1−1)α +k2 +1 +ζ1+2|y(ϑζ1ω)|+2σ +� 0 +ζ1 y(ϑηω)dη∥f(ζ1 + s)∥2 +L2(Rd)dζ1 +� k1 +2 +. +(3.33) + +ASYMPTOTIC AUTONOMY OF RANDOM ATTRACTORS FOR SCBF EQUATIONS +15 +Proof. Let us write the energy inequality (3.3) for u(ζ) = u(ζ, s − t, ϑ−sω, u0), that is, +d +dζ ∥u(ζ)∥2 +H + (α − 2σy(ϑζ−sω))∥u(ζ)∥2 +H + α +2 ∥u(ζ)∥2 +H + 2µ∥∇u(ζ)∥2 +H + 2βe(r−1)y(ϑζ−sω)∥u(ζ)∥r+1 +�Lr+1 +≤ 2e2|y(ϑζ−sω)| +α +∥f(ζ)∥2 +L2(Rd). +(3.34) +In view of the variation of constants formula with respect to ζ ∈ (s − t, ξ), we get +∥u(ξ, s − t, ϑ−sω, u0)∥2 +H + α +2 +� ξ +s−t +eα(ζ−ξ)−2σ � ζ +ξ y(ϑη−sω)dη∥u(ζ, s − t, ϑ−sω, u0)∥2 +Hdζ ++ 2µ +� ξ +s−t +eα(ζ−ξ)−2σ +� ζ +ξ y(ϑη−sω)dη∥∇u(ζ, s − t, ϑ−sω, u0)∥2 +Hdζ ++ 2β +� ξ +s−t +e(r−1)y(ϑζ−sω)+α(ζ−ξ)−2σ +� ζ +ξ y(ϑη−sω)dη∥u(ζ, s − t, ϑ−sω, u0)∥r+1 +�Lr+1dζ +≤ e−α(ξ−s+t)+2σ +� ξ−s +−t +y(ϑηω)dη∥u0∥2 +H + 2 +α +� ξ−s +−t +eα(ζ+s−ξ)+2|y(ϑζω)|+2σ � ξ−s +ζ +y(ϑηω)dη∥f(ζ + s)∥2 +L2(Rd)dζ. +(3.35) +Putting ξ = s in (3.35), we find +∥u(s, s − t, ϑ−sω, u0)∥2 +H + α +2 +� s +s−t +eα(ζ−s)−2σ � ζ +s y(ϑη−sω)dη∥u(ζ, s − t, ϑ−sω, u0)∥2 +Hdζ ++ 2µ +� s +s−t +eα(ζ−s)−2σ � ζ +s y(ϑη−sω)dη∥∇u(ζ, s − t, ϑ−sω, u0)∥2 +Hdζ ++ 2β +� s +s−t +e(r−1)y(ϑζ−sω)+α(ζ−s)−2σ +� ζ +s y(ϑη−sω)dη∥u(ζ, s − t, ϑ−sω, u0)∥r+1 +�Lr+1dζ +≤ e−αt+2σ � 0 +−t y(ϑηω)dη∥u0∥2 +H + 2 +α +� 0 +−∞ +eαζ+2|y(ϑζω)|+2σ � 0 +ζ y(ϑηω)dη∥f(ζ + s)∥2 +L2(Rd)dζ, +(3.36) +for all s ≤ τ. Since u0 ∈ D(s−t, ϑ−tω) and D is backward tempered, it implies from (2.9) and the definition +of backward temperedness (2.12) that there exists a time T = T(τ, ω, D) such that for all t ≥ T, +e−αt+2σ +� 0 +−t y(ϑηω)dη sup +s≤τ +∥u0∥2 +H +≤ e− α +3 t sup +s≤τ +∥D(s − t, ϑ−tω)∥2 +H ≤ 2 +α +� 0 +−∞ +eαζ+2|y(ϑζω)|+2σ +� 0 +ζ y(ϑηω)dη∥f(ζ + s)∥2 +L2(Rd)dζ. +(3.37) +Hence, by using (3.37) and taking supremum on s ∈ (−∞, τ] in (3.36), we reach at (3.32). Now, using +(3.35), we estimate for 2 < k1 < ∞ and k2 > 0 +� s +s−t +ek2|y(ϑζ−sω)|+α(ζ−s)−2σ +� ζ +s y(ϑη−sω)dη∥u(ζ, s − t, ϑ−sω, u0)∥k1 +H dζ +≤ C +� s +s−t +ek2|y(ϑζ−sω)|+α(ζ−s)+2σ +� 0 +ζ−s y(ϑηω)dη +� +e− k1 +2 α(ζ−s+t)+k1σ +� ζ−s +−t +y(ϑηω)dη∥u0∥k1 +H ++ +� ζ−s +� +−t +eα(ζ1+s−ζ)+2|y(ϑζ1ω)|+2σ � ζ−s +ζ1 +y(ϑηω)dη∥f(ζ1 + s)∥2 +L2(Rd)dζ1 +� k1 +2 � +dζ + +16 +KUSH KINRA, MANIL T. MOHAN, AND RENHAI WANG +≤ C +� 0 +−∞ +ek2|y(ϑζω)|+ α +k1 ζ−(k1−2)σ � 0 +ζ y(ϑηω)dηdζ × +� +e− (k1−1)α +k1 +t+k1σ � 0 +−t y(ϑηω)dη∥u0∥k1 +H ++ +� � 0 +−∞ +e +2(k1−1)α +k2 +1 +ζ1+2|y(ϑζ1ω)|+2σ � 0 +ζ1 y(ϑηω)dη∥f(ζ1 + s)∥2 +L2(Rd)dζ1 +� k1 +2 � +. +(3.38) +Hence, using (2.9) and the backward-uniform temperedness property (2.12) of u0 (see (3.37)), we obtain +(3.33), as required. +□ +Proposition 3.7. For all the cases given in Table 1, suppose that f ∈ L2 +loc(R; L2(Rd)) and Assumption 1.1 +is satisfied. For K(τ, ω) same as in (3.32), we have +(i) There is an increasing pullback D-random absorbing set K given by +K(τ, ω) := +� +v ∈ H : ∥v∥2 +H ≤ 4ey(ω) +α +sup +s≤τ +K(s, ω) +� +, +for all τ ∈ R and ω ∈ Ω. +(3.39) +Moreover, K is backward-uniformly tempered with arbitrary rate, that is, K ∈ D. +(ii) There is a B-pullback random absorbing set �K given by +�K(τ, ω) := +� +v ∈ H : ∥v∥2 +H ≤ 4ey(ω) +α +K(τ, ω) +� +∈ B, +for all τ ∈ R and ω ∈ Ω. +(3.40) +Proof. See the proof of in [73, Proposition 4.6]. +□ +3.4. Backward uniform tail-estimates and backward flattening-property. In this subsection, we +show that the solution of the system (3.1) satisfies the backward uniform tail-estimates and backward +flattening-property for d = 2 with r ∈ {1} ∪ [2, ∞), d = 3 with r ∈ (3, ∞) and d = r = 3 with 2βµ ≥ 1. +These estimates help us to obtain the backward uniform pullback D-asymptotic compactness of Φ. We use a +cut-off function technique to obtain backward uniform tail-estimates and backward flattening-property. The +following lemma provides the backward uniform tail-estimates for the solutions of the system (3.1). +Lemma 3.8. For all the cases given in Table 1, suppose that Assumption 1.1 holds. Then, for any (τ, ω, D) ∈ +R × Ω × D, the solution of (3.1) satisfies +lim +k,t→+∞ sup +s≤τ +sup +u0∈D(s−t,ϑ−tω) +∥u(s, s − t, ϑ−sω, u0)∥2 +L2(Oc +k) = 0, +(3.41) +where Ok = {x ∈ Rd : |x| ≤ k}, k ∈ N. +Proof. Let ρ be a smooth function such that 0 ≤ ρ(ξ) ≤ 1, for ξ ∈ R+ and +ρ(ξ) = +�0, for 0 ≤ ξ ≤ 1, +1, for ξ ≥ 2. +Then, there exists a positive constant C such that |ρ′(ξ)| ≤ C, for all ξ ∈ R+. Taking divergence to the first +equation of (3.1), we obtain formally in weak sense +−e−y(ϑtω)∆p = ey(ϑtω)∇ · +�� +u · ∇ +� +u +� ++ βe(r−1)y(ϑtω)∇ · +� +|u|r−1u +� +− e−y(ϑtω)∇ · f + +ASYMPTOTIC AUTONOMY OF RANDOM ATTRACTORS FOR SCBF EQUATIONS +17 += ey(ϑtω)∇ · +� +∇ · +� +u ⊗ u +�� ++ βe(r−1)y(ϑtω)∇ · +� +|u|r−1u +� +− e−y(ϑtω)∇ · f += ey(ϑtω) +d +� +i,j=1 +∂2 +∂xi∂xj +� +uiuj +� ++ βe(r−1)y(ϑtω)∇ · +� +|u|r−1u +� +− e−y(ϑtω)∇ · f, +which implies +p = (−∆)−1 + +e2y(ϑtω) +d +� +i,j=1 +∂2 +∂xi∂xj +� +uiuj +� ++ βery(ϑtω)∇ · +� +|u|r−1u +� +− ∇ · f + +, +(3.42) +in the weak sense. Taking the inner product to the first equation of (3.1) with ρ +� +|x|2 +k2 +� +u, we have +1 +2 +d +dt +� +Rd ρ +�|x|2 +k2 +� +|u|2dx = µ +� +Rd(∆u)ρ +�|x|2 +k2 +� +udx − α +� +Rd ρ +�|x|2 +k2 +� +|u|2dx − ey(ϑtω)b +� +u, u, ρ +�|x|2 +k2 +� +u +� +− βe(r−1)y(ϑtω) +� +Rd|u|r+1ρ +�|x|2 +k2 +� +dx − e−y(ϑtω) +� +Rd(∇p)ρ +�|x|2 +k2 +� +udx ++ e−y(ϑtω) +� +Rd fρ +�|x|2 +k2 +� +udx + σy(ϑtω) +� +Rd ρ +�|x|2 +k2 +� +|u|2dx. +(3.43) +Let us now estimate each term on the right hand side of (3.43). Integration by parts and divergence free +condition of u(·) help us to obtain +µ +� +Rd(∆u)ρ +�|x|2 +k2 +� +udx + µ +� +Rd |∇u|2ρ +�|x|2 +k2 +� +dx += −µ +� +Rd ρ′ +�|x|2 +k2 +� 2 +k2 (x · ∇)u · udx +≤ 2 +√ +2µ +k +� +k≤|x|≤ +√ +2k +|u| +����ρ′ +�|x|2 +k2 +�����|∇u|dx ≤ C +k +� +Rd|u||∇u|dx ≤ C +k +� +∥u∥2 +H + ∥∇u∥2 +H +� +, +(3.44) +and +−ey(ϑtω)b +� +u, u, ρ +�|x|2 +k2 +� +u +� += ey(ϑtω) +� +Rd ρ′ +�|x|2 +k2 +� x +k2 · u|u|2dx +≤ +√ +2e|y(ϑtω)| +k +� +k≤|x|≤ +√ +2k +����ρ′ +�|x|2 +k2 +�����|u|3dx ≤ C +k e|y(ϑtω)|∥u∥2 +�L4∥u∥H +≤ C +k e|y(ϑtω)|∥u∥ +6−d +2 +H +∥∇u∥ +d +2 +H ≤ C +k +� +∥∇u∥2 +H + e +4|y(ϑtω)| +4−d +∥u∥ +2(6−d) +4−d +H +� +, +(3.45) +where we have used Ladyzhenskaya’s (for both d = 2, 3) and Young’s inequalities in the penultimate and +final inequalities, respectively. Using integration by parts, divergence free condition and (3.42), we obtain +−e−y(ϑtω) +� +Rd(∇p)ρ +�|x|2 +k2 +� +udx = e−y(ϑtω) +� +Rd pρ′ +�|x|2 +k2 +� 2 +k2 (x · u)dx +≤ Ce|y(ϑtω)| +k +� +Rd +��(−∆)−1� +∇ · +� +∇ · +� +u ⊗ u +����� · |u|dx + +18 +KUSH KINRA, MANIL T. MOHAN, AND RENHAI WANG ++ Ce(r−1)y(ϑtω) +k +� +Rd +��(−∆)−1� +∇ · +� +|u|r−1u +���� · |u|dx + Ce|y(ϑtω)| +k +� +Rd |(−∆)−1[∇ · f]| · |u|dx +=: C +k +� +S1(d, r) + e(r−1)y(ϑtω)S2(d, r) + S3(d, r) +� +. +(3.46) +Estimate of S1(d, r): Using H¨older’s inequality, Fourier transformation, Ladyzhenskaya’s and Young’s +inequalities, respectively, we get (for d = 2, 3) +|S1(d, r)| ≤ e|y(ϑtω)|��(−∆)−1� +∇ · +� +∇ · +� +u ⊗ u +����� +L2(Rd)∥u∥H ≤ e|y(ϑtω)|∥u∥2 +�L4∥u∥H +≤ Ce|y(ϑtω)|∥u∥ +6−d +2 +H +∥∇u∥ +d +2 +H ≤ C +� +∥∇u∥2 +H + e +4|y(ϑtω)| +4−d +∥u∥ +2(6−d) +4−d +H +� +. +(3.47) +Estimate of S2(d, r): Divergence free condition gives S2(d, r) = 0 for r = 1. Therefore, we will consider +r ∈ [2, ∞) for d = 2 and r ∈ [3, ∞) for d = 3. Applying H¨older’s, Gagliardo-Nirenberg’s, interpolation and +Young’s inequalities, we obtain +|S2(d, r)| ≤ + + + + + + + +∥(−∆)−1� +∇ · +� +|u|r−1u +�� +∥L2(Rd)∥u∥H, +for d = 2 and r ∈ [2, ∞), +∥(−∆)−1� +∇ · +� +|u|r−1u +�� +∥L2(Rd)∥u∥H, +for d = 3 and r ∈ [3, 5], +∥(−∆)−1� +∇ · +� +|u|r−1u +�� +∥ +L +3(r+1) +2r−1 (Rd)∥u∥ +�L +3(r+1) +r+4 , +for d = 3 and r ∈ (5, ∞), +≤ C × + + + + + + + +∥u∥r +�Lr∥u∥H, +for d = 2 and r ∈ [2, ∞), +∥u∥r +�L +6r +5 ∥u∥H, +for d = 3 and r ∈ [3, 5], +∥u∥r +�Lr+1∥u∥ +�L +3(r+1) +r+4 , +for d = 3 and r ∈ (5, ∞), +≤ C × + + + + + + + + + + + +∥u∥ +(r+1)(r−2) +r−1 +�Lr+1 +∥u∥ +r+1 +(r−1) +H +, +for d = 2 and r ∈ [2, ∞), +∥u∥ +(r+1)(3r−5) +3(r−1) +�Lr+1 +∥u∥ +2(r+1) +3(r−1) +H +, +for d = 3 and r ∈ [3, 5], +∥u∥ +(r+1)(3r−5) +3(r−1) +�Lr+1 +∥u∥ +2(r+1) +3(r−1) +H +, +for d = 3 and r ∈ (5, ∞), +≤ C +� +∥u∥r+1 +�Lr+1 + ∥u∥r+1 +H +� +. +(3.48) +Estimate of S3(d, r): Applying H¨older’s, Gagliardo-Nirenberg’s and Young’s inequalities, we find (for +d = 2, 3) +|S3(d, r)| ≤ Ce|y(ϑtω)|∥(−∆)−1[∇ · f]∥ +L +d +d−1 (Rd)∥u∥Ld(Rd) +≤ Ce|y(ϑtω)|∥f∥L1(Rd)∥u∥ +4−d +2 +H +∥∇u∥ +d−2 +2 +H +≤ Ce2|y(ϑtω)|∥f∥2 +L1(Rd) + C∥u∥2 +H + C∥∇u∥2 +H. +(3.49) +Finally, we estimate the penultimate term on right hand side (RHS) of (3.43) by using H¨older’s and +Young’s inequalities as follows: +e−y(ϑtω) +� +Rd f(x)ρ +�|x|2 +k2 +� +udx ≤ α +4 +� +Rd ρ +�|x|2 +k2 +� +|u|2dx + e2|y(ϑtω)| +α +� +Rd ρ +�|x|2 +k2 +� +|f(x)|2dx. +(3.50) + +ASYMPTOTIC AUTONOMY OF RANDOM ATTRACTORS FOR SCBF EQUATIONS +19 +Combining (3.43)-(3.50), we get +d +dt∥u∥2 +L2(Oc +k) + (α − 2σy(ϑtω))∥u∥2 +L2(Oc +k) +≤ C +k +� +∥u∥2 +H + ∥∇u∥2 +H + e(r−1)y(ϑtω)∥u∥r+1 +�Lr+1 + e2|y(ϑtω)|∥f∥2 +L1(Rd) +� ++ C +k +� +e +4|y(ϑtω)| +4−d +∥u∥ +2(6−d) +4−d +H ++ e(r−1)|y(ϑtω)|∥u∥r+1 +H +� ++ 2e2|y(ϑtω)| +α +� +|x|≥k +|f(x)|2dx. +(3.51) +Applying the variation of constants formula to the above equation (3.51) on (s − t, s) and replacing ω by +ϑ−sω, for s ≤ τ, t ≥ 0 and ω ∈ Ω, we find +∥u(s, s − t, ϑ−sω, u0)∥2 +L2(Oc +k) +≤ e−αt+2σ +� 0 +−t y(ϑηω)dη∥u0∥2 +H + C +k +� � s +s−t +eα(ζ−s)−2σ � ζ +s y(ϑη−sω)dη∥u(ζ, s − t, ϑ−sω, u0)∥2 +Hdζ ++ +� s +s−t +eα(ζ−s)−2σ � ζ +s y(ϑη−sω)dη∥∇u(ζ, s − t, ϑ−sω, u0)∥2 +Hdζ ++ +� s +s−t +e(r−1)y(ϑζ−sω)+α(ζ−s)−2σ +� ζ +s y(ϑη−sω)dη∥u(ζ, s − t, ϑ−sω, u0)∥r+1 +�Lr+1dζ ++ +� 0 +−t +eαζ+2|y(ϑζω)|+2σ � 0 +ζ y(ϑηω)dη∥f(ζ + s)∥2 +L1(Rd)dζ +� ++ C +k +� � s +s−t +e +4|y(ϑζ−sω)| +4−d ++α(ζ−s)−2σ � ζ +s y(ϑη−sω)dη∥u(ζ, s − t, ϑ−sω, u0)∥ +2(6−d) +4−d +H +dζ ++ +� s +s−t +e(r−1)|y(ϑζ−sω)|+α(ζ−s)−2σ +� ζ +s y(ϑη−sω)dη∥u(ζ, s − t, ϑ−sω, u0)∥r+1 +H +dζ +� ++ C +� 0 +−t +eαζ+2|y(ϑζω)|+2σ � 0 +ζ y(ϑηω)dη +� +|x|≥k +|f(x, ζ + s)|2dxdζ. +(3.52) +Now using (2.9), the definition of backward-uniform temperedness (2.12) (for the first term on RHS of +(3.52)), Lemma 3.6 ((3.32) and (3.33) for the second and third terms on RHS of (3.52), respectively) and +(1.8) (for the final term on RHS of (3.52)), we immediately complete the proof. +□ +The following lemma provides the backward flattening-property for the solution of the system (3.1). For +each k ≥ 1, we let +̺k(x) := 1 − ρ +�|x|2 +k2 +� +, +x ∈ Rd. +Let ¯u := ̺ku for u := u(s, s − t, ω, uτ) ∈ H. Then ¯u ∈ L2(O√ +2k), which has the orthogonal decomposition: +¯u = Pi¯u ⊕ (I − Pi)¯u =: ¯ui,1 + ¯ui,2, +for eah i ∈ N, +(3.53) +where, Pi : L2(O√ +2k) → Hi := span{e1, e2, · · · , ei} ⊂ L2(O√ +2k) is a canonical projection and {ej}∞ +j=1 is the +family of eigenfunctions for −∆ in L2(O√ +2k) with corresponding positive eigenvalues λ1 ≤ λ2 ≤ · · · ≤ λj → +∞ as j → ∞. We also have that +̺k∆u = ∆¯u − u∆̺k − 2∇̺k · ∇u. + +20 +KUSH KINRA, MANIL T. MOHAN, AND RENHAI WANG +Furthermore, for ψ ∈ H1 +0(O√ +2k), we have +Piψ = +i +� +j=1 +(ψ, ej)ej, ∇Piψ = A1/2Piψ = +i +� +j=1 +λ1/2 +j +(ψ, ej)ej, +(I − Pi)ψ = +∞ +� +j=i+1 +(ψ, ej)ej, ∇(I − Pi)ψ = A1/2(I − Pi)ψ = +∞ +� +j=i+1 +λ1/2 +j +(ψ, ej)ej, +∥∇(I − Pi)ψ∥2 +L2(O√ +2k) = +∞ +� +j=i+1 +λj|(ψ, ej)|2 ≥ λi+1 +∞ +� +j=i+1 +|(ψ, ej)|2 = λi+1∥(I − Pi)ψ∥2 +L2(O√ +2k). +(3.54) +Lemma 3.9. For all the cases given in Table 1, suppose that Assumption 1.1 is satisfied. Let (τ, ω, D) ∈ +R × Ω × D and k ≥ 1 be fixed. Then +lim +i,t→+∞ sup +s≤τ +sup +u0∈D(s−t,ϑ−tω) +∥(I − Pi)¯u(s, s − t, ϑ−sω, ¯u0,2)∥2 +L2(O√ +2k) = 0, +(3.55) +where ¯u0,2 = (I − Pi)(̺ku0). +Proof. Multiplying by ̺k in the first equation of (3.1), we rewrite the equation as: +d¯u +dt − µ∆¯u + ey(ϑtω)̺k(u · ∇)u + α¯u + βe(r−1)y(ϑtω)̺k|u|r−1u += −e−y(ϑtω)̺k∇p + e−y(ϑtω)̺kf + σy(ϑtω)¯u − µu∆̺k − 2µ∇̺k · ∇u. +(3.56) +Applying (I − Pi) to the equation (3.56) and taking the inner product of the resulting equation with ¯ui,2 in +L2(O√ +2k) gives +1 +2 +d +dt∥¯ui,2∥2 +L2(O√ +2k) + µ∥∇¯ui,2∥2 +L2(O√ +2k) + (α − σy(ϑtω))∥¯ui,2∥2 +L2(O√ +2k) + βe(r−1)y(ϑtω)∥|u| +r−1 +2 ¯ui,2∥2 +L2(O√ +2k) += − ey(ϑtω) +d +� +q,q′=1 +� +O√ +2k +(I − Pi) +� +uq +∂uq′ +∂xq +{̺k(x)}2uq′ +� +dx +� +�� +� +=:J1 +− +� +e−y(ϑtω)̺k∇p, ¯ui,2 +� +� +�� +� +=:J2 ++ +�� +e−y(ϑtω)̺kf, ¯ui,2 +� +− µ +� +u∆̺k, ¯ui,2 +� +− µ +� +2∇̺k · ∇u, ¯ui,2 +�� +� +�� +� +=:J3 +. +(3.57) +Next, we estimate each terms of (3.57) as follows: Using integration by parts, divergence free condition +of u(·), (3.54) (without loss of generality (WLOG), one may assume that λi ≥ 1), H¨older’s and Young’s +inequalities, we get +|J1| = e|y(ϑtω)| +����� +� +O√ +2k +(I − Pi) +� +ρ′ +�|x|2 +k2 +� x +k2 · ̺k(x)u|u|2 +� +dx +����� +≤ Ce|y(ϑtω)|∥¯ui,2∥L2(O√ +2k)∥u∥2 +�L4 +≤ Cλ +− (4−d) +8 +i+1 +e|y(ϑtω)|∥∇¯ui,2∥ +4−d +4 +L2(O√ +2k)∥∇u∥ +d +2 +H∥u∥ +8−d +4 +H +≤ µ +20∥∇¯ui,2∥2 +L2(O√ +2k) + Cλ +− 4−d +4+d +i+1 +� +∥∇u∥2 +H + e +8|y(ϑtω)| +4−d +∥u∥ +2(8−d) +4−d +H +� +, +(3.58) + +ASYMPTOTIC AUTONOMY OF RANDOM ATTRACTORS FOR SCBF EQUATIONS +21 +|J3| ≤ C +� +e|y(ϑtω)|∥f∥L2(Rd) + ∥u∥H + ∥∇u∥H +� +∥¯ui,2∥L2(O√ +2k) +≤ Cλ +− 1 +2 +i+1 +� +∥u∥H + ∥∇u∥H + e|y(ϑtω)|∥f∥L2(Rd) +� +∥∇¯ui,2∥L2(O√ +2k) +≤ µ +20∥∇¯ui,2∥2 +L2(O√ +2k) + Cλ−1 +i+1 +� +∥u∥2 +H + ∥∇u∥2 +H + e2|y(ϑtω)|∥f∥2 +L2(Rd) +� +. +(3.59) +Using integration by parts, divergence free condition and (3.42), we obtain +|J2| += +�����e−y(ϑtω) +� +O√ +2k +(I − Pi)pρ′ +�|x|2 +k2 +� 4 +k2 (x · ¯u)dx +����� +≤ Ce|y(ϑtω)| +� +O√ +2k +��(−∆)−1� +∇ · +� +∇ · +� +u ⊗ u +����� · |¯ui,2|dx ++Ce(r−1)y(ϑtω) +� +O√ +2k +��(−∆)−1� +∇ · +� +|u|r−1u +���� · |¯ui,2|dx + Ce|y(ϑtω)| +� +O√ +2k +|(−∆)−1[∇ · f]| · |¯ui,2|dx +=: C +� +�S1(d, r) + �S2(d, r) + �S3(d, r) +� +. +(3.60) +Estimate of �S1(d, r): Using H¨older’s inequality, Fourier transformation, Ladyzhenskaya’s and Young’s +inequalities, respectively, we get for d = 2, 3 (similar to (3.58) above), +|�S1(d, r)| ≤ e|y(ϑtω)|��(−∆)−1� +∇ · +� +∇ · +� +u ⊗ u +����� +L2(Rd)∥¯ui,2∥L2(O√ +2k) +≤ e|y(ϑtω)|∥u∥2 +�L4∥¯ui,2∥L2(O√ +2k) +≤ µ +20∥∇¯ui,2∥2 +L2(O√ +2k) + Cλ +− 4−d +4+d +i+1 +� +∥∇u∥2 +H + e +8|y(ϑtω)| +4−d +∥u∥ +2(8−d) +4−d +H +� +. +(3.61) +Estimate of �S2(d, r): Divergence free condition gives �S2(d, r) = 0 for r = 1. Therefore, we will consider +r ∈ [2, ∞) for d = 2 and r ∈ [3, ∞) for d = 3. Applying H¨older’s, Gagliardo-Nirenberg’s, interpolation and +Young’s inequalities, we obtain +|�S2(d, r)| ≤ e(r−1)|y(ϑtω)| × + + + + + + + + + + + + + +∥(−∆)−1� +∇ · +� +|u|r−1u +�� +∥L2(Rd)∥¯ui,2∥L2(O√ +2k), +for d = 2 and r ∈ [2, ∞), +∥(−∆)−1� +∇ · +� +|u|r−1u +�� +∥L2(Rd)∥¯ui,2∥L2(O√ +2k), +for d = 3 and r ∈ [3, 5], +∥(−∆)−1� +∇ · +� +|u|r−1u +�� +∥ +L +3(r+1) +2r−1 (Rd)∥¯ui,2∥ +L +3(r+1) +r+4 (O√ +2k), +for d = 3 and r ∈ (5, ∞), +≤ Ce(r−1)|y(ϑtω)| × + + + + + + + + + +∥u∥r +�Lr∥¯ui,2∥L2(O√ +2k), +for d = 2 and r ∈ [2, ∞), +∥u∥r +�L +6r +5 ∥¯ui,2∥L2(O√ +2k), +for d = 3 and r ∈ [3, 5], +∥u∥r +�Lr+1∥¯ui,2∥ +L +3(r+1) +r+4 (O√ +2k), +for d = 3 and r ∈ (5, ∞), + +22 +KUSH KINRA, MANIL T. MOHAN, AND RENHAI WANG +≤ Ce(r−1)|y(ϑtω)| × + + + + + + + + + + + +∥u∥ +(r+1)(r−2) +r−1 +�Lr+1 +∥u∥ +2 +(r−1) +H +∥¯ui,2∥L2(O√ +2k), +for d = 2 and r ∈ [2, ∞), +∥u∥ +(r+1)(3r−5) +3(r−1) +�Lr+1 +∥u∥ +5−r +3(r−1) +H +∥¯ui,2∥L2(O√ +2k), +for d = 3 and r ∈ [3, 5], +∥u∥ +(r+1)(3r−5) +3(r−1) +�Lr+1 +∥¯ui,2∥ +2(r+1) +3(r−1) +L2(O√ +2k), +for d = 3 and r ∈ (5, ∞), +≤ Ce(r−1)|y(ϑtω)| × + + + + + + + + + + + + + +λ +− +1 +2(r−1) +i+1 +∥u∥ +(r+1)(r−2) +r−1 +�Lr+1 +∥u∥ +r +r−1 +H +∥∇¯ui,2∥ +1 +r−1 +L2(O√ +2k), +for d = 2 and r ∈ [2, ∞), +λ +− +1 +3(r−1) +i+1 +∥u∥ +(r+1)(3r−5) +3(r−1) +�Lr+1 +∥u∥ +2r +3(r−1) +H +∥∇¯ui,2∥ +2 +3(r−1) +L2(O√ +2k), for d = 3 and r ∈ [3, 5], +λ +− +1 +3(r−1) +i+1 +∥u∥ +(r+1)(3r−5) +3(r−1) +�Lr+1 +∥u∥ +2r +3(r−1) +H +∥∇¯ui,2∥ +2 +3(r−1) +L2(O√ +2k), for d = 3 and r ∈ (5, ∞), +≤ µ +20∥∇¯ui,2∥2 +L2(O√ +2k) + Cλ +− 1 +r2 +i+1 +� +e(r−1)|y(ϑtω)|∥u∥r+1 +�Lr+1 + e2(r−1)|y(ϑtω)|∥u∥2r +H +� +, +(3.62) +where we have used the fact that λi ≥ 1. +Estimate of �S3(d, r): Similar to (3.49), we find (for d = 2, 3) +|�S3(d, r)| ≤ Ce|y(ϑtω)|∥(−∆)−1[∇ · f]∥ +L +d +d−1 (Rd)∥¯ui,2∥Ld(O√ +2k) +≤ Ce|y(ϑtω)|∥f∥L1(Rd)∥¯ui,2∥ +4−d +2 +L2(O√ +2k)∥∇¯ui,2∥ +d−2 +2 +L2(O√ +2k) +≤ Cλ +− 4−d +4 +i+1 +e|y(ϑtω)|∥f∥L1(Rd)∥∇¯ui,2∥L2(O√ +2k) +≤ µ +20∥¯ui,2∥2 +L2(O√ +2k) + Cλ +− 4−d +2 +i+1 +e2|y(ϑtω)|∥f∥2 +L1(Rd). +(3.63) +Now, combining (3.57)-(3.63), we arrive at +d +dt∥¯ui,2∥2 +L2(O√ +2k) + (α − 2σy(ϑtω))∥¯ui,2∥2 +L2(O√ +2k) +≤ Cλ +− 1 +r2 +i+1 +� +∥u∥2 +H + e +8|y(ϑtω)| +4−d +∥u∥ +2(8−d) +4−d +H ++ e2(r−1)|y(ϑtω)|∥u∥2r +H + ∥∇u∥2 +H + e(r−1)y(ϑtω)∥u∥r+1 +�Lr+1 ++ e2|y(ϑtω)|∥f∥2 +L1(Rd) + e2|y(ϑtω)|∥f∥2 +L2(Rd) +� +. +(3.64) +In view of the variation of constant formula, we find +∥(I − Pi)¯u(s, s − t, ϑ−sω, ¯u0,2)∥2 +L2(O√ +2k) +≤ e−αt+2σ � 0 +−t y(ϑηω)dη∥(I − Pi)(̺ku0)∥2 +L2(O√ +2k) ++ Cλ +− 1 +r2 +i+1 +� � s +s−t +eα(ζ−s)−2σ +� ζ +s y(ϑη−sω)dη∥u(ζ, s − t, ϑ−sω, u0)∥2 +Hdζ ++ +� s +s−t +e +8|y(ϑζ−sω)| +4−d ++α(ζ−s)−2σ � ζ +s y(ϑη−sω)dη∥u(ζ, s − t, ϑ−sω, u0)∥ +2(8−d) +4−d +H +dζ ++ +� s +s−t +e2(r−1)|y(ϑζ−sω)|+α(ζ−s)−2σ � ζ +s y(ϑη−sω)dη∥u(ζ, s − t, ϑ−sω, u0)∥2r +H dζ ++ +� s +s−t +eα(ζ−s)−2σ +� ζ +s y(ϑη−sω)dη∥∇u(ζ, s − t, ϑ−sω, u0)∥2 +Hdζ + +ASYMPTOTIC AUTONOMY OF RANDOM ATTRACTORS FOR SCBF EQUATIONS +23 ++ +� s +s−t +e(r−1)y(ϑζ−sω)+α(ζ−s)−2σ � ζ +s y(ϑη−sω)dη∥u(ζ, s − t, ϑ−sω, u0)∥r+1 +�Lr+1dζ ++ +� 0 +−t +eαζ+2|y(ϑζω)|+2σ +� 0 +ζ y(ϑηω)dη� +∥f(ζ + s)∥2 +L1(Rd) + ∥f(ζ + s)∥2 +L2(Rd) +� +dζ +� +. +(3.65) +Since, ∥(I − Pi)(̺ku0)∥2 +L2(O√ +2k) ≤ C∥u0∥2 +H, for all u0 ∈ D(s − t, ϑ−tω) and s ≤ τ. Now, using the definition +of backward temperedness (2.12), (2.9), (1.7), Lemma 3.6 (in particular, (3.32) and (3.33)) and the fact that +λi → ∞ as i → ∞, we obtain (3.55), as desired, which completes the proof. +□ +3.5. Proof of Theorem 1.2. This subsection is devoted to the main result of this section, that is, the +existence of pullback D-random attractors and their asymptotic autonomy for the solution of the system +(2.6) with S(v) = v. For all the cases given in Table 1, the existence of pullback random attractors for +non-autonomous SCBF equations driven by multiplicative noise on the whole space is established in [37]. +For all the cases given in Table 1, as the existence of a unique pullback random attractor is known for each +τ, one can obtain the existence of a unique random attractor for autonomous SCBF equations driven by +multiplicative noise on the whole space (cf. [37]). +In view of Propositions 3.5 and 3.7, and Lemmas 3.8-3.9, the proof of Theorem 1.2 can be completed by +applying similar arguments as in the proof of [73, Theorem 1.6] ([73, Subsection 3.5]) and [9, Theorem 5.2]. +4. 2D and 3D SCBF equations: Additive noise +In this section, we consider SCBF equations driven by additive white noise, that is, S(v) is independent +of v and establish the asymptotic autonomy of pullback random attractors. Let us consider the following +SCBF equations: +(4.1) + + + +dv(t) +dt ++ µAv(t) + B(v(t)) + αv(t) + βC(v(t)) = Pf(t) + g(x)dW(t) +dt +, +t > τ, τ ∈ R, +v(x)|t=τ = vτ(x), +x ∈ Rd, +where g ∈ D(A) and W(t, ω) is the standard scalar Wiener process on the probability space (Ω, F, P) (see +Section 3 above). Let us define u(t, τ, ω, uτ) := v(t, τ, ω, vτ) − g(x)y(ϑtω), where y is given in (2.7) and +satisfies (2.8), and v is the solution of (1.1) with S(v) = g(x). Then u satisfies: +(4.2) + + + + + + + + + + + + + + + + + + + + + +du +dt − µ∆u + +� +(u + gy(ϑtω)) · ∇ +� +(u + gy(ϑtω)) + αu + β|u + gy(ϑtω)|r−1(u + gy(ϑtω)) += −∇p + f + (σ − α)gy(ϑtω) + µy(ϑtω)∆g, +in Rd × (τ, ∞), +∇ · u = 0, +in +Rd × (τ, ∞), +u(x)|t=τ = uτ(x) = vτ(x) − g(x)y(ϑτω), +x ∈ Rd and τ ∈ R, +u(x)|t=τ → 0, +as |x| → ∞, + +24 +KUSH KINRA, MANIL T. MOHAN, AND RENHAI WANG +as well as (projected form) +(4.3) + + + + + + + + + +du +dt + µAu + B(u + gy(ϑtω)) + αu + βC(u + gy(ϑtω)) += Pf + (σ − α)gy(ϑtω) + µy(ϑtω)∆g, +t > τ, +τ ∈ R, +u(x)|t=τ = uτ(x) = v0(x) − g(x)y(ϑτω), +x ∈ Rd, +in V′ + �L +r+1 +r , where r ≥ 1. For r = 1, we obtain SNSEs with linear damping. The results of this section can +be proven in a similar way as it has been done for SNSEs in [73] under Assumption 1.9. Since, Assumption +1.9 is not required for r > 1, we provide a different treatment for r > 1. +4.1. NRDS. The following lemma will be frequently used. +Lemma 4.1. For all the cases given in Table 1 (excluding d = 2 with r = 1), assume that f ∈ L2 +loc(R; L2(Rd)). +Then, the solution of (4.3) satisfies the following inequality: +d +dt∥u(t)∥2 +H + α∥u(t)∥2 +H + µ∥∇u(t)∥2 +H + β∥u(t) + gy(ϑtω)∥r+1 +�Lr+1 +≤ R +� +∥f(t)∥2 +L2(Rd) + |y(ϑtω)|2 + |y(ϑtω)|r+1 + |y(ϑtω)| +2(r+1) +r−1 +� +, +(4.4) +for a.e. t, where R > 0 is some constant. +Proof. We find from (4.3) that +1 +2 +d +dt∥u∥2 +H = − µ∥∇u∥2 +H − α∥u∥2 +H − β∥u + gy∥r+1 +�Lr+1 + b(u + gy, u + gy, gy) ++ β⟨C(u + gy), gy⟩ + (f, u) + y((σ − α)g − µAg, u), +(4.5) +for a.e. t ∈ [τ, τ + T] with T > 0. Using g ∈ D(A), H¨older’s and Young’s inequalities, there exist constants +R1, R2, R3, R4 > 0 such that +β⟨C(u + gy), gy⟩ ≤ β|y|∥u + gy∥r +�Lr+1∥g∥�Lr+1 ≤ β +4 ∥u + gy∥r+1 +�Lr+1 + R1|y|r+1, +(4.6) +(f, u) ≤ ∥f∥L2(Rd)∥u∥H ≤ α +12∥u∥2 +H + R2∥f∥2 +L2(Rd), +(4.7) +y +� +(σ − α)g − µAg, u +� +≤ α +12∥u∥2 +H + R3|y|2, +(4.8) +|b(u + gy, u + gy, gy)| = |b(u + gy, u, gy)| +≤ |y|∥u + gy∥�Lr+1∥∇u∥H∥g∥ +�L +2(r+1) +r−1 +≤ β +4 ∥u + gy∥r+1 +�Lr+1 + µ +2 ∥∇u∥2 +H + R4|y| +2(r+1) +r−1 . +(4.9) +Combining (4.5)-(4.9), we reach at (4.4) with R = max{2R1, 2R2, 2R3, 2R4}, as required. +□ +Lemma 4.2. For all the cases given in Table 1 (excluding d = 2 with r = 1), assume that f ∈ L2 +loc(R; L2(Rd)). +For each (τ, ω, uτ) ∈ R × Ω × H, the system (4.3) has a unique solution u(·, τ, ω, uτ) ∈ C([τ, +∞); H) ∩ +L2 +loc(τ, +∞; V) ∩ Lr+1 +loc (τ, +∞; �Lr+1) such that u is continuous with respect to the initial data. + +ASYMPTOTIC AUTONOMY OF RANDOM ATTRACTORS FOR SCBF EQUATIONS +25 +Proof. One can prove the existence and uniqueness of solution by a standard Faedo-Galerkin approximation +method, see the works [31, 34, 47], etc. For the continuity with respect to the initial data uτ, see the proof +of Theorem 3.9 in [34]. +□ +Next result shows the Lusin continuity of the mapping of the solution to the system (4.3) in sample points. +Proposition 4.3. For all the cases given in Table 1 (excluding d = 2 with r = 1), suppose that f ∈ +L2 +loc(R; L2(Rd)). For each N ∈ N, the mapping ω �→ u(t, τ, ω, uτ) (solution of (4.3)) is continuous from +(ΩN, dΩN ) to H, uniformly in t ∈ [τ, τ + T] with T > 0. +Proof. Let us assume ωk, ω0 ∈ ΩN be such that dΩN (ωk, ω0) → 0 as k → ∞. Let U k(·) := uk(·) − u0(·), +where uk(·) = u(·, τ, ωk, uτ) and u0(·) = u(·, τ, ω0, uτ). Then, U k(·) satisfies: +dU k +dt += −µAU k − αU k − +� +B +� +uk + y(ϑtωk)g +� +− B +� +u0 + y(ϑtω0)g +�� +− +� +βC +� +uk + y(ϑtωk)g +� +− βC +� +u0 + y(ϑtω0)g +�� ++ {(σ − α)g + µ∆g}[y(ϑtωk) − y(ϑtω0)], +(4.10) +in V′ + �L +r+1 +r . Taking the inner product with U k(·) in (4.10), using (2.4) and rearranging the terms, we +obtain +1 +2 +d +dt∥U k∥2 +H + µ∥∇U k∥2 +H + α∥U k∥2 +H += −b +� +U k + [y(ϑtωk) − y(ϑtω0)]g, U k + [y(ϑtωk) − y(ϑtω0)]g, u0 + y(ϑtω0)g +� ++ [y(ϑtωk) − y(ϑtω0)] +� +b +� +uk + y(ϑtωk)g, uk, g +� +− b +� +u0 + y(ϑtω0)g, u0, g +�� +− β +� +C +� +uk + y(ϑtωk)g +� +− C +� +u0 + y(ϑtω0)g +� +, +� +uk + y(ϑtωk)g +� +− +� +u0 + y(ϑtω0)g +�� ++ β[y(ϑtωk) − y(ϑtω0)] +� +C +� +uk + y(ϑtωk)g +� +− C +� +u0 + y(ϑtω0)g +� +, g +� ++ [y(ϑtωk) − y(ϑtω0)] +� +(σ − α)g + µ∆g, U k� +. +(4.11) +From (2.5), we get +−β +� +C +� +uk + y(ϑtωk)g +� +− C +� +u0 + y(ϑtω0)g +� +, +� +uk + y(ϑtωk)g +� +− +� +u0 + y(ϑtω0)g +�� +≤ −β +2 +���� +��� +� +uk + y(ϑtωk)g +���� +r−1 +2 � +U k + (y(ϑtωk) − y(ϑtω0))g +����� +2 +H +− β +2 +���� +��� +u0 + y(ϑtω0)g +��� +r−1 +2 +� +U k + (y(ϑtωk) − y(ϑtω0))g +����� +2 +H +. +(4.12) +For r > 1 and g ∈ D(A), in view of H¨older’s and Young’s inequalities, we obtain +� +b +� +uk + y(ϑtωk)g, uk, g +� +− b +� +u0 + y(ϑtω0)g, u0, g +�� +≤ +� +∥uk + y(ϑtωk)g∥�Lr+1∥∇uk∥H + ∥u0 + y(ϑtω0)g∥�Lr+1∥∇u0∥H +� +∥g∥ +�L +2(r+1) +r−1 +≤ C +� +∥uk + y(ϑtωk)g∥r+1 +�Lr+1 + ∥∇uk∥2 +H + ∥u0 + y(ϑtω0)g∥r+1 +�Lr+1 + ∥∇u0∥2 +H + 1 +� +. +(4.13) + +26 +KUSH KINRA, MANIL T. MOHAN, AND RENHAI WANG +Moreover, we have +β +� +C +� +uk + y(ϑtωk)g +� +− C +� +u0 + y(ϑtω0)g +� +, g +� +≤ +� +∥uk + y(ϑtωk)g∥r +�Lr+1 + ∥u0 + y(ϑtω0)g∥r +�Lr+1 +� +∥g∥�Lr+1 +≤ C +� +∥uk + y(ϑtωk)g∥r+1 +�Lr+1 + ∥u0 + y(ϑtω0)g∥r+1 +�Lr+1 + 1 +� +, +(4.14) +� +(σ − α)g + µ∆g, U k� +≤ C∥uk − u0∥H ≤ C +� +∥uk∥2 +H + ∥u0∥2 +H + 1 +� +. +(4.15) +Next, we estimate the remaining terms of (4.11) separately. +Case I: d = 2 and r ≥ 2. Applying (2.2), (2.3) and Young’s inequality, we estimate +���b +� +U k + [y(ϑtωk) − y(ϑtω0)]g, U k + [y(ϑtωk) − y(ϑtω0)]g, u0 + y(ϑtω0)g +���� +≤ C∥U k + (y(ϑtωk) − y(ϑtω0))g∥H∥∇U k + (y(ϑtωk) − y(ϑtω0))∇g∥H +× ∥∇u0 + y(ϑtω0)∇g∥H +≤ C∥∇u0 + y(ϑtω0)∇g∥2 +H∥U k∥2 +H + C|y(ϑtωk) − y(ϑtω0)|2 +� +∥∇u0∥2 +H + |y(ϑtω0)|2 + 1 +� ++ µ +2 ∥∇U k∥2 +H. +(4.16) +Case II: d = 3 and r > 3. Using H¨older’s and Young’s inequalities, we infer +���b +� +U k + [y(ϑtωk) − y(ϑtω0)]g, U k + [y(ϑtωk) − y(ϑtω0)]g, u0 + y(ϑtω0)g +���� +≤ µ +2 ∥∇U k∥2 +H + C∥U k∥2 +H + C|y(ϑtωk) − y(ϑtω0)|2 ++ β +4 +���� +��u0 + y(ϑtω0)g +�� +r−1 +2 +� +U k + (y(ϑtωk) − y(ϑtω0))g +����� +2 +H +. +(4.17) +Case III: When d = r = 3 with 2βµ ≥ 1. Applying (2.2), H¨older’s and Young’s inequalities, we obtain +���b +� +U k + [y(ϑtωk) − y(ϑtω0)]g, U k + [y(ϑtωk) − y(ϑtω0)]g, u0 + y(ϑtω0)g +���� +≤ +���b +� +U k, U k + [y(ϑtωk) − y(ϑtω0)]g, u0 + y(ϑtω0)g +���� ++ |y(ϑtωk) − y(ϑtω0)| +���b +� +g, U k + [y(ϑtωk) − y(ϑtω0)]g, uk + y(ϑtωk)g +���� +≤ 1 +2β ∥∇U k∥H + β +2 +��� +��u0 + y(ϑtω0)g +�� +� +U k + (y(ϑtωk) − y(ϑtω0))g +���� +2 +H ++ C|y(ϑtωk) − y(ϑtω0)|2 + β +2 +��� +���uk + y(ϑtωk)g +��� +� +U k + (y(ϑtωk) − y(ϑtω0))g +���� +2 +H. +(4.18) +Combining (4.11)-(4.18), we arrive at +d +dt∥U k(t)∥2 +H ≤ C +� +�P(t)∥U k(t)∥2 +H + �Q(t) +� +, +(4.19) + +ASYMPTOTIC AUTONOMY OF RANDOM ATTRACTORS FOR SCBF EQUATIONS +27 +for a.e. t ∈ [τ, τ + T], T > 0, and where +�P = + + + + + +∥∇u0 + y(ϑtω0)∇g∥2 +H, +for d = 2 and r > 1, +1, +for d = 3 and r > 3, +0, +for d = r = 3 and 2βµ ≥ 1, +�Q = |y(ϑtωk) − y(ϑtω0)| +� +∥uk + y(ϑtωk)g∥r+1 +�Lr+1 + ∥uk∥2 +V + ∥u0 + y(ϑtω0)g∥r+1 +�Lr+1 + ∥u0∥2 +V + 1 +� ++ |y(ϑtωk) − y(ϑtω0)|2 × + + + + + +∥∇u0∥2 +H + |y(ϑtω0)|2 + 1, for d = 2 and r > 1, +1, +for d = 3 and r > 3, +1, +for d = r = 3 and 2βµ ≥ 1. +We infer from (4.4) that +� τ+T +τ +� +∥uk(t) + y(ϑtωk)g∥r+1 +�Lr+1 + ∥uk(t)∥2 +H + ∥∇uk(t)∥2 +H +� +dt +≤ ∥uτ∥2 +L2(Rd) + C +� τ+T +τ +� +∥f(t)∥2 +H + |y(ϑtωk)|2 + |y(ϑtωk)|r+1 + |y(ϑtωk)| +2(r+1) +r−1 +� +dt, +which gives +sup +k∈N +� τ+T +τ +� +∥uk(t) + y(ϑtωk)g∥r+1 +�Lr+1 + ∥uk(t)∥2 +H + ∥∇uk(t)∥2 +H +� +dt ≤ C(τ, T, ω0), +(4.20) +where we have used (2.11) and the fact that f ∈ L2 +loc(R; L2(Rd)). +It implies from (2.11) and u0 ∈ +L2 +loc(τ, +∞; V) that +� τ+T +τ +�P(t)dt ≤ C(τ, T, ω0). +(4.21) +Now, from f ∈ L2 +loc(R; L2(Rd)), u0 ∈ C([τ, +∞); H) ∩ L2 +loc(τ, +∞; V) ∩ Lr+1 +loc (τ, +∞; �Lr+1), Lemma 2.4 and +(4.20), we conclude that +lim +k→+∞ +� τ+T +τ +�Q(t)dt = 0. +(4.22) +In view of the Gronwall inequality in (4.19) and making use of (4.21)-(4.22), one can complete the proof. +□ +Lemma 4.2 ensures us that we can define a mapping �Φ : R+ × R × Ω × H → H by +�Φ(t, τ, ω, vτ) := v(t + τ, τ, ϑ−τω, vτ) = u(t + τ, τ, ϑ−τω, uτ) + gy(ϑtω). +(4.23) +The Lusin continuity in Proposition 4.3 provides the F-measurability of �Φ. Consequently, �Φ defined by +(4.23) is a NRDS on H. +4.2. Backward convergence of NRDS. Consider the autonomous SCBF equations driven by the additive +white noise: +(4.24) + + + +d�v(t) +dt ++ µA�v(t) + B(�v(t)) + α�v(t) + βC(�v(t)) = Pf ∞ + g(x)dW(t) +dt +, +t > 0, +�v(x, 0) = �v0(x), +x ∈ Rd. + +28 +KUSH KINRA, MANIL T. MOHAN, AND RENHAI WANG +Let �u(t, ω) = �v(t, ω) − g(x)y(ϑtω). +Then, the system (4.24) can be written in the following pathwise +deterministic system: +(4.25) + + + + + + + + + +d�u(t) +dt ++ µA�u(t) + B(�u(t) + gy(ϑtω)) + α�u(t) + βC(�u(t) + gy(ϑtω)) += Pf ∞ + (σ − α)gy(ϑtω) − µy(ϑtω)Ag, +t > 0, +�u(x, 0) = �u0(x) = �v0(x) − g(x)y(ω), +x ∈ Rd, +in V′ + �L +r+1 +r . +Proposition 4.4. For all the cases given in Table 1 (excluding d = 2 with r = 1), suppose that Assumption +1.1 is satisfied and +lim +τ→−∞ ∥uτ − �u0∥H = 0. Then, the solution u of the system (4.3) backward converges to +the solution �u of the system (4.25), that is, +lim +τ→−∞ ∥u(T + τ, τ, ϑ−τω, uτ) − �u(t, ω, �u0)∥H = 0, +for all T > 0 and ω ∈ Ω. +(4.26) +Proof. Let U τ(·) := u(· + τ, τ, ϑ−τω, uτ) − �u(·, ω, �u0). From (4.3) and (4.25), we get +dU τ +dt += −µAU τ − αU τ − +� +B +� +u + gy(ϑtω) +� +− B +� +�u + gy(ϑtω) +�� +− β +� +C +� +u + gy(ϑtω) +� +− C +� +�u + gy(ϑtω) +�� ++ [Pf(t + τ) − Pf ∞], +(4.27) +in V′ + �L +r+1 +r . In view of (4.27), we obtain +d +dt∥U τ∥2 +H = −µ∥∇U τ∥2 +H − α∥U τ∥2 +H − +� +B +� +u + gy(ϑtω) +� +− B +� +�u + gy(ϑtω) +� +, u − �u +� +− β +� +C +� +u + gy(ϑtω) +� +− C +� +�u + gy(ϑtω) +� +, u − �u +� ++ (f(t + τ) − f ∞, U τ). +(4.28) +From (2.5), one can rewrite +−β +� +C +� +u + gy(ϑtω) +� +− C +��u + gy(ϑtω) +� +, (u + gy(ϑtω)) − (�u + gy(ϑtω)) +� +≤ −β +2 ∥|u + gy(ϑtω)| +r−1 +2 |U τ|∥2 +H − β +2 ∥|�u + gy(ϑtω)| +r−1 +2 |U τ|∥2 +H +(4.29) +Applying (2.2), (2.4), H¨older’s and Young’s inequalities, we infer +��� +B +� +u + gy(ϑtω) +� +− B +� +�u + gy(ϑtω) +� +, (u + gy(ϑtω)) − (�u + gy(ϑtω)) +��� += |b(U τ, U τ, �u + gy(ϑtω))| +≤ + + + + + + + +C∥∇�u + ∇gy(ϑtω)∥2 +H∥U τ∥2 +H + µ +2 ∥∇U τ∥2 +H, for d = 2 and r ≥ 1, +µ +2 ∥∇U τ∥2 +H + β +4 ∥|�u + gy(ϑtω)| +r−1 +2 |U τ|∥2 +H + C∥U τ∥2 +H, for d = 3 and r > 3, +1 +2β ∥∇U τ∥2 +H + β +2 ∥|�u + gy(ϑtω)||U τ|∥2 +H, for d = r = 3 and 2βµ ≥ 1, +(4.30) +and +|(f(t + τ) − f∞, U τ)| ≤ C∥f(t + τ) − f ∞∥2 +L2(Rd) + α +2 ∥U τ∥2 +H. +(4.31) + +ASYMPTOTIC AUTONOMY OF RANDOM ATTRACTORS FOR SCBF EQUATIONS +29 +Combining (4.28)-(4.31), we achieve +d +dt∥U τ∥2 +H ≤ C × + + + + + + + +∥∇�u + gy(ϑtω)∥2 +H∥U τ∥2 +H + ∥f(t + τ) − f ∞∥2 +L2(Rd), +for d = 2 and r ≥ 1, +∥U τ∥2 +H + ∥f(t + τ) − f ∞∥2 +L2(Rd), +for d = 3 and r > 3, +∥f(t + τ) − f ∞∥2 +L2(Rd), +for d = r = 3 and 2βµ ≥ 1. +(4.32) +Applying similar steps as in Proposition 3.5, we complete the proof. +□ +4.3. Increasing random absorbing sets. This subsection provides the existence of increasing D-random +absorbing set for non-autonomous SCBF equations (4.1). +Lemma 4.5. For all the cases given in Table 1 (excluding d = 2 with r = 1) and for each (τ, ω, D) ∈ +R × Ω × D, there exists a time �T := �T(τ, ω, D) > 0 such that +sup +s≤τ +sup +t≥�T +sup +u0∈D(s−t,ϑ−tω) +� +∥u(s, s − t, ϑ−sω, u0)∥2 +H + α +2 +� s +s−t +eα(ζ−s)∥u(ζ, s − t, ϑ−sω, u0)∥2 +Hdζ ++ µ +� s +s−t +eα(ζ−s)∥∇u(ζ, s − t, ϑ−sω, u0)∥2 +Hdζ ++ β +� s +s−t +eα(ζ−s)∥u(ζ, s − t, ϑ−sω, u0) + gy(ϑζ−sω)∥r+1 +�Lr+1dζ +� +≤ 2R sup +s≤τ +�K(s, ω), +(4.33) +where R is the same as in (4.4) and �K(s, ω) is given by +�K(s, ω) := +� 0 +−∞ +eαζ +� +∥f(ζ + s)∥2 +L2(Rd) + |y(ϑζω)|2 + |y(ϑζω)|r+1 + |y(ϑζω)| +2(r+1) +r−1 +� +dζ. +(4.34) +Furthermore, for 2 < k1 < ∞, there exists a time �T∗ := �T∗(τ, ω, D, k1) > 0 such that +sup +s≤τ +sup +t≥�T∗ +sup +u0∈D(s−t,ϑ−tω) +� s +s−t +eα(ζ−s)∥u(ζ, s − t, ϑ−sω, u0)∥k1 +H dζ +≤ Ck1 +α +� � 0 +−∞ +e +2(k1−1)α +k2 +1 +ζ� +∥f(ζ + s)∥2 +L2(Rd) + |y(ϑζω)|2 + |y(ϑζω)|r+1 + |y(ϑζω)| +2(r+1) +r−1 +� +dζ +� k1 +2 +. +(4.35) +Proof. Let us consider the energy inequality (4.4) for u(ζ) = u(ζ, s − t, ϑ−sω, u0), that is, +d +dζ ∥u(ζ)∥2 +H + α∥u(ζ)∥2 +H + α +2 ∥u(ζ)∥2 +H + µ∥∇u(ζ)∥2 +H + β∥u(ζ) + gy(ϑζ−sω)∥r+1 +�Lr+1 +≤ R +� +∥f(ζ)∥2 +L2(Rd) + |y(ϑζ−sω)|2 + |y(ϑζ−sω)|r+1 + |y(ϑζ−sω)| +2(r+1) +r−1 +� +, +In view of the variation of constants formula with respect to ζ ∈ (s − t, ξ), +∥u(ξ, s − t, ϑ−sω, u0)∥2 +H + α +2 +� ξ +s−t +eα(ζ−ξ)∥u(ζ, s − t, ϑ−sω, u0)∥2 +Hdζ ++µ +� ξ +s−t +eα(ζ−ξ)∥∇u(ζ, s − t, ϑ−sω, u0)∥2 +Hdζ + β +� ξ +s−t +eα(ζ−ξ)∥u(ζ, s − t, ϑ−sω, u0) + gy(ϑζ−sω)∥r+1 +�Lr+1dζ +≤ e−α(ξ−s+t)∥u0∥2 +H + +30 +KUSH KINRA, MANIL T. MOHAN, AND RENHAI WANG ++R +� ξ−s +−t +eα(ζ+s−ξ) +� +∥f(ζ + s)∥2 +L2(Rd) + |y(ϑζω)|2 + |y(ϑζω)|r+1 + |y(ϑζω)| +2(r+1) +r−1 +� +dζ. +(4.36) +Putting ξ = s in (4.36), we find +∥u(s, s − t, ϑ−sω, u0)∥2 +H + α +2 +� s +s−t +eα(ζ−s)∥u(ζ, s − t, ϑ−sω, u0)∥2 +Hdζ ++ µ +� s +s−t +eα(ζ−s)∥∇u(ζ, s − t, ϑ−sω, u0)∥2 +Hdζ ++ β +� s +s−t +eα(ζ−s)∥u(ζ, s − t, ϑ−sω, u0) + gy(ϑζ−sω)∥r+1 +�Lr+1dζ +≤ e−αt∥u0∥2 +H + R +� 0 +−∞ +eαζ +� +∥f(ζ + s)∥2 +L2(Rd) + |y(ϑζω)|2 + |y(ϑζω)|r+1 + |y(ϑζω)| +2(r+1) +r−1 +� +dζ, +(4.37) +for all s ≤ τ. Since u0 ∈ D(s−t, ϑ−tω) and D is backward tempered, the definition of backward temperedness +(2.12) ensures that there exists a time �T = �T(τ, ω, D) such that for all t ≥ �T, +e−αt sup +s≤τ +∥u0∥2 +H ≤ R +� 0 +−∞ +eαζ +� +∥f(ζ + s)∥2 +L2(Rd) + |y(ϑζω)|2 + |y(ϑζω)|r+1 + |y(ϑζω)| +2(r+1) +r−1 +� +dζ. +(4.38) +Hence, Using (4.38) and taking supremum on s over (−∞, τ] in (4.37), we arrive at (4.33). Furthermore, +the inequality (4.35) can be obtained by using (4.36) and following the similar arguments as in (3.38). +□ +Proposition 4.6. For all the cases given in Table 1 (excluding d = 2 with r = 1), suppose that f ∈ +L2 +loc(R; L2(Rd)). For R and �K(s, ω), the same as in (4.4) and (4.34), respectively, we have +(i) There is an increasing pullback D-random absorbing set R given by +R(τ, ω) := +� +v ∈ H : ∥v∥2 +H ≤ 4R sup +s≤τ +�K(s, ω) + 2∥g∥2 +H|y(ω)|2 +� +, for all τ ∈ R and ω ∈ Ω. +(4.39) +Moreover, R is backward-uniformly tempered with arbitrary rate, that is, R ∈ D. +(ii) There is a B-pullback random absorbing set �R given by +�R(τ, ω) := +� +v ∈ H : ∥v∥2 +H ≤ 4R �K(τ, ω) + 2∥g∥2 +H|y(ω)|2� +∈ B, for all τ ∈ R and ω ∈ Ω. +(4.40) +Proof. See the proof of [73, Proposition 3.6]. +□ +4.4. Backward uniform tail-estimates and backward flattening-property. In this subsection, we +prove the backward tail-estimates and backward flattening-property for the solution of (4.2) for all the cases +given in Table 1 (excluding d = 2 with r = 1). These estimates help us to prove the backward uniform +pullback D-asymptotic compactness of the solution of (4.3). We will use the cut-off function (same as in +Lemma 3.8) to obtain these estimates. The following lemma provides the backward uniform tail-estimates +for the solution of the system (4.2). +Lemma 4.7. For all the cases given in Table 1 (excluding d = 2 with r = 1), suppose that f ∈ L2 +loc(R; L2(Rd)). +Then, for any (τ, ω, D) ∈ R × Ω × D, the solution of (4.1) satisfies +lim +k,t→+∞ sup +s≤τ +sup +u0∈D(s−t,ϑ−tω) +∥u(s, s − t, ϑ−sω, u0)∥2 +L2(Oc +k) = 0, +(4.41) + +ASYMPTOTIC AUTONOMY OF RANDOM ATTRACTORS FOR SCBF EQUATIONS +31 +where Ok = {x ∈ Rd : |x| ≤ k}. +Proof. Let ρ be a smooth function defined in Lemma 3.8. Taking divergence to the first equation in (4.2), +formally we obtain (see the proof of Lemma 3.8 the detailed calculations) +p = (−∆)−1� +∇ · +� +∇ · +� +(u + gy) ⊗ (u + gy) +�� ++ β∇ · +� +|u + gy|r−1(u + gy) +� +− ∇ · f +� +. +(4.42) +Taking the inner product to the first equation of (4.2) with ρ +� +|x|2 +k2 +� +u, we have +1 +2 +d +dt +� +Rd ρ +�|x|2 +k2 +� +|u|2dx += µ +� +Rd(∆u)ρ +�|x|2 +k2 +� +udx − α +� +Rd ρ +�|x|2 +k2 +� +|u|2dx − b +� +u + gy, u + gy, ρ +�|x|2 +k2 +� +(u + gy) +� ++ b +� +u + gy, u + gy, ρ +�|x|2 +k2 +� +gy +� +− β +� +Rd ρ +�|x|2 +k2 +� +|u + gy|r+1dx ++ β +� +Rd|u + gy|r−1(u + gy)ρ +�|x|2 +k2 +� +gydx − +� +Rd(∇p)ρ +�|x|2 +k2 +� +udx + +� +Rd fρ +�|x|2 +k2 +� +udx ++ (σ − α)y +� +Rd gρ +�|x|2 +k2 +� +udx + µy +� +Rd(∆g)ρ +�|x|2 +k2 +� +udx. +(4.43) +Let us now estimate each terms on right hand side of (4.43). Using integration by parts, divergence free +condition of u(·) and g ∈ D(A), we infer (see inequalities (3.44)-(3.50)) +µ +� +Rd(∆u)ρ +�|x|2 +k2 +� +udx ≤ −µ +� +Rd |∇u|2ρ +�|x|2 +k2 +� +dx + C +k +� +∥u∥2 +H + ∥∇u∥2 +H +� +, +(4.44) +y2 b +� +u + gy, g, ρ +�|x|2 +k2 +� +g +� +≤ C +k +� +∥u + gy∥2 +H + |y|4∥g∥4 +�L4 +� +≤ C +k +� +∥u∥2 +H + |y|2 + |y|4� +, +(4.45) +−b +� +u + gy, u + gy, ρ +�|x|2 +k2 +� +(u + gy) +� +≤ C +k ∥u + gy∥3 +�L3 ≤ C +k +� +∥u + gy∥2 +H + ∥u + gy∥r+1 +�Lr+1 +� +≤ C +k +� +∥u∥2 +H + |y|2 + ∥u + gy∥r+1 +�Lr+1 +� +, +for r ≥ 2, +(4.46) +where we have used interpolation and Young’s inequalities. +Using integration by parts, divergence free +condition and (4.42), we obtain for r ≥ 2, +− +� +Rd(∇p)ρ +�|x|2 +k2 +� +udx = +� +Rd pρ′ +�|x|2 +k2 +� 2 +k2 (x · u)dx +≤ C +k +� +Rd +��(−∆)−1� +∇ · +� +∇ · +� +(u + gy) ⊗ (u + gy) +����� · |u|dx ++ C +k +� +Rd +��(−∆)−1� +∇ · +� +|u + gy|r−1(u + gy) +���� · |u|dx + C +k +� +Rd |(−∆)−1[∇ · f]| · |u|dx +=: C +k [Q1(d, r) + Q2(d, r) + Q3(d, r)]. +(4.47) + +32 +KUSH KINRA, MANIL T. MOHAN, AND RENHAI WANG +Estimate of Q1(d, r): Using g ∈ D(A), H¨older’s inequality, Fourier transformation, Ladyzhenskaya’s and +Young’s inequalities, respectively, we get for d = 2, 3, +|Q1(d, r)| ≤ +��(−∆)−1� +∇ · +� +∇ · +� +(u + gy) ⊗ (u + gy) +����� +L2(Rd)∥u∥H ≤ ∥u + gy∥2 +�L4∥u∥H +≤ C∥u∥2 +�L4∥u∥H + C|y|2∥g∥2 +�L4∥u∥H +≤ C∥u∥ +6−d +2 +H +∥∇u∥ +d +2 +H + C|y|4 + C∥u∥2 +H +≤ C +� +∥∇u∥2 +H + ∥u∥2 +H + ∥u∥ +2(6−d) +4−d +H ++ |y|4 +� +. +(4.48) +Estimate of Q2(d, r): Applying H¨older’s (see (3.48)), Gagliardo-Nirenberg’s (see (3.48)), interpolation and +Young’s inequalities, we obtain +|Q2(d, r)| ≤ C × + + + + + + + +∥u + gy∥r +�Lr∥u∥H, +for d = 2 and r ∈ [2, ∞), +∥u + gy∥r +�L +6r +5 ∥u∥H, +for d = 3 and r ∈ [3, 5], +∥u + gy∥r +�Lr+1∥u∥ +�L +3(r+1) +r+4 , +for d = 3 and r ∈ (5, ∞), +≤ C × + + + + + + + + + + + +∥u + gy∥ +(r+1)(r−2) +r−1 +�Lr+1 +∥u + gy∥ +2 +r−1 +H +∥u∥H, +for d = 2 and r ∈ [2, ∞), +∥u + gy∥ +(r+1)(3r−5) +3(r−1) +�Lr+1 +∥u + gy∥ +5−r +3(r−1) +H +∥u∥H, +for d = 3 and r ∈ [3, 5], +∥u + gy∥ +(r+1)(3r−5) +3(r−1) +�Lr+1 +∥u∥ +2(r+1) +3(r−1) +H +, +for d = 3 and r ∈ (5, ∞), +≤ C +� +∥u + gy∥r+1 +�Lr+1 + ∥u∥r+1 +H ++ |y|r+1� +, +(4.49) +where we have used interpolation and Young’s inequalities. +Estimate of Q3(d, r): Similar to (3.49), we find (for d = 2, 3) +|Q3(d, r)| ≤ C∥(−∆)−1[∇ · f]∥ +L +d +d−1 (Rd)∥u∥Ld(Rd) ≤ C +� +∥f∥2 +L1(Rd) + ∥u∥2 +H + ∥∇u∥2 +H +� +. +(4.50) +Finally, we estimate the remaining terms of (4.43) by using H¨older’s and Young’s inequalities as follows, +yb +� +u + gy, u, ρ +�|x|2 +k2 +� +g +� ++ βy +� +Rd|u + gy|r−1(u + gy)ρ +�|x|2 +k2 +� +gdx ++ +� +Rd f(x)ρ +�|x|2 +k2 +� +udx + (ℓ − α)y +� +Rd gρ +�|x|2 +k2 +� +udx + µy +� +Rd(∆g)ρ +�|x|2 +k2 +� +udx +≤ β +2 +� +Rd ρ +�|x|2 +k2 +� +|u + gy|r+1dx + µ +2 +� +Rd ρ +�|x|2 +k2 +� +|∇u|2dx + α +2 +� +Rd ρ +�|x|2 +k2 +� +|u|2dx ++ C +� +Rd ρ +�|x|2 +k2 +�� +|y| +2(r+1) +r−1 |g| +2(r+1) +r−1 + |y|r+1|g|r+1 + |f|2 + |y|2|g|2 + |y|2|∆g|2 +� +dx. +(4.51) +Combining (4.43)-(4.51), we get +d +dt∥u∥2 +L2(Oc +k) ≤ −α∥u∥2 +L2(Oc +k) + C +k +� +∥u∥2 +H + ∥∇u∥2 +H + ∥u + gy∥r+1 +�Lr+1 + ∥u∥ +2(6−d) +4−d +H ++ ∥u∥r+1 +H ++ ∥f∥2 +L1(Rd) ++ |y|2 + |y|4 + |y|r+1 +� ++ C|y| +2(r+1) +r−1 +� +|x|≥k +|g(x)| +2(r+1) +r−1 dx + C|y|r+1 +� +|x|≥k +|g(x)|r+1dx + +ASYMPTOTIC AUTONOMY OF RANDOM ATTRACTORS FOR SCBF EQUATIONS +33 ++ C +� +|x|≥k +|f(x)|2dx + C|y|2 +� +|x|≥k +|g(x)|2dx + C|y|2 +� +|x|≥k +|∆g(x)|2dx. +(4.52) +Making use of the variation of constant formula to the above equation (4.52) on (s − t, s) and replacing ω +by ϑ−sω, we find that, for s ≤ τ, t ≥ 0 and ω ∈ Ω, +∥u(s, s − t, ϑ−sω, u0)∥2 +L2(Oc +k) +≤ e−αt∥u0∥2 +H + C +k +� � s +s−t +eα(ζ−s) +� +∥u(ζ, s − t, ϑ−sω, u0)∥2 +H + ∥∇u(ζ, s − t, ϑ−sω, u0)∥2 +H ++ ∥u(ζ, s − t, ϑ−sω, u0) + gy(ϑζ−sω)∥r+1 +�Lr+1 + ∥u(ζ, s − t, ϑ−sω, u0)∥ +2(6−d) +4−d +H ++ ∥u(ζ, s − t, ϑ−sω, u0)∥r+1 +H +� +dζ + +� 0 +−∞ +eαζ +� +∥f(ζ + s)∥2 +L2(Rd) + |y(ϑζω)|2 + |y(ϑζω)|4 ++ |y(ϑζω)|r+1 +� +dζ +� ++ C +� 0 +−∞ +eαζ|y(ϑζω)| +2(r+1) +r−1 dζ +� +|x|≥k +|g(x)| +2(r+1) +r−1 dx ++ C +� 0 +−∞ +eαζ|y(ϑζω)|r+1dζ +� +|x|≥k +|g(x)|r+1dx + C +� 0 +−∞ +eαζ|y(ϑζω)|2dζ +� +|x|≥k +|g(x)|2dx ++ C +� 0 +−∞ +eαζ|y(ϑζω)|2dζ +� +|x|≥k +|∆g(x)|2dx + C +� 0 +−∞ +eαζ +� +|x|≥k +|f(x, ζ + s)|2dxdζ. +(4.53) +Now, using the definition of backward temperedness (2.12), (2.9), (1.8), g ∈ D(A) and Lemma 4.5 (both +(4.33) and (4.35)), one can complete the proof. +□ +Lemma 4.8. For all the cases given in Table 1 (excluding d = 2 with r = 1), suppose that f ∈ L2 +loc(R; H). +Let (τ, ω, D) ∈ R × Ω × D and k ≥ 1 be fixed. Then +lim +i,t→+∞ sup +s≤τ +sup +u0∈D(s−t,ϑ−tω) +∥(I − Pi)¯u(s, s − t, ϑ−sω, ¯u0,2)∥2 +L2(O√ +2k) = 0, +(4.54) +where ¯u0,2 = (I − Pi)(̺ku0). +Proof. The first equation of (4.2) can be rewritten as (multiplying by ̺k): +d¯u +dt − µ∆¯u + ̺k +� +(u + gy) · ∇ +� +(u + gy) + α¯u + ̺k|u + gy|r−1(u + gy) + ̺k∇p += −µu∆̺k − 2µ∇̺k · ∇u + ̺kf + (σ − α)̺kgy + µy̺k∆g. +(4.55) +Applying the projection (I − Pi) and taking the inner product with ¯ui,2 in L2(O√ +2k) to the equation (4.55), +we get +1 +2 +d +dt∥¯ui,2∥2 +L2(O√ +2k) + µ∥∇¯ui,2∥2 +L2(O√ +2k) + α∥¯ui,2∥2 +L2(O√ +2k) + β∥|u + gy| +r−1 +2 (¯ui,2 + ¯gi,2y)∥2 +L2(O√ +2k) += − +2 +� +q,m=1 +� +O√ +2k +(I − Pi) +� +(uq + gqy)∂(um + gmy) +∂xq +{̺k(x)}2(um + gmy) +� +dx +� +�� +� +:=L1 + +34 +KUSH KINRA, MANIL T. MOHAN, AND RENHAI WANG ++ y +2 +� +q,m=1 +� +O√ +2k +(I − Pi) +� +(uq + gqy)∂(um + gmy) +∂xq +{̺k(x)}2gm +� +dx +� +�� +� +:=L2 ++ y +� +O√ +2k +� +|u + gy|r−1(¯ui,2 + ¯gi,2y)¯gi,2 +� +dx +� +�� +� +:=L3 +− +� +̺k(x)∇p, ¯ui,2 +� +� +�� +� +:=L4 +− +� +µ +� +u∆̺k + 2∇̺k · ∇u, ¯ui,2 +� +− +� +̺kf, ¯ui,2 +� +− (σ − α)y +� +̺kg, ¯ui,2 +� +− µy +� +̺k∆g, ¯ui,2 +�� +� +�� +� +:=L5 +. +(4.56) +Next, we estimate each terms on the RHS of (4.56) as follows: +Estimate of L1: Using integration by parts, divergence free condition of u(·), (3.54) (WLOG we assume +that λi ≥ 1), H¨older’s, Ladyzhenskaya’s (for d = 2, 3) and Young’s inequalities, we find +|L1| = +����� +� +O√ +2k +(I − Pi) +� +ρ′ +�|x|2 +k2 +� x +k2 · {¯u + ¯gy}|u + gy|2 +� +dx +����� +≤ C∥¯ui,2 + ¯gi,2y∥L4(O√ +2k)∥u + gy∥�L4∥u + gy∥H +≤ C∥¯ui,2 + ¯gi,2y∥ +4−d +4 +L2(O√ +2k)∥∇(¯ui,2 + ¯gi,2y)∥ +d +4 +L2(O√ +2k)∥∇(u + gy)∥ +d +4 +H∥u + gy∥ +8−d +4 +H +≤ Cλ +− 4−d +8 +i+1 +∥u + gy∥ +4+d +4 +V +∥u + gy∥ +8−d +4 +H +≤ Cλ +− 4−d +8 +i+1 +� +∥u + gy∥2 +V + ∥u + gy∥ +2(8−d) +4−d +H +� +≤ Cλ +− 4−d +8 +i+1 +� +∥u∥2 +H + ∥∇u∥2 +H + ∥u∥ +2(8−d) +4−d +H ++ |y|2 + |y| +2(8−d) +4−d +� +. +(4.57) +Estimate of L2 and L3: Using g ∈ D(A), H¨older’s, Agmon’s, (3.54), interpolation and Young’s inequalities, +respectively, we get for d = 2 with r ≥ 2 and d = 3 with r ≥ 3, +|L2 + L3| ≤ |y|∥¯gi,2∥L∞(O√ +2k) +� +∥u + gy∥H∥∇(u + gy)∥H + ∥u + gy∥r +�Lr +� +≤ C∥¯gi,2∥ +4−d +4 +L2(O√ +2k)∥¯gi,2∥ +d +4 +H2(O√ +2k) +� +|y|∥u + gy∥H∥∇(u + gy)∥H ++ |y|∥u + gy∥ +2 +r−1 +H +∥u + gy∥ +(r+1)(r−2) +r−1 +�Lr+1 +� +≤ Cλ +− 4−d +8 +i+1 +∥∇¯gi,2∥ +4−d +4 +L2(O√ +2k)∥¯gi,2∥ +d +4 +H2(O√ +2k) +� +∥u + gy∥4 +H + ∥∇(u + gy)∥2 +H ++ ∥u + gy∥r+1 +�Lr+1 + |y|4 + |y|2(r−1) +� +≤ Cλ +− 4−d +8 +i+1 +� +∥∇u∥2 +H + ∥u∥4 +H + ∥u + gy∥r+1 +�Lr+1 + |y|2 + |y|4 + |y|2(r−1) +� +. +(4.58) +Estimate of L4: Using integration by parts, divergence free condition for u(·) and (4.42), we obtain +|L4| = +����� +� +O√ +2k +(I − Pi) +� +pρ′ +�|x|2 +k2 +� 4 +k2 (x · ¯u) +� +dx +����� + +ASYMPTOTIC AUTONOMY OF RANDOM ATTRACTORS FOR SCBF EQUATIONS +35 +≤ C +� +O√ +2k +��(−∆)−1� +∇ · +� +∇ · +� +(u + gy) ⊗ (u + gy) +����� · |¯ui,2|dx ++ C +� +O√ +2k +��(−∆)−1� +∇ · +� +|u + gy|r−1(u + gy) +���� · |¯ui,2|dx + C +� +O√ +2k +|(−∆)−1[∇ · f]| · |¯ui,2|dx +=: C +� +�Q1(d, r) + �Q2(d, r) + �Q3(d, r) +� +. +(4.59) +Estimate of �Q1(d, r): Using H¨older’s inequality, Fourier transformation, Ladyzhenskaya’s and Young’s in- +equalities, we get for d = 2, 3 +| �Q1(d, r)| ≤ +��(−∆)−1� +∇ · +� +∇ · +� +(u + gy) ⊗ (u + gy) +����� +L2(Rd)∥¯ui,2∥L2(O√ +2k) ≤ ∥u + gy∥2 +�L4∥¯ui,2∥L2(O√ +2k) +≤ Cλ +− 4−d +8 +i+1 +∥∇(u + gy)∥ +d +2 +H∥u + gy∥ +4−d +2 +H +∥∇¯ui,2∥ +4−d +4 +L2(O√ +2k)∥¯ui,2∥ +d +4 +L2(O√ +2k) +≤ Cλ +− 4−d +8 +i+1 +� +∥u + gy∥2 +V + ∥u + gy∥ +2(4−d) +2−d +H ++ ∥u∥2 +V + ∥u∥2 +H +� +≤ Cλ +− 4−d +8 +i+1 +� +∥u∥2 +H + ∥∇u∥2 +H + ∥u∥ +2(4−d) +2−d +H ++ |y|2 + |y| +2(4−d) +2−d � +. +(4.60) +Estimate of �Q2(d, r): Applying H¨older’s (see (3.62)), Gagliardo-Nirenberg’s (see (3.62)), interpolation and +Young’s inequalities, we find +| �Q2(d, r)| ≤ C × + + + + + + + + + +∥u + gy∥r +�Lr∥¯ui,2∥L2(O√ +2k), +for d = 2 and r ∈ [2, ∞), +∥u + gy∥r +�L +6r +5 ∥¯ui,2∥L2(O√ +2k), +for d = 3 and r ∈ [3, 5], +∥u + gy∥r +�Lr+1∥¯ui,2∥ +L +3(r+1) +r+4 (O√ +2k), +for d = 3 and r ∈ (5, ∞), +≤ C × + + + + + + + + + + + +∥u + gy∥ +(r+1)(r−2) +r−1 +�Lr+1 +∥u + gy∥ +2 +(r−1) +H +∥¯ui,2∥L2(O√ +2k), +for d = 2 and r ∈ [2, ∞), +∥u + gy∥ +(r+1)(3r−5) +3(r−1) +�Lr+1 +∥u + gy∥ +5−r +3(r−1) +H +∥¯ui,2∥L2(O√ +2k), +for d = 3 and r ∈ [3, 5], +∥u + gy∥r +�Lr+1∥u∥ +r−5 +3(r−1) +�Lr+1 ∥¯ui,2∥ +2(r+1) +3(r−1) +L2(O√ +2k), +for d = 3 and r ∈ (5, ∞), +≤ C × + + + + + + + + + + + + + + + + + + + + + + + + + + + +λ +− +1 +2(r−1) +i+1 +∥u + gy∥ +(r+1)(r−2) +r−1 +�Lr+1 +∥u + gy∥ +2 +r−1 +H +∥u∥ +r−2 +r−1 +H +∥∇¯ui,2∥ +1 +r−1 +L2(O√ +2k), +for d = 2 and r ∈ [2, ∞), +λ +− 1 +12 +i+1 ∥u + gy∥ +(r+1)(3r−5) +3(r−1) +�Lr+1 +∥u + gy∥ +5−r +3(r−1) +H +∥u∥ +5 +6 +H∥∇¯ui,2∥ +1 +6 +L2(O√ +2k), +for d = 3 and r ∈ [3, 5], +λ +− +r+1 +3r(r−1) +i+1 +∥u + gy∥r +�Lr+1∥u∥ +r−5 +3(r−1) +�Lr+1 ∥u∥ +2(r+1) +3r +H +∥∇¯ui,2∥ +2(r+1) +3r(r−1) +L2(O√ +2k), +for d = 3 and r ∈ (5, ∞), + +36 +KUSH KINRA, MANIL T. MOHAN, AND RENHAI WANG +≤ C × + + + + + + + + + + + + + + + + + + + +λ +− +1 +2(r−1) +i+1 +� +∥u + gy∥r+1 +�Lr+1 + ∥u + gy∥4(r−1) +H ++ ∥u∥2(r−1) +H ++ ∥u∥2 +V +� +, +for d = 2 and r ∈ [2, ∞), +λ +− 1 +12 +i+1 +� +∥u + gy∥r+1 +�Lr+1 + ∥u + gy∥2 +H + ∥u∥10 +H + ∥u∥2 +V +� +, +for d = 3 and r ∈ [3, 5], +λ +− +r+1 +3r(r−1) +i+1 +� +∥u + gy∥r+1 +�Lr+1 + ∥u∥r+1 +�Lr+1 + ∥u∥r+1 +H ++ ∥u∥2 +V +� +, for d = 3 and r ∈ (5, ∞). +(4.61) +Estimate of �Q3(d, r): Similar to (3.63), we find (for d = 2, 3) +| �Q3(d, r)| ≤ C∥(−∆)−1[∇ · f]∥ +L +d +d−1 (Rd)∥¯ui,2∥Ld(O√ +2k) +≤ C∥f∥L1(Rd)∥¯ui,2∥ +4−d +2 +L2(O√ +2k)∥∇¯ui,2∥ +d−2 +2 +L2(O√ +2k) +≤ Cλ +− 4−d +4 +i+1 +∥f∥L1(Rd)∥∇¯ui,2∥L2(O√ +2k) +≤ µ +4 ∥¯ui,2∥2 +L2(O√ +2k) + Cλ +− 4−d +2 +i+1 +∥f∥2 +L1(Rd). +(4.62) +Estimate of L5: Applying H¨older’s and Young’s inequalities, we deduce +|L5| ≤ C +� +∥u∥H + ∥∇u∥H + ∥f∥L2(Rd) + |y| +� +∥¯ui,2∥L2(O√ +2k) +≤ Cλ−1/2 +i+1 +� +∥u∥H + ∥∇u∥H + ∥f∥L2(Rd) + |y| +� +∥∇¯ui,2∥L2(O√ +2k) +≤ µ +4∥∇¯ui,2∥L2(O√ +2k) + Cλ−1 +i+1 +� +∥u∥2 +H + ∥∇u∥2 +H + ∥f∥2 +L2(Rd) + |y|2 +� +. +(4.63) +Now, combining (4.56)-(4.63), applying the variation of constant formula, using Lemma 4.5 (both (4.33) +and (4.35)) and passing limit i → ∞, (λi+1 → 0 as i → ∞), we demonstrate (4.54), as desired (see the proof +of Lemma 3.9), which completes the proof. +□ +4.5. Proof of Theorem 1.3. This subsection is devoted to the proof of main result of this section, that +is, the existence of pullback D-random attractors and their asymptotic autonomy for the solution of the +system (2.6) with S(v) = g ∈ D(A). For all the cases given in Table 1 (excluding d = 2 with r = 1), the +existence of pullback D-random attractors for non-autonomous SCBF equations driven by additive noise on +the whole space is established in [38]. For all the cases given in Table 1 (excluding d = 2 with r = 1), as +the existence of a unique pullback random attractor is known for each τ, one can obtain the existence of a +unique random attractor for an autonomous SCBF equations driven by additive noise on the whole space +(cf. [38]). +In view of Propositions 4.4 and 4.6, and Lemmas 4.7 and 4.8, the proof of Theorem 1.3 can be obtained +by applying similar arguments as in the proof of [73, Theorem 1.6] (Subsection 3.5 in [73]) and [9, Theorem +5.2]. +Acknowledgments: The first author would like to thank the Council of Scientific & Industrial Research +(CSIR), India for financial assistance (File No. 09/143(0938)/2019-EMR-I). M. T. Mohan would like to + +ASYMPTOTIC AUTONOMY OF RANDOM ATTRACTORS FOR SCBF EQUATIONS +37 +thank the Department of Science and Technology (DST), Govt of India for Innovation in Science Pursuit +for Inspired Research (INSPIRE) Faculty Award (IFA17-MA110). Renhai Wang was supported by China +Postdoctoral Science Foundation under grant numbers 2020TQ0053 and 2020M680456. +Declarations: +Ethical Approval: Not applicable +Competing interests: +The authors declare no competing interests. +Authors’ contributions: +All authors have contributed equally. +Funding: +CSIR, India, 09/143(0938)/2019-EMR-I (K. Kinra), DST, India, IFA17-MA110 (M. T. Mohan). +Availability of data and materials: +Not applicable. +References +[1] S.N. Antontsev and H.B. de Oliveira, The Navier-Stokes problem modified by an absorption term, Appl. Anal., 89(12), +2010, 1805–1825. +[2] L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, Heidelberg, New York, 1998. +[3] J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10(1-2) (2004), 31–52. +[4] M. C. Bortolan, A.N. Carvalho and J. A. Langa, Attractors under autonomous and non- autonomous perturbations, Math- +ematical Surveys and Monographs, AMS, 2020. +[5] Z. 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Wang, Random dynamics of fractional nonclassical diffusion equations driven by colored noise, +Discrete Contin. Dyn. Syst., 39(7) (2019), 4091–4126. +[70] R. Wang, B. Guo, B. Wang, Well-posedness and dynamics of fractional FitzHugh-Nagumo systems on RN driven by +nonlinear noise, Sci. China Math., 64(11) (2021), 2395-2436. +[71] R. Wang, L. Shi and B. Wang, Asymptotic behavior of fractional nonclassical diffusion equations driven by nonlinear +colored noise on RN, Nonlinearity, 32(11) (2019), 4524–4556. +[72] S. Wang and Y. Li, Longtime robustness of pullback random attractors for stochastic magneto-hydrodynamics equations, +Physica D, 382 (2018), 46–57. +[73] R. Wang, K. Kinra and M. T. Mohan, Asymptotically autonomous robustness in probability of random attractors +for stochastic Navier-Stokes equations on unbounded Poincar´e domains, Accepted in SIAM J. Math. Anal., (2022), +https://arxiv.org/pdf/2208.06808.pdf. +[74] S. Wang, M. Si and R. Yang, Random attractors for non-autonomous stochastic Brinkman-Forchheimer equations on +unbounded domains, Commun. Pure Appl. Anal., 21(5) (2022), 1621–1636. +[75] J. Xu and T. Caraballo, Long time behavior of stochastic nonlocal partial differential equations and Wong-Zakai approxi- +mations, SIAM J. Math. Anal., 54(3) (2022), 2792–2844. +[76] Q. Zhang and Y. Li, Regular attractors of asymptotically autonomous stochastic 3D Brinkman-Forchheimer equations with +delays, Commun. Pure Appl. Anal., 20(10) (2021), p.3515. + +40 +KUSH KINRA, MANIL T. MOHAN, AND RENHAI WANG +(Kush Kinra) Department of Mathematics, Indian Institute of Technology Roorkee-IIT Roorkee, Haridwar +Highway, Roorkee, Uttarakhand 247667, INDIA. +Email address, K. Kinra: kkinra@ma.iitr.ac.in +(Manil T. Mohan) Corresponding author, Department of Mathematics, Indian Institute of Technology Roorkee- +IIT Roorkee, Haridwar Highway, Roorkee, Uttarakhand 247667, INDIA. +Email address, M. T. Mohan: maniltmohan@ma.iitr.ac.in, maniltmohan@gmail.com +(Renhai Wang) School of Mathematics and Statistics, Southwest University, Chongqing 400715, CHINA. +Email address, R. Wang: rwang-math@outlook.com + diff --git a/ndAyT4oBgHgl3EQfYvfO/content/tmp_files/load_file.txt b/ndAyT4oBgHgl3EQfYvfO/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..23c885296e5d44d1c05a670eca974d648bb274cd --- /dev/null +++ b/ndAyT4oBgHgl3EQfYvfO/content/tmp_files/load_file.txt @@ -0,0 +1,1842 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf,len=1841 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='00211v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='PR] 31 Dec 2022 ASYMPTOTICALLY AUTONOMOUS ROBUSTNESS IN PROBABILITY OF NON-AUTONOMOUS RANDOM ATTRACTORS FOR STOCHASTIC CONVECTIVE BRINKMAN-FORCHHEIMER EQUATIONS ON Rd KUSH KINRA, MANIL T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' MOHAN, AND RENHAI WANG Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' This article is concerned with the asymptotically autonomous robustness (almost surely and in proba- bility) of non-autonomous random attractors for two stochastic versions of convective Brinkman-Forchheimer (CBF) equations defined on the whole space Rd: ∂v ∂t − µ∆v + (v · ∇)v + αv + β|v|r−1v + ∇p = f(t) + “stochastic terms”, ∇ · v = 0, with initial and boundary vanishing conditions, where d = 2, 3, µ, α, β > 0, r ≥ 1 and f(t) is a given time- dependent external force field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' By the asymptotically autonomous robustness of a non-autonomous random attractor A = {A (τ, ω) : τ ∈ R, ω ∈ Ω} we mean its time-section A (τ, ω) is robust to a time-independent random set as time τ tends to negative infinity according to the Hausdorff semi-distance of the underlying space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Our goal is to study this topic, almost surely and in probability, for the non-autonomous CBF equations when the stochastic term is a linear multiplicative or additive noise, and the time-dependent forcing converges towards a time-independent function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Our main results contain three cases: i) d = 2 and r ∈ {1} ∪ [2, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ii) d = 3 and r ∈ (3, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' iii) d = 3, r = 3 and 2βµ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' The main procedure to achieve our goal is how to justify that the usual pullback asymptotic compactness of the solution operators is uniform on some uniformly tempered universes over an infinite time-interval (−∞, τ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' This can be done by a method based on Kuratowski’s measure of noncompactness by showing the backward uniform “tail-smallness” and “flattening-property” of the solutions over (−∞, τ] in order to overcome the lack of compact Sobolev embeddings on unbounded domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Several rigorous calculations dealing the pressure term p and the fast growing term β|v|r−1v play key role in the whole analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' When α = β = 0, the present result can be viewed as a generation of the authors’s recent work [73] for the standard Navier-Stokes equations on unbounded Poincar´e domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' The model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' In this article, we consider a stochastic fluid dynamic model concerning the convective Brinkman-Forchheimer (CBF) equation driven by stochastic and non-autonomous forcing simultaneously defined on the whole space Rd (d = 2, 3): (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1) \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ∂v ∂t − µ∆v + (v · ∇)v + αv + β|v|r−1v + ∇p = f + S(v) ◦ dW(t) dt , in Rd × (τ, ∞), ∇ · v = 0, in Rd × (τ, ∞), v(x)|t=τ = v0(x), x ∈ Rd and τ ∈ R, v(x)|t=τ → 0, as |x| → ∞, 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Primary 37L55;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Secondary 37B55, 35B41, 35B40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Asymptotically autonomous robustness, pullback random attractor, stochastic convective Brinkman- Forchheimer equations, backward uniform-tail estimate, backward flattening-property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 1 2 KUSH KINRA, MANIL T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' MOHAN, AND RENHAI WANG where v(x, t) ∈ Rd, p(x, t) ∈ R and f(x, t) ∈ Rd represent the velocity field, pressure field and external forcing, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' The constants µ, α, β > 0 stand for the Brinkman (effective viscosity), Darcy (perme- ability of the porous medium) and Forchheimer coefficients, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Note that r ∈ [1, ∞) is called the absorption exponent and r = 3 is called the critical absorption exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' The one-dimensional two-sided Wiener W(t) is defined on a probability space (Ω, F, P) (see Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' The diffusion coefficient of the noise is S(v) = v (multiplicative noise) or independent of v (additive noise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' The symbol ◦ means that the stochastic integral should be understood in the sense of Stratonovich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' The CBF equations are also referred to as the tamed Navier-Stokes equations with a modified damping term αv +β|v|r−1v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' If α = β = 0, then the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1) is reduced to the standard Navier-Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' It has been proved by Hajduk and Robinson [31, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1] that the CBF equations and the Navier- Stokes equations have the same scaling only when r = 3 and α = 0, but have no scaling invariance for other values of α and r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' From the physical point of views, system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1) is applied to the flows when the velocities are sufficiently high and porosities are not too small, that is, when the Darcy law for a porous medium does not apply, see [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' In this case, system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1) is also referred as the non-Darcy model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Literature results for CBF equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For deterministic 2D/3D CBF equations, the existence and uniqueness of weak/strong solutions on bounded, periodic and unbounded domains has been investigated in [1, 26, 31, 47, 48];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' the existence, uniqueness, regularity, stability and Hausdorff/fractal dimension of global/pullback/exponential attractors have studied in [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=', 37, 50, 51] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For sto- chastic 2D/3D CBF equations, the existence of a global in time pathwise mild solution was justified in [52] when the equations are defined on the whole space and driven by fractional Brownian noise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' the existence and uniqueness of strong solutions (in probabilistic sense) was established in [34] when the equations are defined on unbounded Poincar´e domains and forced by Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' However, similar to the 3D Navier-Stokes equations, the existence of the unique global weak solution and unique pathwise strong solution of 3D deterministic and stochastic CBF equations are still open problems for r ∈ [1, 3) with β, µ > 0, and r = 3 with 2βµ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Literature survey for random attractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' The theory of various types of attractors, such as global,pullback,exponential and trajectory attractors, of deterministic dynamical systems has been exten- sively studied in [3, 4, 10, 11, 16, 17, 18, 56, 57, 60] and many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For the long term dynamics of stochastic ordinary/partial differential equations which generates a random dynamical system (RDS) [2], the extension of global attractors to the random attractors was introduced in [5, 20, 21, 58] and success- fully applied to 2D stochastic Navier-Stokes equations and other stochastic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Since the evolution equations arriving from physics and other fields of science are often driven by non-autonomous and sto- chastic forcing simultaneously, random attractors of autonomous RDS are generalized to pullback random attractors in [65] under the framework of non-autonomous RDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' In view of these abstract results, there is a large number of literature on the random attractors for autonomous and non-autonomous stochastic ASYMPTOTIC AUTONOMY OF RANDOM ATTRACTORS FOR SCBF EQUATIONS 3 equations, see [6, 14, 15, 29, 37, 38, 39, 64, 73, 74, 75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' As per the existing literature, the existence of ran- dom attractors for stochastic systems is based on some transformation which converts the stochastic system into a pathwise deterministic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' This transformation is available in the literature only when the noise is either linear multiplicative or additive one, see [6, 27, 37, 43, 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' In order to deal with the nonlinear diffusion term of the noise, the concept of mean random attractors was introduced in [66] and applied to stochastic Navier-Stokes and CBF equations (Itˆo sense) with Lipschitz nonlinear diffusion term in [67] and [36], respectively, see [68, 70] for other physically relevant stochastic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Another different approach in the direction of random attractors, when the diffusion term is nonlinear, is the Wong-Zakai approximation of pathwise random attractors, see [29, 30, 38, 75] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Motivation, assumptions and main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' In general, a non-autonomous random attractor car- ries the form Aς = {Aς(τ, ω) : τ ∈ R, ω ∈ Ω}, where ς stands for some external perturbation parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' In the literature, the robustness of pullback random attractors of stochastic CBF equations have been established in [37, 38] with respect to the external parameter ς.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For the robustness with respect to the ex- ternal/internal parameters of pullback random attractors of 2D stochastic Navier-Stokes, we refer interested readers to the works [22, 29, 30, 39, 73] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Currently, the questions of robustness of pullback random attractors of stochastic CBF equations defined on unbounded domains with respect to the internal parameter τ, however, is still unsolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' As per our expectations, if the time-dependent forcing term f(x, t) converges to some time-independent forcing term f ∞(x) in some sense, the non-autonomous random dynamics of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1) becomes more and more autonomous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Our main motivation is to examine the asymptotically autonomous robustness of pullback random attractors of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1) when the parameters in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1) are discussed in the following three cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Cases d r conditions on µ & β I d = 2 r ∈ {1} ∪ [2, ∞) for any µ > 0 and β > 0 II d = 3 r ∈ (3, ∞) for any µ > 0 and β > 0 III d = 3 r = 3 for µ > 0 and β > 0 with 2βµ ≥ 1 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Values of µ, β and r for d = 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' f ∈ L2 loc(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' L2(Rd)) converges to f∞ ∈ L2(Rd) : lim τ→−∞ � τ −∞ ∥f(t) − f ∞∥2 L2(Rd)dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2) Moreover, f(·, ·) satisfies sup s≤τ � s −∞ eκ(t−s)∥f(t)∥2 L1(Rd)dt < +∞, ∀ κ > 0, τ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='3) Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2 (Multiplicative noise case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Let Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1 be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Then, for all the cases given in Table 1, the non-autonomous RDS Φ generated by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1) with S(v) = v has a unique pullback 4 KUSH KINRA, MANIL T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' MOHAN, AND RENHAI WANG random attractor A = {A (τ, ω) : τ ∈ R, ω ∈ Ω} such that � s∈(−∞,τ] A (s, ω) is precompact in H (the definition of H is given below, see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1) and lim t→+∞ e−γt sup s∈(−∞,τ] ∥A (s−t, ϑ−tω)∥H = 0, for any γ > 0, τ ∈ R and ω ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' In addition, the time-section A (τ, ω) is asymptotically autonomous robust in H, and the limiting set of A (τ, ω) as τ → −∞ is just determined by the random attractor A∞ = {A (ω) : ω ∈ Ω} of stochastic CBF equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1) with the autonomous forcing f ∞, that is, lim τ→−∞ distH(A (τ, ω), A∞(ω)) = 0, P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ω ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='4) Moreover, the asymptotically autonomous robustness in probability is also justified: lim τ→−∞ P � ω ∈ Ω : distH(A (τ, ω), A∞(ω)) ≥ δ � = 0, ∀ δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='5) In addition, for any ε > 0 and sequence τn → −∞, there exists Ωε ∈ F with P(Ωε) > 1 − ε such that lim n→∞ sup ω∈Ωε distH(A (τn, ω), A∞(ω)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='6) Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='3 (Additive noise case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Under the Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1 and for all the cases given in Table 1 (excluding d = 2 with r = 1), all results in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2 hold for the non-autonomous RDS generated by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1) with S(v) = g with g ∈ D(A), where D(A) is the domain of the Stokes operator A defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (i) An example of Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1 is f(x, t) = f∞(x)et +f ∞(x) with f ∞ ∈ L2(Rd)∩L1(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (ii) Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1 implies the following conditions (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' [9]): Uniform integrability: sup s≤τ � s −∞ eκ(ξ−s)∥f(ξ)∥2 L2(Rd)dξ < +∞, ∀ κ > 0, τ ∈ R, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='7) Uniform tails-smallness: lim k→∞ sup s≤τ � s −∞ eκ(ξ−s) � |x|≥k |f(x, ξ)|2dxdξ = 0, ∀ κ > 0, τ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='8) (iii) We only use Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1 for f in the whole paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (iv) In Poincar´e domains (bounded or unbounded), we can relax the condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='3) (see [73]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (iv) Due to technical difficulties, we are not able to establish the present results for d = 2 and r ∈ (1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (v) In the additive noise case, we do not need to assume, as in [73, Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='3], that there exists a constant ℵ > 0 such that g ∈ D(A) satisfies ���� d � i,j=1 � Rd vi(x)∂gj(x) ∂xi vj(x)dx ���� ≤ ℵ∥v∥2 L2(Rd), ∀ v ∈ L2(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='9) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Novelties, difficulties and approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' In order to prove Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='3, the uniform pre- compactness of � s∈(−∞,τ] A (s, ω) in H is a pivotal point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' The well-known abstract theory of pullback random attractors from [65] tells us that the pullback asymptotic compactness of Φ gives the compactness of A (τ, ω) for each τ ∈ R, but it cannot provide the precompactness of � s∈(−∞,τ] A (s, ω) in H, since (−∞, τ] is an in- finite interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' However, motivated by the ideas of [65], this can be done if one is able to show that the ASYMPTOTIC AUTONOMY OF RANDOM ATTRACTORS FOR SCBF EQUATIONS 5 usual pullback asymptotic compactness of Φ is uniform with respect to a uniformly tempered universe (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='13)) over (−∞, τ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Note that in the bounded domain case, one can obtain the uniform pullback asymptotic compactness of Φ over (−∞, τ] via a compact uniform pullback absorbing set by using compact Sobolev embeddings (see [39, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Moreover, the same idea is used for several stochastic Navier-Stokes, g-Navier-Stokes, magneto-hydrodynamics, Brinkman-Forchheimer equations on bounded domains, see [39, 44, 72, 76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Due to the lack of compact Sobolev embeddings in unbounded domains as considered in the present work, to demonstrate such backward uniform pullback asymptotic compactness is therefore harder than that in the bounded domain case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' We mention that the criteria of Kuratowski’s measure of noncompactness ([41, 55]) is useful to resolve the difficulty created by the noncompactness of Sobolev embeddings on unbounded domains (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' [42, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' In order to apply such criteria, we use the idea of uniform tail-estimates introduced by Wang [62] and flattening-properties introduced by Ma et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' [46](deterministic case) and Kloeden and Langa [33](random case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Using the cut-off technique, we show that the solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1) are sufficiently small in L2(Oc k) uniformly over (−∞, τ], when k is large enough, where Ok = {x ∈ Rd : |x| ≤ k} and Oc k = Rd \\ Ok, that is, we obtain the backward uniform tail-estimates for the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Furthermore, using the same cut-off function, we can also establish the backward flattening-properties of the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Note that parabolic and hyperbolic stochastic models as considered in the works [12, 13, 9, 19, 42, 59, 62, 69] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=', do not contain pressure term p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' But some physically relevant models such as Navier-Stokes (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' [73]), Brinkman-Forchheimer equations (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' [76]) and many others, contain the pressure term p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' While proving the backward uniform tail-estimates as well as backward flattening-properties of the solutions, when we take a suitable inner product, the pressure term p does not vanish with the help of divergence free condition (or incompressibility condition) of the solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' However, by taking the divergence in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1) formally and using the divergence free condition, we end up with the rigorous expression of the pressure term p = (−∆)−1 � d � i,j=1 ∂2 ∂xi∂xj (vivj) + ∇ · {|v|r−1v} − ∇ · f � , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='10) in the weak sense, which is the most difficult term to handle in an appropriate way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Then it is possible to obtain these backward uniform tail-estimates as well as backward flattening-property with the help of Gagliardo-Nirenberg ([54, Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' In this paper it has been used very carefully in each case to get appropriate estimates by using H¨older, interpolation and Young inequalities (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='8-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='9 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='7- 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' It is worth mentioning here that we are able to prove the backward uniform tail-estimate as well as the backward flattening-property for d = 2 and d = 3 with r ∈ {1} ∪ [2, ∞) and r ∈ [3, ∞), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' But establishing the backward uniform tail-estimate as well as the backward flattening-properties for d = 2 with 1 < r < 2 on the whole space is still not yet resolved (that is, the difficulty in estimating the pressure term (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='10) on the whole space is not resolved for d = 2 with 1 < r < 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Note that one can estimate the pressure term (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='10) for d = 2 with 1 < r < 2 on unbounded Poincar´e domains O by using the elliptic regularity (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' [35, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' As a result of these backward uniform tail-estimates and backward flattening-property 6 KUSH KINRA, MANIL T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' MOHAN, AND RENHAI WANG of the solutions to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1), the backward uniform pullback asymptotic compactness of Φ in H follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' The wide-spread idea of energy equations introduced in [3] can be used to overcome the noncompactness of Sobolev embeddings on unbounded domains, see the works [6, 7, 29, 38, 63, 64, 71], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' and many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' A remark is that we are currently unable to use the idea of energy equations to prove the backward uniform pullback asymptotic compactness of Φ in H since (−∞, τ] is an infinite time-interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Since we have to consider the uniformly tempered universe to prove the backward uniform pullback asymptotic compactness of Φ, we shall establish the measurability of the uniformly compact attractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' This is not straightforward compared with the usual case since the radii of the uniform pullback absorbing set is taken as the supremum over an uncountable set (−∞, τ] (see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' In order to overcome the difficulty, we first observe that the measurability of the usual random attractor is known in the literature, see for example, [6, 7, 29, 65], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=', and then prove that such a uniformly compact attractor is just equal to the usual random attractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' This idea has been successfully used by the authors in [9, 72, 73] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=', for different stochastic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Advantages of the damping term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' CBF equations are also known as damped Navier-Stokes equa- tions (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' [32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' The damping arises from the resistance to the motion of the flow or by friction effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Due to the presence of the damping term αv + β|v|r−1v, we are able to establish better results than which are available for the Navier-Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' The existence of global as well as random attractors for the Navier-Stokes equations on the whole space or general unbounded domains is an interesting and challeng- ing open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' In the literature, for Navier-Stokes equations, these types of results are available on unbounded Poincar´e domains only (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' [39, 73]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For 2D Navier-Stokes equations forced by a linear mul- tiplicative noise, we refer to [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For stochastic CBF equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1), we are considering the whole space, where the linear damping term αv plays a crucial role to establish the required results on the whole space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' This is different from the 2D Navier-Stokes equations on unbounded Poincar´e domains, see [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Outline of the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' In the next section, we provide the necessary function spaces and abstract formulation of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1), and discuss the Ornstein-Uhlenbeck process with its properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' In Section 3, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2 for the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1) driven by multiplicative noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' In the final section, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='3 for the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1) driven by additive noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Mathematical formulation We start this section with some necessary function spaces whose elements satisfy the divergence free conditions, that is, ∇ · v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Next, in order to obtain the abstract formulation of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1), we define linear, bilinear and nonlinear operators along with their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Finally, we discuss the Ornstein- Uhlenbeck process with some of its properties and the backward tempered random sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Function spaces and operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Let C∞ 0 (Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Rd) denote the space of all Rd-valued, infinitely dif- ferentiable functions with compact support in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Let Ls(Rd) := Ls(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Rd) and Hk(Rd) := Hk(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Rd) for ASYMPTOTIC AUTONOMY OF RANDOM ATTRACTORS FOR SCBF EQUATIONS 7 s ∈ [2, ∞) and k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Define the spaces H := {v ∈ C∞ 0 (Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Rd) : ∇ · v = 0} L2(Rd), V := {v ∈ C∞ 0 (Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Rd) : ∇ · v = 0} H1(Rd), �Lp := {v ∈ C∞ 0 (Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Rd) : ∇ · v = 0} Lp(Rd), p > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' The spaces H, V and �Lp are endowed with the norms ∥v∥2 H := � Rd |v(x)|2dx, ∥v∥2 V = � Rd |v(x)|2dx + � Rd |∇v(x)|2dx and ∥v∥p �Lp := � Rd |v(x)|pdx, for p ∈ (2, ∞), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' The inner product in the Hilbert space H is represented by (·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' The duality pairing between the spaces V and V′, and �Lp and its dual �L p p−1 is denoted by ⟨·, ·⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Also, the space H can be identified with its own dual H′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' We endow the space V ∩ �Lp with the norm ∥v∥V + ∥v∥�Lp, for v ∈ V ∩ �Lp and its dual V′ + �Lp′ with the norm (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' [24, Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1]) inf � ∥u1∥V′ + ∥u2∥�Lp′ : u = u1 + u2, u1 ∈ V′, u2 ∈ �Lp′� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Linear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Let P : L2(Rd) → H be the Helmholtz-Hodge (or Leray) projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Note that the projection operator P can be expressed in terms of the Riesz transform (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' [53]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' We define the Stokes operator (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1) Av := −P∆v, v ∈ D(A) := V ∩ H2(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Moreover, P and ∆ commutes in Rd, that is, P∆ = ∆P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Bilinear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Let us define the trilinear form b(·, ·, ·) : V × V × V → R by b(v1, v2, v3) = � Rd(v1(x) · ∇)v2(x) · v3(x)dx = d � i,j=1 � Rd v1,i(x)∂v2,j(x) ∂xi v3,j(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' If v1, v2 are such that the linear map b(v1, v2, ·) is continuous on V, the corresponding element of V′ is denoted by B(v1, v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' We also denote B(v) = B(v, v) = P[(v · ∇)v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' An integration by parts yields (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2) �b(v1, v2, v2) = 0, for all v1, v2 ∈ V, b(v1, v2, v3) = −b(v1, v3, v2), for all v1, v2, v3 ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1 ([61, Chapter 2, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For all v1, v2, v3 ∈ V, |b(v1, v2, v3)| ≤ C × � ∥v1∥1/2 H ∥∇v1∥1/2 H ∥∇v2∥H∥v3∥1/2 H ∥∇v3∥1/2 H , for d = 2, ∥v1∥1/4 H ∥∇v1∥3/4 H ∥∇v2∥H∥v3∥1/4 H ∥∇v3∥3/4 H , for d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='3) Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Note that ⟨B(v1, v1 − v2), v1 − v2⟩ = 0 (for all v1, v2 ∈ V) gives us (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='4) ⟨B(v1) − B(v2), v1 − v2⟩ = ⟨B(v1 − v2, v2), v1 − v2⟩ = −⟨B(v1 − v2, v1 − v2), v2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 8 KUSH KINRA, MANIL T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' MOHAN, AND RENHAI WANG 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Nonlinear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Let us consider the nonlinear operator C(v) := P(|v|r−1v), for v ∈ V ∩ �Lr+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' The map C(·) : V ∩ �Lr+1 → V′ + �L r+1 r and ⟨C(v), v⟩ = ∥v∥r+1 �Lr+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For any v1, v2 ∈ V ∩ �Lr+1, we have (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' [49, Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='4]) ⟨C(v1) − C(v2), v1 − v2⟩ ≥ 1 2∥|v1| r−1 2 (v1 − v2)∥2 H + 1 2∥|v2| r−1 2 (v1 − v2)∥2 H ≥ 0, for all r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='5) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Abstract formulation and Ornstein-Uhlenbeck process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' By taking the projection P on the SCBF equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1), we obtain the following abstract formulation by linear, bilinear and nonlinear oper- ators: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='6) \uf8f1 \uf8f2 \uf8f3 dv dt + µAv + B(v) + αv + βC(v) = Pf + S(v) ◦ dW dt , t > τ, v(x)|t=τ = vτ(x), x ∈ Rd, where S(v) = v (multiplicative noise) or S(v) is independent of v (additive noise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Here, the symbol ◦ represents that the stochastic integral is understood in the sense of Stratonovich and W(t, ω) is the standard scalar Wiener process on the probability space (Ω, F, P), where Ω = {ω ∈ C(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' R) : ω(0) = 0}, endowed with the compact-open topology given by the metric dΩ(ω, ω′) := ∞ � m=1 1 2m ∥ω − ω′∥m 1 + ∥ω − ω′∥m , where ∥ω − ω′∥m := sup −m≤t≤m |ω(t) − ω′(t)|, F is the Borel sigma-algebra induced by the compact-open topology of (Ω, dΩ) and P is the two-sided Wiener measure on (Ω, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' From [28], it is clear that the measure P is ergodic and invariant under the translation-operator group {ϑt}t∈R on Ω defined by ϑtω(·) = ω(· + t) − ω(t), for all t ∈ R, ω ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' The operator ϑ(·) is known as Wiener shift operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Ornstein-Uhlenbeck process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Consider for some σ > 0 y(ϑtω) = � t −∞ e−σ(t−ξ)dW(ξ), ω ∈ Ω, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='7) which is the stationary solution of the one-dimensional Ornstein-Uhlenbeck equation dy(ϑtω) + σy(ϑtω)dt = dW(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='8) It is known from [25] that there exists a ϑ-invariant subset �Ω ⊂ Ω of full measure such that y(ϑtω) is continuous in t for every ω ∈ �Ω, and lim t→±∞ |y(ϑtω)| |t| = lim t→±∞ 1 t � t 0 y(ϑξω)dξ = lim t→∞ e−δt|y(ϑ−tω)| = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='9) for all δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For further analysis of this work, we do not distinguish between �Ω and Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Since, ω(·) has sub-exponential growth (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' [8, Lemma 11]), Ω can be written as Ω = � N∈N ΩN, where ΩN := {ω ∈ Ω : |ω(t)| ≤ Ne|t|, for all t ∈ R}, for all N ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Moreover, for each N ∈ N, (ΩN, dΩN ) is a polish space (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' [8, Lemma 17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ASYMPTOTIC AUTONOMY OF RANDOM ATTRACTORS FOR SCBF EQUATIONS 9 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For each N ∈ N, suppose ωk, ω0 ∈ ΩN are such that dΩ(ωk, ω0) → 0 as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Then, for each τ ∈ R, T ∈ R+ and a ∈ R, sup t∈[τ,τ+T] � |y(ϑtωk) − y(ϑtω0)| + |eay(ϑtωk) − eay(ϑtω0)| � → 0 as k → ∞, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='10) sup k∈N sup t∈[τ,τ+T] |y(ϑtωk)| ≤ C(τ, T, ω0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='11) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' See the proofs of [23, Corollary 22] and [45, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Backward-uniformly tempered random set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' A bi-parametric set D = {D(τ, ω)} in a Banach space X is said to be backward-uniformly tempered if lim t→+∞ e−ct sup s≤τ ∥D(s − t, ϑ−tω)∥2 X = 0 ∀ (τ, ω, c) ∈ R × Ω × R+, where ∥D∥X = sup x∈D ∥x∥X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='12) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Class of random sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Let D be the collection of subsets of H defined as: D = � D = {D(τ, ω) : (τ, ω) ∈ R × Ω} : lim t→+∞ e−ct sup s≤τ ∥D(s − t, ϑ−tω)∥2 H = 0, ∀ c > 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='13) Let B be the collection of subsets of H defined as: B = � B = {B(τ, ω) : (τ, ω) ∈ R × Ω} : lim t→+∞ e−ct∥B(τ − t, ϑ−tω)∥2 H = 0, ∀ c > 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Let D∞ be the collection of subsets of H defined as: D∞ = � �D = {�D(ω) : ω ∈ Ω} : lim t→+∞ e−ct∥�D(ϑ−tω)∥2 H = 0, ∀ c > 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 2D and 3D SCBF equations: Multiplicative noise In this section, we consider 2D and 3D SCBF equations driven by a linear multiplicative white noise, that is, S(v) = v and establish the asymptotic autonomy of pullback random attractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Let us define u(t, τ, ω, uτ) := e−y(ϑtω)v(t, τ, ω, vτ) with uτ = e−y(ϑτ ω)vτ, where y satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='8) and v(·) := v(·, τ, ω, vτ) is the solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1) with S(v) = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Then u(·) := u(·, τ, ω, uτ) satisfies: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1) \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 du(t) dt − µ∆u(t) + ey(ϑtω)(u(t) · ∇)u(t) + αu(t) + βe(r−1)y(ϑtω)|u(t)|r−1u(t) = −e−y(ϑtω)∇p(t) + f(t)e−y(ϑtω) + σy(ϑtω)u(t), in Rd × (τ, ∞), ∇ · u = 0, in Rd × (τ, ∞), u(x)|t=τ = u0(x) = e−y(ϑτ ω)v0(x), x ∈ Rd and τ ∈ R, u(x)|t=τ → 0 as |x| → ∞, 10 KUSH KINRA, MANIL T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' MOHAN, AND RENHAI WANG as well as (projected form) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2) \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 du(t) dt + µAu(t) + ey(ϑtω)B � u(t) � + αu(t) + βe(r−1)y(ϑtω)C � u(t) � = e−y(ϑtω)Pf(t) + σy(ϑtω)u(t), t > τ, τ ∈ R, u(x)|t=τ = u0(x) = e−y(ϑτ ω)v0(x), x ∈ Rd, in V′ + �L r+1 r , where r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Due to some technical difficulties, we restrict ourselves to all the cases given in Table 1 (see Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='8 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Non-autonomous random dynamical system (NRDS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Lusin continuity helps us to define the NRDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' The following lemma (energy inequality) will be frequently used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For all the cases given in Table 1, assume that f ∈ L2 loc(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' L2(Rd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Then, the solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2) satisfies the following energy inequality: d dt∥u∥2 H + � α − 2σy(ϑtω) + α 2 � ∥u∥2 H + 2µ∥∇u∥2 H + 2βe(r−1)y(ϑtω)∥u∥r+1 �Lr+1 ≤ 2e2|y(ϑtω)| α ∥f∥2 L2(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='3) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' From the first equation of the system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2), using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2) and the Cauchy-Schwarz inequality, one can obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='3) immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For all the cases given in Table 1, let f ∈ L2 loc(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' L2(Rd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For each (τ, ω, uτ) ∈ R×Ω×H, the system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2) has a unique weak solution u(·, τ, ω, uτ) ∈ C([τ, +∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' H) ∩ L2 loc(τ, +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' V) ∩ Lr+1 loc (τ, +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' �Lr+1) such that u is continuous with respect to the initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' One can prove the existence and uniqueness of solutions by a standard Faedo-Galerkin approximation method, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' [31, 34, 47], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For continuity with respect to initial data uτ, see [38, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' □ Next result shows the Lusin continuity of the mapping of solution to the system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2) in sample points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For all the cases given in Table 1, suppose that f ∈ L2 loc(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' L2(Rd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For each N ∈ N, the mapping ω �→ u(t, τ, ω, uτ) (solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2)) is continuous from (ΩN, dΩN ) to H, uniformly in t ∈ [τ, τ +T] with T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Assume that ωk, ω0 ∈ ΩN, N ∈ N such that dΩN (ωk, ω0) → 0 as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Let U k(·) := uk(·) − u0(·), where uk(·) := u(·, τ, ωk, uτ) and u0 := u(·, τ, ω0, uτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Then, U k(·) satisfies: dU k dt = −µAU k − (α − σy(ϑtωk))U k − ey(ϑtωk)� B � uk� − B � u0�� − � ey(ϑtωk) − ey(ϑtω0)� B � u0� − βe(r−1)y(ϑtωk)� C � uk� − C � u0�� − β � e(r−1)y(ϑtωk) − e(r−1)y(ϑtω0)� C � u0� + f � e−y(ϑtωk) − e−y(ϑtω0)� + σ[y(ϑtωk) − y(ϑtω0)]u0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='4) in V′ + �L r+1 r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Taking the inner product with U k(·) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='4), and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='4), we obtain 1 2 d dt∥U k∥2 H = −µ∥∇U k∥2 H − (α − σy(ϑtωk))∥U k∥2 H − βe(r−1)y(ϑtωk)� C � uk� − C � u0� , uk − u0� ASYMPTOTIC AUTONOMY OF RANDOM ATTRACTORS FOR SCBF EQUATIONS 11 + ey(ϑtωk)b(U k, U k, u0) + � ey(ϑtωk) − ey(ϑtω0)� b(u0, U k, u0) − β � e(r−1)y(ϑtωk) − e(r−1)y(ϑtω0)�� C � u0� , U k� + � e−y(ϑtωk) − e−y(ϑtω0)� (f, U k) + σ[y(ϑtωk) − y(ϑtω0)](u0, U k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='5) We know by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='5) that − � C � uk� − C � u0� , uk − u0� ≤ −1 2∥|uk| r−1 2 (uk − u0)∥2 H − 1 2∥|u0| r−1 2 (uk − u0)∥2 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='6) Using H¨older’s and Young’s inequalities, we obtain ��� � e−y(ϑtωk) − e−y(ϑtω0)� (f, U k) ��� ≤ C ���e−y(ϑtωk) − e−y(ϑtω0)��� 2 ∥f∥2 L2(Rd) + α 4 ∥U k∥2 H, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='7) ���σ[y(ϑtωk) − y(ϑtω0)](u0, U k) ��� ≤ C|y(ϑtωk) − y(ϑtω0)|2∥u0∥2 H + α 4 ∥U k∥2 H, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='8) ��� � e(r−1)y(ϑtωk) − e(r−1)y(ϑtω0)�� C � u0� , U k���� ≤ C ���e(r−1)y(ϑtωk) − e(r−1)y(ϑtω0)��� � ∥u0∥r+1 �Lr+1 + ∥uk∥r+1 �Lr+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='9) Next, we estimate the remaining terms of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='5) separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Case I: d = 2 and r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Applying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='3), H¨older’s and Young’s inequalities, we estimate ���ey(ϑtωk)b(U k, U k, u0) ��� ≤ Cey(ϑtωk)∥U k∥H∥∇U k∥H∥∇u0∥H ≤ Ce2y(ϑtωk)∥∇u0∥2 H∥U k∥2 H + µ 4 ∥∇U k∥2 H, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='10) and ��� � ey(ϑtωk) − ey(ϑtω0)� b(u0, U k, u0) ��� ≤ C ���ey(ϑtωk) − ey(ϑtω0)��� 2 ∥u0∥2 H∥∇u0∥2 H + µ 4 ∥∇U k∥2 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='11) Case II: d = 3 and r > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Using H¨older’s and Young’s inequalities, we infer ���ey(ϑtωk)b(U k, U k, u0) ��� ≤ µ 4 ∥∇U k∥2 H + β 4 e(r−1)y(ϑtωk)∥|U k||u0| r−1 2 ∥2 H + C∥U k∥2 H, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='12) and ��� � ey(ϑtωk) − ey(ϑtω0)� b(u0, U k, u0) ��� ≤ ���1 − ey(ϑtω0)−y(ϑtωk)���ey(ϑtωk)∥∇u0∥H∥|U k||u0|∥H ≤ β 4 e(r−1)y(ϑtωk)∥|U k||u0| r−1 2 ∥2 H + C ���1 − ey(ϑtω0)−y(ϑtωk)��� 2 ∥∇u0∥2 H + C∥U k∥2 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='13) Case III: When d = r = 3 with 2βµ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Applying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2), H¨older’s and Young’s inequalities, we obtain ���ey(ϑtωk)b(U k, U k, u0) ��� = ���ey(ϑtωk)b(U k, U k, uk) ��� ≤ 1 2β ∥∇U k∥2 H + β 2 e2y(ϑtωk)∥|U k||uk|∥2 H, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='14) and ���ey(ϑtωk) − ey(ϑtω0)||b(u0, U k, u0) ��� ≤ C ���1 − ey(ϑtω0)−y(ϑtωk)��� 2 ∥∇u0∥2 H + β 2 e2y(ϑtωk)∥|U k||u0|∥2 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='15) Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='5)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='15), we arrive at d dt∥U k(t)∥2 H ≤ P(t)∥U k(t)∥2 H + Q(t), for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' t ∈ [τ, τ + T] with T > 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='16) 12 KUSH KINRA, MANIL T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' MOHAN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' AND RENHAI WANG where P = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 y(ϑtωk) + Ce2y(ϑtωk)∥∇u0∥2 H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 2 with r ≥ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' y(ϑtωk) + C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 3 with r > 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' y(ϑtωk),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = r = 3 with 2βµ ≥ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Q = C ���e−y(ϑtωk) − e−y(ϑtω0)��� 2 ∥f∥2 L2(Rd) + C|y(ϑtωk) − y(ϑtω0)|2∥u0∥2 H + C ���e(r−1)y(ϑtωk) − e(r−1)y(ϑtω0)��� × � ∥u0∥r+1 �Lr+1 + ∥uk∥r+1 �Lr+1 � + C × \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 ��ey(ϑtωk) − ey(ϑtω0)��2∥u0∥2 H∥∇u0∥2 H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 2 with r ≥ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ��1 − ey(ϑtω0)−y(ϑtωk)��2∥∇u0∥2 H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 3 with r > 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ��1 − ey(ϑtω0)−y(ϑtωk)��2∥∇u0∥2 H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = r = 3 with 2βµ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='3), we deduce � τ+T τ 2βe(r−1)y(ϑtωk)∥uk(t)∥r+1 �Lr+1dt ≤ ∥uτ∥2 H + 2 α � τ+T τ e−2y(ϑtωk)∥f(t)∥2 L2(Rd)dt ≤ ∥uτ∥2 H + 2 α sup t∈[τ,τ+T] � e−2y(ϑtωk)� � τ+T τ ∥f(t)∥2 L2(Rd)dt, which gives sup k∈N � τ+T τ e(r−1)y(ϑtωk)∥uk(t)∥r+1 �Lr+1dt ≤ C(τ, T, ω0, uτ, f), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='17) where we have used (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='11) and the fact f ∈ L2 loc(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' L2(Rd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='11) and u0 ∈ L2 loc(τ, +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' V), we deduce � τ+T τ P(t)dt ≤ C(τ, T, ω0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='18) Now, from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='17), f ∈ L2 loc(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' H), u0 ∈ C([τ, +∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' H)∩L2 loc(τ, +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' V)∩Lr+1 loc (τ, +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' �Lr+1) and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='4, we conclude lim k→+∞ � τ+T τ Q(t)dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='19) Making use of the Gronwall inequality in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='16), we get ∥U k(t)∥2 H ≤ e � τ+T τ P (t)dt �� τ+T τ Q(t)dt � , for all t ∈ [τ, τ + T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='20) In view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='18)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='20), we complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2 ensures that we can define a mapping Φ : R+ × R × Ω × H → H by Φ(t, τ, ω, vτ) := v(t + τ, τ, ϑ−τω, vτ) = ey(ϑtω)u(t + τ, τ, ϑ−τω, uτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='21) The Lusin continuity in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='3 provides the F-measurability of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Consequently, in view of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='3, we have the following result for NRDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' The mapping Φ defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='21) is an NRDS on H, that is, Φ has the following properties: (i) Φ is (B(R+) × B(R) × F × B(H);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' B(H))-measurable, ASYMPTOTIC AUTONOMY OF RANDOM ATTRACTORS FOR SCBF EQUATIONS 13 (ii) Φ satisfies the cocycle property: Φ(0, τ, ω, ·) = I, and Φ(t + s, τ, ω, vτ) = Φ(t, τ + s, ϑsω, Φ(s, τ, ω, vτ)), t, s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Backward convergence of NRDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Consider the autonomous SCBF equations driven by linear multiplicative white noise: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='22) \uf8f1 \uf8f2 \uf8f3 d�v(t) dt + µA�v(t) + B(�v(t)) + α�v(t) + βC(�v(t)) = Pf ∞ + �v(t) ◦ dW(t) dt , t > 0, �v(x, 0) = �v0(x), x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Let �u(t, ω) = e−y(ϑtω)�v(t, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Then, �u(·) satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='23) \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 d�u(t) dt + µA�u(t) + ey(ϑtω)B ��u(t) � + α�u(t) + βe(r−1)y(ϑtω)C ��u(t) � = Pf ∞e−y(ϑtω) + σy(ϑtω)�u(t), t > 0, �u(x, 0) = �u0(x) = e−y(ω)�v0(x), x ∈ Rd, in V′ + �L r+1 r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For all the cases given in Table 1, suppose that Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Then, lim τ→−∞ ∥uτ − �u0∥H = 0 implies that the solution u of the system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2) backward converges to the solution �u of the system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='23) , that is, lim τ→−∞ ∥u(T + τ, τ, ϑ−τω, uτ) − �u(t, ω, �u0)∥H = 0, for all T > 0 and ω ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='24) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Let U τ(·) := u(· + τ, τ, ϑ−τω, uτ) − �u(·, ω, �u0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='23), we obtain dU τ dt = −µAU τ − αU τ − ey(ϑtω)� B � u � − B ��u �� − βe(r−1)y(ϑtω)� C � u � − C ��u �� + e−y(ϑtω)[Pf(t + τ) − Pf ∞] + σy(ϑtω)U τ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='25) in V′ + �L r+1 r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Taking the inner product with U τ(·) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='25), and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='4), we get 1 2 d dt∥U τ∥2 H = −µ∥∇U τ∥2 H − (α − σy(ϑtω))∥U τ∥2 H − βe(r−1)y(ϑtω)� C � u � − C ��u � , u − �u � + ey(ϑtω)b(U τ, U τ, �u) + e−y(ϑtω)(f(t + τ) − f∞, U τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='26) From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='5), one can write − � C � u � − C ��u � , u − �u � ≤ −1 2∥|u| r−1 2 (u − �u)∥2 H − 1 2∥|�u| r−1 2 (u − �u)∥2 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='27) Applying H¨older’s and Young’s inequalities, we infer ���e−y(ϑtω)(f(t + τ) − f ∞, U τ) ��� ≤ ∥f(t + τ) − f ∞∥2 L2(Rd) + Ce−2y(ϑtω)∥U τ∥2 H, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='28) and ey(ϑtω)b(U τ, U τ, �u) ≤ \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 Ce2y(ϑtω)∥∇�u∥2 H∥U τ∥2 H + µ 2 ∥∇U τ∥2 H, for d = 2 and r ≥ 1, µ 2∥∇U τ∥2 H + β 4 e(r−1)y(ϑtω)∥|U τ||�u| r−1 2 ∥2 H + C∥U τ∥2 H, for d = 3 and r > 3, 1 2β∥∇U τ∥2 H + β 2 e2y(ϑtω)∥|U τ||�u|∥2 H, for d = r = 3 and 2βµ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='29) 14 KUSH KINRA, MANIL T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' MOHAN, AND RENHAI WANG Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='26)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='29), we achieve d dt∥U τ(t)∥2 H ≤ S(t)∥U τ(t)∥2 H + ∥f(t + τ) − f∞∥2 L2(Rd), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='30) where S(t) = C × \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 e2y(ϑtω)∥∇�u(t)∥2 H + e−2y(ϑtω) + |y(ϑtω)|, for d = 2 and r ≥ 1, e−2y(ϑtω) + |y(ϑtω)| + 1, for d = 3 and r > 3, e−2y(ϑtω) + |y(ϑtω)|, for d = r = 3 and 2βµ ≥ 1, for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' t ∈ [τ, τ + T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Making use of Gronwall’s inequality in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='30) over (0, T), we obtain ∥U τ(T)∥2 H ≤ � ∥U τ(0)∥2 H + � T 0 ∥f(t + τ) − f ∞∥2 L2(Rd)dt � e � T 0 S(t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Since y is continuous and �u ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' V), it implies that � T 0 S(t)dt is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' From Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1 (particularly, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2)), we deduce � T 0 ∥f(t + τ) − f ∞∥2 L2(Rd)dt ≤ � τ+T −∞ ∥f(t) − f ∞∥2 L2(Rd)dt → 0 as τ → −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='31) Using the fact that � T 0 S(t)dt is bounded, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='31) and lim τ→∞ ∥U τ(0)∥2 H = 0, we conclude the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Increasing random absorbing sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' In this subsection, we prove the existence of a pullback D- random absorbing set for the system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='6) with S(v) = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For all the cases given in Table 1, suppose that f ∈ L2 loc(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' L2(Rd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for each (τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' D) ∈ R × Ω × D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' there exists a time T := T(τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' D) > 0 such that sup s≤τ sup t≥T sup u0∈D(s−t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='ϑ−tω) � ∥u(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' s − t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ϑ−sω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' u0)∥2 H + α 2 � s s−t eα(ζ−s)−2σ � ζ s y(ϑη−sω)dη∥u(ζ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' s − t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ϑ−sω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' u0)∥2 Hdζ + 2µ � s s−t eα(ζ−s)−2σ � ζ s y(ϑη−sω)dη∥∇u(ζ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' s − t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ϑ−sω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' u0)∥2 Hdζ + 2β � s s−t e(r−1)y(ϑζ−sω)+α(ζ−s)−2σ � ζ s y(ϑη−sω)dη∥u(ζ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' s − t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ϑ−sω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' u0)∥r+1 �Lr+1dζ � ≤ 4 α sup s≤τ � 0 −∞ eαζ+2|y(ϑζω)|+2σ � 0 ζ y(ϑηω)dη∥f(ζ + s)∥2 L2(Rd)dζ =: 4 α sup s≤τ K(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='32) Furthermore, for 2 < k1 < ∞ and k2 > 0, there exists a time T∗ := T∗(τ, ω, D, k1) > 0 such that sup s≤τ sup t≥T∗ sup u0∈D(s−t,ϑ−tω) � s s−t ek2|y(ϑζ−sω)|+α(ζ−s)−2σ � ζ s y(ϑη−sω)dη∥u(ζ, s − t, ϑ−sω, u0)∥k1 H dζ ≤ C 0 � −∞ ek2|y(ϑζω)|+ α k1 ζ−(k1−2)σ � 0 ζ y(ϑηω)dηdζ × � 0 � −∞ e 2(k1−1)α k2 1 ζ1+2|y(ϑζ1ω)|+2σ � 0 ζ1 y(ϑηω)dη∥f(ζ1 + s)∥2 L2(Rd)dζ1 � k1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='33) ASYMPTOTIC AUTONOMY OF RANDOM ATTRACTORS FOR SCBF EQUATIONS 15 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Let us write the energy inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='3) for u(ζ) = u(ζ, s − t, ϑ−sω, u0), that is, d dζ ∥u(ζ)∥2 H + (α − 2σy(ϑζ−sω))∥u(ζ)∥2 H + α 2 ∥u(ζ)∥2 H + 2µ∥∇u(ζ)∥2 H + 2βe(r−1)y(ϑζ−sω)∥u(ζ)∥r+1 �Lr+1 ≤ 2e2|y(ϑζ−sω)| α ∥f(ζ)∥2 L2(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='34) In view of the variation of constants formula with respect to ζ ∈ (s − t, ξ), we get ∥u(ξ, s − t, ϑ−sω, u0)∥2 H + α 2 � ξ s−t eα(ζ−ξ)−2σ � ζ ξ y(ϑη−sω)dη∥u(ζ, s − t, ϑ−sω, u0)∥2 Hdζ + 2µ � ξ s−t eα(ζ−ξ)−2σ � ζ ξ y(ϑη−sω)dη∥∇u(ζ, s − t, ϑ−sω, u0)∥2 Hdζ + 2β � ξ s−t e(r−1)y(ϑζ−sω)+α(ζ−ξ)−2σ � ζ ξ y(ϑη−sω)dη∥u(ζ, s − t, ϑ−sω, u0)∥r+1 �Lr+1dζ ≤ e−α(ξ−s+t)+2σ � ξ−s −t y(ϑηω)dη∥u0∥2 H + 2 α � ξ−s −t eα(ζ+s−ξ)+2|y(ϑζω)|+2σ � ξ−s ζ y(ϑηω)dη∥f(ζ + s)∥2 L2(Rd)dζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='35) Putting ξ = s in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='35), we find ∥u(s, s − t, ϑ−sω, u0)∥2 H + α 2 � s s−t eα(ζ−s)−2σ � ζ s y(ϑη−sω)dη∥u(ζ, s − t, ϑ−sω, u0)∥2 Hdζ + 2µ � s s−t eα(ζ−s)−2σ � ζ s y(ϑη−sω)dη∥∇u(ζ, s − t, ϑ−sω, u0)∥2 Hdζ + 2β � s s−t e(r−1)y(ϑζ−sω)+α(ζ−s)−2σ � ζ s y(ϑη−sω)dη∥u(ζ, s − t, ϑ−sω, u0)∥r+1 �Lr+1dζ ≤ e−αt+2σ � 0 −t y(ϑηω)dη∥u0∥2 H + 2 α � 0 −∞ eαζ+2|y(ϑζω)|+2σ � 0 ζ y(ϑηω)dη∥f(ζ + s)∥2 L2(Rd)dζ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='36) for all s ≤ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Since u0 ∈ D(s−t, ϑ−tω) and D is backward tempered, it implies from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='9) and the definition of backward temperedness (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='12) that there exists a time T = T(τ, ω, D) such that for all t ≥ T, e−αt+2σ � 0 −t y(ϑηω)dη sup s≤τ ∥u0∥2 H ≤ e− α 3 t sup s≤τ ∥D(s − t, ϑ−tω)∥2 H ≤ 2 α � 0 −∞ eαζ+2|y(ϑζω)|+2σ � 0 ζ y(ϑηω)dη∥f(ζ + s)∥2 L2(Rd)dζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='37) Hence, by using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='37) and taking supremum on s ∈ (−∞, τ] in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='36), we reach at (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Now, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='35), we estimate for 2 < k1 < ∞ and k2 > 0 � s s−t ek2|y(ϑζ−sω)|+α(ζ−s)−2σ � ζ s y(ϑη−sω)dη∥u(ζ, s − t, ϑ−sω, u0)∥k1 H dζ ≤ C � s s−t ek2|y(ϑζ−sω)|+α(ζ−s)+2σ � 0 ζ−s y(ϑηω)dη � e− k1 2 α(ζ−s+t)+k1σ � ζ−s −t y(ϑηω)dη∥u0∥k1 H + � ζ−s � −t eα(ζ1+s−ζ)+2|y(ϑζ1ω)|+2σ � ζ−s ζ1 y(ϑηω)dη∥f(ζ1 + s)∥2 L2(Rd)dζ1 � k1 2 � dζ 16 KUSH KINRA, MANIL T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' MOHAN, AND RENHAI WANG ≤ C � 0 −∞ ek2|y(ϑζω)|+ α k1 ζ−(k1−2)σ � 0 ζ y(ϑηω)dηdζ × � e− (k1−1)α k1 t+k1σ � 0 −t y(ϑηω)dη∥u0∥k1 H + � � 0 −∞ e 2(k1−1)α k2 1 ζ1+2|y(ϑζ1ω)|+2σ � 0 ζ1 y(ϑηω)dη∥f(ζ1 + s)∥2 L2(Rd)dζ1 � k1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='38) Hence, using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='9) and the backward-uniform temperedness property (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='12) of u0 (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='37)), we obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='33), as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For all the cases given in Table 1, suppose that f ∈ L2 loc(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' L2(Rd)) and Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For K(τ, ω) same as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='32), we have (i) There is an increasing pullback D-random absorbing set K given by K(τ, ω) := � v ∈ H : ∥v∥2 H ≤ 4ey(ω) α sup s≤τ K(s, ω) � , for all τ ∈ R and ω ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='39) Moreover, K is backward-uniformly tempered with arbitrary rate, that is, K ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (ii) There is a B-pullback random absorbing set �K given by �K(τ, ω) := � v ∈ H : ∥v∥2 H ≤ 4ey(ω) α K(τ, ω) � ∈ B, for all τ ∈ R and ω ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='40) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' See the proof of in [73, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Backward uniform tail-estimates and backward flattening-property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' In this subsection, we show that the solution of the system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1) satisfies the backward uniform tail-estimates and backward flattening-property for d = 2 with r ∈ {1} ∪ [2, ∞), d = 3 with r ∈ (3, ∞) and d = r = 3 with 2βµ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' These estimates help us to obtain the backward uniform pullback D-asymptotic compactness of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' We use a cut-off function technique to obtain backward uniform tail-estimates and backward flattening-property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' The following lemma provides the backward uniform tail-estimates for the solutions of the system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For all the cases given in Table 1, suppose that Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Then, for any (τ, ω, D) ∈ R × Ω × D, the solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1) satisfies lim k,t→+∞ sup s≤τ sup u0∈D(s−t,ϑ−tω) ∥u(s, s − t, ϑ−sω, u0)∥2 L2(Oc k) = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='41) where Ok = {x ∈ Rd : |x| ≤ k}, k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Let ρ be a smooth function such that 0 ≤ ρ(ξ) ≤ 1, for ξ ∈ R+ and ρ(ξ) = �0, for 0 ≤ ξ ≤ 1, 1, for ξ ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Then, there exists a positive constant C such that |ρ′(ξ)| ≤ C, for all ξ ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Taking divergence to the first equation of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' we obtain formally in weak sense −e−y(ϑtω)∆p = ey(ϑtω)∇ · �� u · ∇ � u � + βe(r−1)y(ϑtω)∇ · � |u|r−1u � − e−y(ϑtω)∇ · f ASYMPTOTIC AUTONOMY OF RANDOM ATTRACTORS FOR SCBF EQUATIONS 17 = ey(ϑtω)∇ · � ∇ · � u ⊗ u �� + βe(r−1)y(ϑtω)∇ · � |u|r−1u � − e−y(ϑtω)∇ · f = ey(ϑtω) d � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='j=1 ∂2 ∂xi∂xj � uiuj � + βe(r−1)y(ϑtω)∇ · � |u|r−1u � − e−y(ϑtω)∇ · f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' which implies p = (−∆)−1 \uf8ee \uf8f0e2y(ϑtω) d � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='j=1 ∂2 ∂xi∂xj � uiuj � + βery(ϑtω)∇ · � |u|r−1u � − ∇ · f \uf8f9 \uf8fb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='42) in the weak sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Taking the inner product to the first equation of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1) with ρ � |x|2 k2 � u, we have 1 2 d dt � Rd ρ �|x|2 k2 � |u|2dx = µ � Rd(∆u)ρ �|x|2 k2 � udx − α � Rd ρ �|x|2 k2 � |u|2dx − ey(ϑtω)b � u, u, ρ �|x|2 k2 � u � − βe(r−1)y(ϑtω) � Rd|u|r+1ρ �|x|2 k2 � dx − e−y(ϑtω) � Rd(∇p)ρ �|x|2 k2 � udx + e−y(ϑtω) � Rd fρ �|x|2 k2 � udx + σy(ϑtω) � Rd ρ �|x|2 k2 � |u|2dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='43) Let us now estimate each term on the right hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Integration by parts and divergence free condition of u(·) help us to obtain µ � Rd(∆u)ρ �|x|2 k2 � udx + µ � Rd |∇u|2ρ �|x|2 k2 � dx = −µ � Rd ρ′ �|x|2 k2 � 2 k2 (x · ∇)u · udx ≤ 2 √ 2µ k � k≤|x|≤ √ 2k |u| ����ρ′ �|x|2 k2 �����|∇u|dx ≤ C k � Rd|u||∇u|dx ≤ C k � ∥u∥2 H + ∥∇u∥2 H � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='44) and −ey(ϑtω)b � u, u, ρ �|x|2 k2 � u � = ey(ϑtω) � Rd ρ′ �|x|2 k2 � x k2 · u|u|2dx ≤ √ 2e|y(ϑtω)| k � k≤|x|≤ √ 2k ����ρ′ �|x|2 k2 �����|u|3dx ≤ C k e|y(ϑtω)|∥u∥2 �L4∥u∥H ≤ C k e|y(ϑtω)|∥u∥ 6−d 2 H ∥∇u∥ d 2 H ≤ C k � ∥∇u∥2 H + e 4|y(ϑtω)| 4−d ∥u∥ 2(6−d) 4−d H � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='45) where we have used Ladyzhenskaya’s (for both d = 2, 3) and Young’s inequalities in the penultimate and final inequalities, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Using integration by parts, divergence free condition and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='42), we obtain −e−y(ϑtω) � Rd(∇p)ρ �|x|2 k2 � udx = e−y(ϑtω) � Rd pρ′ �|x|2 k2 � 2 k2 (x · u)dx ≤ Ce|y(ϑtω)| k � Rd ��(−∆)−1� ∇ · � ∇ · � u ⊗ u ����� · |u|dx 18 KUSH KINRA, MANIL T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' MOHAN, AND RENHAI WANG + Ce(r−1)y(ϑtω) k � Rd ��(−∆)−1� ∇ · � |u|r−1u ���� · |u|dx + Ce|y(ϑtω)| k � Rd |(−∆)−1[∇ · f]| · |u|dx =: C k � S1(d, r) + e(r−1)y(ϑtω)S2(d, r) + S3(d, r) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='46) Estimate of S1(d, r): Using H¨older’s inequality, Fourier transformation, Ladyzhenskaya’s and Young’s inequalities, respectively, we get (for d = 2, 3) |S1(d, r)| ≤ e|y(ϑtω)|��(−∆)−1� ∇ · � ∇ · � u ⊗ u ����� L2(Rd)∥u∥H ≤ e|y(ϑtω)|∥u∥2 �L4∥u∥H ≤ Ce|y(ϑtω)|∥u∥ 6−d 2 H ∥∇u∥ d 2 H ≤ C � ∥∇u∥2 H + e 4|y(ϑtω)| 4−d ∥u∥ 2(6−d) 4−d H � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='47) Estimate of S2(d, r): Divergence free condition gives S2(d, r) = 0 for r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Therefore, we will consider r ∈ [2, ∞) for d = 2 and r ∈ [3, ∞) for d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Applying H¨older’s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Gagliardo-Nirenberg’s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' interpolation and Young’s inequalities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' we obtain |S2(d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' r)| ≤ \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 ∥(−∆)−1� ∇ · � |u|r−1u �� ∥L2(Rd)∥u∥H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 2 and r ∈ [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∥(−∆)−1� ∇ · � |u|r−1u �� ∥L2(Rd)∥u∥H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 3 and r ∈ [3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 5],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∥(−∆)−1� ∇ · � |u|r−1u �� ∥ L 3(r+1) 2r−1 (Rd)∥u∥ �L 3(r+1) r+4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 3 and r ∈ (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ≤ C × \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 ∥u∥r �Lr∥u∥H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 2 and r ∈ [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∥u∥r �L 6r 5 ∥u∥H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 3 and r ∈ [3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 5],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∥u∥r �Lr+1∥u∥ �L 3(r+1) r+4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 3 and r ∈ (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ≤ C × \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ∥u∥ (r+1)(r−2) r−1 �Lr+1 ∥u∥ r+1 (r−1) H ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 2 and r ∈ [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∥u∥ (r+1)(3r−5) 3(r−1) �Lr+1 ∥u∥ 2(r+1) 3(r−1) H ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 3 and r ∈ [3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 5],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∥u∥ (r+1)(3r−5) 3(r−1) �Lr+1 ∥u∥ 2(r+1) 3(r−1) H ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 3 and r ∈ (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ≤ C � ∥u∥r+1 �Lr+1 + ∥u∥r+1 H � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='48) Estimate of S3(d, r): Applying H¨older’s, Gagliardo-Nirenberg’s and Young’s inequalities, we find (for d = 2, 3) |S3(d, r)| ≤ Ce|y(ϑtω)|∥(−∆)−1[∇ · f]∥ L d d−1 (Rd)∥u∥Ld(Rd) ≤ Ce|y(ϑtω)|∥f∥L1(Rd)∥u∥ 4−d 2 H ∥∇u∥ d−2 2 H ≤ Ce2|y(ϑtω)|∥f∥2 L1(Rd) + C∥u∥2 H + C∥∇u∥2 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='49) Finally, we estimate the penultimate term on right hand side (RHS) of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='43) by using H¨older’s and Young’s inequalities as follows: e−y(ϑtω) � Rd f(x)ρ �|x|2 k2 � udx ≤ α 4 � Rd ρ �|x|2 k2 � |u|2dx + e2|y(ϑtω)| α � Rd ρ �|x|2 k2 � |f(x)|2dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='50) ASYMPTOTIC AUTONOMY OF RANDOM ATTRACTORS FOR SCBF EQUATIONS 19 Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='43)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='50), we get d dt∥u∥2 L2(Oc k) + (α − 2σy(ϑtω))∥u∥2 L2(Oc k) ≤ C k � ∥u∥2 H + ∥∇u∥2 H + e(r−1)y(ϑtω)∥u∥r+1 �Lr+1 + e2|y(ϑtω)|∥f∥2 L1(Rd) � + C k � e 4|y(ϑtω)| 4−d ∥u∥ 2(6−d) 4−d H + e(r−1)|y(ϑtω)|∥u∥r+1 H � + 2e2|y(ϑtω)| α � |x|≥k |f(x)|2dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='51) Applying the variation of constants formula to the above equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='51) on (s − t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' s) and replacing ω by ϑ−sω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for s ≤ τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' t ≥ 0 and ω ∈ Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' we find ∥u(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' s − t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ϑ−sω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' u0)∥2 L2(Oc k) ≤ e−αt+2σ � 0 −t y(ϑηω)dη∥u0∥2 H + C k � � s s−t eα(ζ−s)−2σ � ζ s y(ϑη−sω)dη∥u(ζ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' s − t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ϑ−sω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' u0)∥2 Hdζ + � s s−t eα(ζ−s)−2σ � ζ s y(ϑη−sω)dη∥∇u(ζ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' s − t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ϑ−sω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' u0)∥2 Hdζ + � s s−t e(r−1)y(ϑζ−sω)+α(ζ−s)−2σ � ζ s y(ϑη−sω)dη∥u(ζ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' s − t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ϑ−sω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' u0)∥r+1 �Lr+1dζ + � 0 −t eαζ+2|y(ϑζω)|+2σ � 0 ζ y(ϑηω)dη∥f(ζ + s)∥2 L1(Rd)dζ � + C k � � s s−t e 4|y(ϑζ−sω)| 4−d +α(ζ−s)−2σ � ζ s y(ϑη−sω)dη∥u(ζ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' s − t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ϑ−sω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' u0)∥ 2(6−d) 4−d H dζ + � s s−t e(r−1)|y(ϑζ−sω)|+α(ζ−s)−2σ � ζ s y(ϑη−sω)dη∥u(ζ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' s − t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ϑ−sω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' u0)∥r+1 H dζ � + C � 0 −t eαζ+2|y(ϑζω)|+2σ � 0 ζ y(ϑηω)dη � |x|≥k |f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ζ + s)|2dxdζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='52) Now using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='9), the definition of backward-uniform temperedness (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='12) (for the first term on RHS of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='52)), Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='6 ((3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='32) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='33) for the second and third terms on RHS of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='52), respectively) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='8) (for the final term on RHS of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='52)), we immediately complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' □ The following lemma provides the backward flattening-property for the solution of the system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For each k ≥ 1, we let ̺k(x) := 1 − ρ �|x|2 k2 � , x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Let ¯u := ̺ku for u := u(s, s − t, ω, uτ) ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Then ¯u ∈ L2(O√ 2k), which has the orthogonal decomposition: ¯u = Pi¯u ⊕ (I − Pi)¯u =: ¯ui,1 + ¯ui,2, for eah i ∈ N, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='53) where, Pi : L2(O√ 2k) → Hi := span{e1, e2, · · · , ei} ⊂ L2(O√ 2k) is a canonical projection and {ej}∞ j=1 is the family of eigenfunctions for −∆ in L2(O√ 2k) with corresponding positive eigenvalues λ1 ≤ λ2 ≤ · · · ≤ λj → ∞ as j → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' We also have that ̺k∆u = ∆¯u − u∆̺k − 2∇̺k · ∇u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 20 KUSH KINRA, MANIL T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' MOHAN, AND RENHAI WANG Furthermore, for ψ ∈ H1 0(O√ 2k), we have Piψ = i � j=1 (ψ, ej)ej, ∇Piψ = A1/2Piψ = i � j=1 λ1/2 j (ψ, ej)ej, (I − Pi)ψ = ∞ � j=i+1 (ψ, ej)ej, ∇(I − Pi)ψ = A1/2(I − Pi)ψ = ∞ � j=i+1 λ1/2 j (ψ, ej)ej, ∥∇(I − Pi)ψ∥2 L2(O√ 2k) = ∞ � j=i+1 λj|(ψ, ej)|2 ≥ λi+1 ∞ � j=i+1 |(ψ, ej)|2 = λi+1∥(I − Pi)ψ∥2 L2(O√ 2k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='54) Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For all the cases given in Table 1, suppose that Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Let (τ, ω, D) ∈ R × Ω × D and k ≥ 1 be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Then lim i,t→+∞ sup s≤τ sup u0∈D(s−t,ϑ−tω) ∥(I − Pi)¯u(s, s − t, ϑ−sω, ¯u0,2)∥2 L2(O√ 2k) = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='55) where ¯u0,2 = (I − Pi)(̺ku0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Multiplying by ̺k in the first equation of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1), we rewrite the equation as: d¯u dt − µ∆¯u + ey(ϑtω)̺k(u · ∇)u + α¯u + βe(r−1)y(ϑtω)̺k|u|r−1u = −e−y(ϑtω)̺k∇p + e−y(ϑtω)̺kf + σy(ϑtω)¯u − µu∆̺k − 2µ∇̺k · ∇u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='56) Applying (I − Pi) to the equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='56) and taking the inner product of the resulting equation with ¯ui,2 in L2(O√ 2k) gives 1 2 d dt∥¯ui,2∥2 L2(O√ 2k) + µ∥∇¯ui,2∥2 L2(O√ 2k) + (α − σy(ϑtω))∥¯ui,2∥2 L2(O√ 2k) + βe(r−1)y(ϑtω)∥|u| r−1 2 ¯ui,2∥2 L2(O√ 2k) = − ey(ϑtω) d � q,q′=1 � O√ 2k (I − Pi) � uq ∂uq′ ∂xq {̺k(x)}2uq′ � dx � �� � =:J1 − � e−y(ϑtω)̺k∇p, ¯ui,2 � � �� � =:J2 + �� e−y(ϑtω)̺kf, ¯ui,2 � − µ � u∆̺k, ¯ui,2 � − µ � 2∇̺k · ∇u, ¯ui,2 �� � �� � =:J3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='57) Next, we estimate each terms of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='57) as follows: Using integration by parts, divergence free condition of u(·), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='54) (without loss of generality (WLOG), one may assume that λi ≥ 1), H¨older’s and Young’s inequalities, we get |J1| = e|y(ϑtω)| ����� � O√ 2k (I − Pi) � ρ′ �|x|2 k2 � x k2 · ̺k(x)u|u|2 � dx ����� ≤ Ce|y(ϑtω)|∥¯ui,2∥L2(O√ 2k)∥u∥2 �L4 ≤ Cλ − (4−d) 8 i+1 e|y(ϑtω)|∥∇¯ui,2∥ 4−d 4 L2(O√ 2k)∥∇u∥ d 2 H∥u∥ 8−d 4 H ≤ µ 20∥∇¯ui,2∥2 L2(O√ 2k) + Cλ − 4−d 4+d i+1 � ∥∇u∥2 H + e 8|y(ϑtω)| 4−d ∥u∥ 2(8−d) 4−d H � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='58) ASYMPTOTIC AUTONOMY OF RANDOM ATTRACTORS FOR SCBF EQUATIONS 21 |J3| ≤ C � e|y(ϑtω)|∥f∥L2(Rd) + ∥u∥H + ∥∇u∥H � ∥¯ui,2∥L2(O√ 2k) ≤ Cλ − 1 2 i+1 � ∥u∥H + ∥∇u∥H + e|y(ϑtω)|∥f∥L2(Rd) � ∥∇¯ui,2∥L2(O√ 2k) ≤ µ 20∥∇¯ui,2∥2 L2(O√ 2k) + Cλ−1 i+1 � ∥u∥2 H + ∥∇u∥2 H + e2|y(ϑtω)|∥f∥2 L2(Rd) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='59) Using integration by parts, divergence free condition and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='42), we obtain |J2| = �����e−y(ϑtω) � O√ 2k (I − Pi)pρ′ �|x|2 k2 � 4 k2 (x · ¯u)dx ����� ≤ Ce|y(ϑtω)| � O√ 2k ��(−∆)−1� ∇ · � ∇ · � u ⊗ u ����� · |¯ui,2|dx +Ce(r−1)y(ϑtω) � O√ 2k ��(−∆)−1� ∇ · � |u|r−1u ���� · |¯ui,2|dx + Ce|y(ϑtω)| � O√ 2k |(−∆)−1[∇ · f]| · |¯ui,2|dx =: C � �S1(d, r) + �S2(d, r) + �S3(d, r) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='60) Estimate of �S1(d, r): Using H¨older’s inequality, Fourier transformation, Ladyzhenskaya’s and Young’s inequalities, respectively, we get for d = 2, 3 (similar to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='58) above), |�S1(d, r)| ≤ e|y(ϑtω)|��(−∆)−1� ∇ · � ∇ · � u ⊗ u ����� L2(Rd)∥¯ui,2∥L2(O√ 2k) ≤ e|y(ϑtω)|∥u∥2 �L4∥¯ui,2∥L2(O√ 2k) ≤ µ 20∥∇¯ui,2∥2 L2(O√ 2k) + Cλ − 4−d 4+d i+1 � ∥∇u∥2 H + e 8|y(ϑtω)| 4−d ∥u∥ 2(8−d) 4−d H � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='61) Estimate of �S2(d, r): Divergence free condition gives �S2(d, r) = 0 for r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Therefore, we will consider r ∈ [2, ∞) for d = 2 and r ∈ [3, ∞) for d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Applying H¨older’s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Gagliardo-Nirenberg’s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' interpolation and Young’s inequalities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' we obtain |�S2(d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' r)| ≤ e(r−1)|y(ϑtω)| × \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ∥(−∆)−1� ∇ · � |u|r−1u �� ∥L2(Rd)∥¯ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2∥L2(O√ 2k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 2 and r ∈ [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∥(−∆)−1� ∇ · � |u|r−1u �� ∥L2(Rd)∥¯ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2∥L2(O√ 2k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 3 and r ∈ [3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 5],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∥(−∆)−1� ∇ · � |u|r−1u �� ∥ L 3(r+1) 2r−1 (Rd)∥¯ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2∥ L 3(r+1) r+4 (O√ 2k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 3 and r ∈ (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ≤ Ce(r−1)|y(ϑtω)| × \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ∥u∥r �Lr∥¯ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2∥L2(O√ 2k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 2 and r ∈ [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∥u∥r �L 6r 5 ∥¯ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2∥L2(O√ 2k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 3 and r ∈ [3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 5],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∥u∥r �Lr+1∥¯ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2∥ L 3(r+1) r+4 (O√ 2k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 3 and r ∈ (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 22 KUSH KINRA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' MANIL T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' MOHAN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' AND RENHAI WANG ≤ Ce(r−1)|y(ϑtω)| × \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ∥u∥ (r+1)(r−2) r−1 �Lr+1 ∥u∥ 2 (r−1) H ∥¯ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2∥L2(O√ 2k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 2 and r ∈ [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∥u∥ (r+1)(3r−5) 3(r−1) �Lr+1 ∥u∥ 5−r 3(r−1) H ∥¯ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2∥L2(O√ 2k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 3 and r ∈ [3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 5],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∥u∥ (r+1)(3r−5) 3(r−1) �Lr+1 ∥¯ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2∥ 2(r+1) 3(r−1) L2(O√ 2k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 3 and r ∈ (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ≤ Ce(r−1)|y(ϑtω)| × \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 λ − 1 2(r−1) i+1 ∥u∥ (r+1)(r−2) r−1 �Lr+1 ∥u∥ r r−1 H ∥∇¯ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2∥ 1 r−1 L2(O√ 2k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 2 and r ∈ [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' λ − 1 3(r−1) i+1 ∥u∥ (r+1)(3r−5) 3(r−1) �Lr+1 ∥u∥ 2r 3(r−1) H ∥∇¯ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2∥ 2 3(r−1) L2(O√ 2k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 3 and r ∈ [3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 5],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' λ − 1 3(r−1) i+1 ∥u∥ (r+1)(3r−5) 3(r−1) �Lr+1 ∥u∥ 2r 3(r−1) H ∥∇¯ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2∥ 2 3(r−1) L2(O√ 2k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 3 and r ∈ (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ≤ µ 20∥∇¯ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2∥2 L2(O√ 2k) + Cλ − 1 r2 i+1 � e(r−1)|y(ϑtω)|∥u∥r+1 �Lr+1 + e2(r−1)|y(ϑtω)|∥u∥2r H � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='62) where we have used the fact that λi ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Estimate of �S3(d, r): Similar to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='49), we find (for d = 2, 3) |�S3(d, r)| ≤ Ce|y(ϑtω)|∥(−∆)−1[∇ · f]∥ L d d−1 (Rd)∥¯ui,2∥Ld(O√ 2k) ≤ Ce|y(ϑtω)|∥f∥L1(Rd)∥¯ui,2∥ 4−d 2 L2(O√ 2k)∥∇¯ui,2∥ d−2 2 L2(O√ 2k) ≤ Cλ − 4−d 4 i+1 e|y(ϑtω)|∥f∥L1(Rd)∥∇¯ui,2∥L2(O√ 2k) ≤ µ 20∥¯ui,2∥2 L2(O√ 2k) + Cλ − 4−d 2 i+1 e2|y(ϑtω)|∥f∥2 L1(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='63) Now, combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='57)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='63), we arrive at d dt∥¯ui,2∥2 L2(O√ 2k) + (α − 2σy(ϑtω))∥¯ui,2∥2 L2(O√ 2k) ≤ Cλ − 1 r2 i+1 � ∥u∥2 H + e 8|y(ϑtω)| 4−d ∥u∥ 2(8−d) 4−d H + e2(r−1)|y(ϑtω)|∥u∥2r H + ∥∇u∥2 H + e(r−1)y(ϑtω)∥u∥r+1 �Lr+1 + e2|y(ϑtω)|∥f∥2 L1(Rd) + e2|y(ϑtω)|∥f∥2 L2(Rd) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='64) In view of the variation of constant formula,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' we find ∥(I − Pi)¯u(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' s − t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ϑ−sω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ¯u0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2)∥2 L2(O√ 2k) ≤ e−αt+2σ � 0 −t y(ϑηω)dη∥(I − Pi)(̺ku0)∥2 L2(O√ 2k) + Cλ − 1 r2 i+1 � � s s−t eα(ζ−s)−2σ � ζ s y(ϑη−sω)dη∥u(ζ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' s − t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ϑ−sω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' u0)∥2 Hdζ + � s s−t e 8|y(ϑζ−sω)| 4−d +α(ζ−s)−2σ � ζ s y(ϑη−sω)dη∥u(ζ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' s − t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ϑ−sω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' u0)∥ 2(8−d) 4−d H dζ + � s s−t e2(r−1)|y(ϑζ−sω)|+α(ζ−s)−2σ � ζ s y(ϑη−sω)dη∥u(ζ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' s − t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ϑ−sω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' u0)∥2r H dζ + � s s−t eα(ζ−s)−2σ � ζ s y(ϑη−sω)dη∥∇u(ζ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' s − t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ϑ−sω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' u0)∥2 Hdζ ASYMPTOTIC AUTONOMY OF RANDOM ATTRACTORS FOR SCBF EQUATIONS 23 + � s s−t e(r−1)y(ϑζ−sω)+α(ζ−s)−2σ � ζ s y(ϑη−sω)dη∥u(ζ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' s − t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ϑ−sω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' u0)∥r+1 �Lr+1dζ + � 0 −t eαζ+2|y(ϑζω)|+2σ � 0 ζ y(ϑηω)dη� ∥f(ζ + s)∥2 L1(Rd) + ∥f(ζ + s)∥2 L2(Rd) � dζ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='65) Since, ∥(I − Pi)(̺ku0)∥2 L2(O√ 2k) ≤ C∥u0∥2 H, for all u0 ∈ D(s − t, ϑ−tω) and s ≤ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Now, using the definition of backward temperedness (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='12), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='9), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='7), Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='6 (in particular, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='32) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='33)) and the fact that λi → ∞ as i → ∞, we obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='55), as desired, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' This subsection is devoted to the main result of this section, that is, the existence of pullback D-random attractors and their asymptotic autonomy for the solution of the system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='6) with S(v) = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For all the cases given in Table 1, the existence of pullback random attractors for non-autonomous SCBF equations driven by multiplicative noise on the whole space is established in [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For all the cases given in Table 1, as the existence of a unique pullback random attractor is known for each τ, one can obtain the existence of a unique random attractor for autonomous SCBF equations driven by multiplicative noise on the whole space (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' [37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' In view of Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='5 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='7, and Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='8-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='9, the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2 can be completed by applying similar arguments as in the proof of [73, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='6] ([73, Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='5]) and [9, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 2D and 3D SCBF equations: Additive noise In this section, we consider SCBF equations driven by additive white noise, that is, S(v) is independent of v and establish the asymptotic autonomy of pullback random attractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Let us consider the following SCBF equations: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1) \uf8f1 \uf8f2 \uf8f3 dv(t) dt + µAv(t) + B(v(t)) + αv(t) + βC(v(t)) = Pf(t) + g(x)dW(t) dt , t > τ, τ ∈ R, v(x)|t=τ = vτ(x), x ∈ Rd, where g ∈ D(A) and W(t, ω) is the standard scalar Wiener process on the probability space (Ω, F, P) (see Section 3 above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Let us define u(t, τ, ω, uτ) := v(t, τ, ω, vτ) − g(x)y(ϑtω), where y is given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='7) and satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='8), and v is the solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1) with S(v) = g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Then u satisfies: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2) \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 du dt − µ∆u + � (u + gy(ϑtω)) · ∇ � (u + gy(ϑtω)) + αu + β|u + gy(ϑtω)|r−1(u + gy(ϑtω)) = −∇p + f + (σ − α)gy(ϑtω) + µy(ϑtω)∆g, in Rd × (τ, ∞), ∇ · u = 0, in Rd × (τ, ∞), u(x)|t=τ = uτ(x) = vτ(x) − g(x)y(ϑτω), x ∈ Rd and τ ∈ R, u(x)|t=τ → 0, as |x| → ∞, 24 KUSH KINRA, MANIL T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' MOHAN, AND RENHAI WANG as well as (projected form) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='3) \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 du dt + µAu + B(u + gy(ϑtω)) + αu + βC(u + gy(ϑtω)) = Pf + (σ − α)gy(ϑtω) + µy(ϑtω)∆g, t > τ, τ ∈ R, u(x)|t=τ = uτ(x) = v0(x) − g(x)y(ϑτω), x ∈ Rd, in V′ + �L r+1 r , where r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For r = 1, we obtain SNSEs with linear damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' The results of this section can be proven in a similar way as it has been done for SNSEs in [73] under Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Since, Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='9 is not required for r > 1, we provide a different treatment for r > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' NRDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' The following lemma will be frequently used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For all the cases given in Table 1 (excluding d = 2 with r = 1), assume that f ∈ L2 loc(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' L2(Rd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Then, the solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='3) satisfies the following inequality: d dt∥u(t)∥2 H + α∥u(t)∥2 H + µ∥∇u(t)∥2 H + β∥u(t) + gy(ϑtω)∥r+1 �Lr+1 ≤ R � ∥f(t)∥2 L2(Rd) + |y(ϑtω)|2 + |y(ϑtω)|r+1 + |y(ϑtω)| 2(r+1) r−1 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='4) for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' t, where R > 0 is some constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' We find from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='3) that 1 2 d dt∥u∥2 H = − µ∥∇u∥2 H − α∥u∥2 H − β∥u + gy∥r+1 �Lr+1 + b(u + gy, u + gy, gy) + β⟨C(u + gy), gy⟩ + (f, u) + y((σ − α)g − µAg, u), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='5) for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' t ∈ [τ, τ + T] with T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Using g ∈ D(A), H¨older’s and Young’s inequalities, there exist constants R1, R2, R3, R4 > 0 such that β⟨C(u + gy), gy⟩ ≤ β|y|∥u + gy∥r �Lr+1∥g∥�Lr+1 ≤ β 4 ∥u + gy∥r+1 �Lr+1 + R1|y|r+1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='6) (f, u) ≤ ∥f∥L2(Rd)∥u∥H ≤ α 12∥u∥2 H + R2∥f∥2 L2(Rd), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='7) y � (σ − α)g − µAg, u � ≤ α 12∥u∥2 H + R3|y|2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='8) |b(u + gy, u + gy, gy)| = |b(u + gy, u, gy)| ≤ |y|∥u + gy∥�Lr+1∥∇u∥H∥g∥ �L 2(r+1) r−1 ≤ β 4 ∥u + gy∥r+1 �Lr+1 + µ 2 ∥∇u∥2 H + R4|y| 2(r+1) r−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='9) Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='5)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='9), we reach at (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='4) with R = max{2R1, 2R2, 2R3, 2R4}, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For all the cases given in Table 1 (excluding d = 2 with r = 1), assume that f ∈ L2 loc(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' L2(Rd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For each (τ, ω, uτ) ∈ R × Ω × H, the system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='3) has a unique solution u(·, τ, ω, uτ) ∈ C([τ, +∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' H) ∩ L2 loc(τ, +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' V) ∩ Lr+1 loc (τ, +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' �Lr+1) such that u is continuous with respect to the initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ASYMPTOTIC AUTONOMY OF RANDOM ATTRACTORS FOR SCBF EQUATIONS 25 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' One can prove the existence and uniqueness of solution by a standard Faedo-Galerkin approximation method, see the works [31, 34, 47], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For the continuity with respect to the initial data uτ, see the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='9 in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' □ Next result shows the Lusin continuity of the mapping of the solution to the system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='3) in sample points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For all the cases given in Table 1 (excluding d = 2 with r = 1), suppose that f ∈ L2 loc(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' L2(Rd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For each N ∈ N, the mapping ω �→ u(t, τ, ω, uτ) (solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='3)) is continuous from (ΩN, dΩN ) to H, uniformly in t ∈ [τ, τ + T] with T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Let us assume ωk, ω0 ∈ ΩN be such that dΩN (ωk, ω0) → 0 as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Let U k(·) := uk(·) − u0(·), where uk(·) = u(·, τ, ωk, uτ) and u0(·) = u(·, τ, ω0, uτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Then, U k(·) satisfies: dU k dt = −µAU k − αU k − � B � uk + y(ϑtωk)g � − B � u0 + y(ϑtω0)g �� − � βC � uk + y(ϑtωk)g � − βC � u0 + y(ϑtω0)g �� + {(σ − α)g + µ∆g}[y(ϑtωk) − y(ϑtω0)], (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='10) in V′ + �L r+1 r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Taking the inner product with U k(·) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='10), using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='4) and rearranging the terms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' we obtain 1 2 d dt∥U k∥2 H + µ∥∇U k∥2 H + α∥U k∥2 H = −b � U k + [y(ϑtωk) − y(ϑtω0)]g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' U k + [y(ϑtωk) − y(ϑtω0)]g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' u0 + y(ϑtω0)g � + [y(ϑtωk) − y(ϑtω0)] � b � uk + y(ϑtωk)g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' uk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' g � − b � u0 + y(ϑtω0)g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' u0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' g �� − β � C � uk + y(ϑtωk)g � − C � u0 + y(ϑtω0)g � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' � uk + y(ϑtωk)g � − � u0 + y(ϑtω0)g �� + β[y(ϑtωk) − y(ϑtω0)] � C � uk + y(ϑtωk)g � − C � u0 + y(ϑtω0)g � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' g � + [y(ϑtωk) − y(ϑtω0)] � (σ − α)g + µ∆g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' U k� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='11) From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='5), we get −β � C � uk + y(ϑtωk)g � − C � u0 + y(ϑtω0)g � , � uk + y(ϑtωk)g � − � u0 + y(ϑtω0)g �� ≤ −β 2 ���� ��� � uk + y(ϑtωk)g ���� r−1 2 � U k + (y(ϑtωk) − y(ϑtω0))g ����� 2 H − β 2 ���� ��� u0 + y(ϑtω0)g ��� r−1 2 � U k + (y(ϑtωk) − y(ϑtω0))g ����� 2 H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='12) For r > 1 and g ∈ D(A), in view of H¨older’s and Young’s inequalities, we obtain � b � uk + y(ϑtωk)g, uk, g � − b � u0 + y(ϑtω0)g, u0, g �� ≤ � ∥uk + y(ϑtωk)g∥�Lr+1∥∇uk∥H + ∥u0 + y(ϑtω0)g∥�Lr+1∥∇u0∥H � ∥g∥ �L 2(r+1) r−1 ≤ C � ∥uk + y(ϑtωk)g∥r+1 �Lr+1 + ∥∇uk∥2 H + ∥u0 + y(ϑtω0)g∥r+1 �Lr+1 + ∥∇u0∥2 H + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='13) 26 KUSH KINRA, MANIL T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' MOHAN, AND RENHAI WANG Moreover, we have β � C � uk + y(ϑtωk)g � − C � u0 + y(ϑtω0)g � , g � ≤ � ∥uk + y(ϑtωk)g∥r �Lr+1 + ∥u0 + y(ϑtω0)g∥r �Lr+1 � ∥g∥�Lr+1 ≤ C � ∥uk + y(ϑtωk)g∥r+1 �Lr+1 + ∥u0 + y(ϑtω0)g∥r+1 �Lr+1 + 1 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='14) � (σ − α)g + µ∆g, U k� ≤ C∥uk − u0∥H ≤ C � ∥uk∥2 H + ∥u0∥2 H + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='15) Next, we estimate the remaining terms of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='11) separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Case I: d = 2 and r ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Applying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='3) and Young’s inequality, we estimate ���b � U k + [y(ϑtωk) − y(ϑtω0)]g, U k + [y(ϑtωk) − y(ϑtω0)]g, u0 + y(ϑtω0)g ���� ≤ C∥U k + (y(ϑtωk) − y(ϑtω0))g∥H∥∇U k + (y(ϑtωk) − y(ϑtω0))∇g∥H × ∥∇u0 + y(ϑtω0)∇g∥H ≤ C∥∇u0 + y(ϑtω0)∇g∥2 H∥U k∥2 H + C|y(ϑtωk) − y(ϑtω0)|2 � ∥∇u0∥2 H + |y(ϑtω0)|2 + 1 � + µ 2 ∥∇U k∥2 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='16) Case II: d = 3 and r > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Using H¨older’s and Young’s inequalities, we infer ���b � U k + [y(ϑtωk) − y(ϑtω0)]g, U k + [y(ϑtωk) − y(ϑtω0)]g, u0 + y(ϑtω0)g ���� ≤ µ 2 ∥∇U k∥2 H + C∥U k∥2 H + C|y(ϑtωk) − y(ϑtω0)|2 + β 4 ���� ��u0 + y(ϑtω0)g �� r−1 2 � U k + (y(ϑtωk) − y(ϑtω0))g ����� 2 H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='17) Case III: When d = r = 3 with 2βµ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Applying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2), H¨older’s and Young’s inequalities, we obtain ���b � U k + [y(ϑtωk) − y(ϑtω0)]g, U k + [y(ϑtωk) − y(ϑtω0)]g, u0 + y(ϑtω0)g ���� ≤ ���b � U k, U k + [y(ϑtωk) − y(ϑtω0)]g, u0 + y(ϑtω0)g ���� + |y(ϑtωk) − y(ϑtω0)| ���b � g, U k + [y(ϑtωk) − y(ϑtω0)]g, uk + y(ϑtωk)g ���� ≤ 1 2β ∥∇U k∥H + β 2 ��� ��u0 + y(ϑtω0)g �� � U k + (y(ϑtωk) − y(ϑtω0))g ���� 2 H + C|y(ϑtωk) − y(ϑtω0)|2 + β 2 ��� ���uk + y(ϑtωk)g ��� � U k + (y(ϑtωk) − y(ϑtω0))g ���� 2 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='18) Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='11)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='18), we arrive at d dt∥U k(t)∥2 H ≤ C � �P(t)∥U k(t)∥2 H + �Q(t) � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='19) ASYMPTOTIC AUTONOMY OF RANDOM ATTRACTORS FOR SCBF EQUATIONS 27 for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' t ∈ [τ, τ + T], T > 0, and where �P = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 ∥∇u0 + y(ϑtω0)∇g∥2 H, for d = 2 and r > 1, 1, for d = 3 and r > 3, 0, for d = r = 3 and 2βµ ≥ 1, �Q = |y(ϑtωk) − y(ϑtω0)| � ∥uk + y(ϑtωk)g∥r+1 �Lr+1 + ∥uk∥2 V + ∥u0 + y(ϑtω0)g∥r+1 �Lr+1 + ∥u0∥2 V + 1 � + |y(ϑtωk) − y(ϑtω0)|2 × \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 ∥∇u0∥2 H + |y(ϑtω0)|2 + 1, for d = 2 and r > 1, 1, for d = 3 and r > 3, 1, for d = r = 3 and 2βµ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' We infer from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='4) that � τ+T τ � ∥uk(t) + y(ϑtωk)g∥r+1 �Lr+1 + ∥uk(t)∥2 H + ∥∇uk(t)∥2 H � dt ≤ ∥uτ∥2 L2(Rd) + C � τ+T τ � ∥f(t)∥2 H + |y(ϑtωk)|2 + |y(ϑtωk)|r+1 + |y(ϑtωk)| 2(r+1) r−1 � dt, which gives sup k∈N � τ+T τ � ∥uk(t) + y(ϑtωk)g∥r+1 �Lr+1 + ∥uk(t)∥2 H + ∥∇uk(t)∥2 H � dt ≤ C(τ, T, ω0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='20) where we have used (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='11) and the fact that f ∈ L2 loc(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' L2(Rd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' It implies from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='11) and u0 ∈ L2 loc(τ, +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' V) that � τ+T τ �P(t)dt ≤ C(τ, T, ω0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='21) Now, from f ∈ L2 loc(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' L2(Rd)), u0 ∈ C([τ, +∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' H) ∩ L2 loc(τ, +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' V) ∩ Lr+1 loc (τ, +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' �Lr+1), Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='4 and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='20), we conclude that lim k→+∞ � τ+T τ �Q(t)dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='22) In view of the Gronwall inequality in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='19) and making use of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='21)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='22), one can complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2 ensures us that we can define a mapping �Φ : R+ × R × Ω × H → H by �Φ(t, τ, ω, vτ) := v(t + τ, τ, ϑ−τω, vτ) = u(t + τ, τ, ϑ−τω, uτ) + gy(ϑtω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='23) The Lusin continuity in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='3 provides the F-measurability of �Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Consequently, �Φ defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='23) is a NRDS on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Backward convergence of NRDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Consider the autonomous SCBF equations driven by the additive white noise: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='24) \uf8f1 \uf8f2 \uf8f3 d�v(t) dt + µA�v(t) + B(�v(t)) + α�v(t) + βC(�v(t)) = Pf ∞ + g(x)dW(t) dt , t > 0, �v(x, 0) = �v0(x), x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 28 KUSH KINRA, MANIL T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' MOHAN, AND RENHAI WANG Let �u(t, ω) = �v(t, ω) − g(x)y(ϑtω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Then, the system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='24) can be written in the following pathwise deterministic system: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='25) \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 d�u(t) dt + µA�u(t) + B(�u(t) + gy(ϑtω)) + α�u(t) + βC(�u(t) + gy(ϑtω)) = Pf ∞ + (σ − α)gy(ϑtω) − µy(ϑtω)Ag, t > 0, �u(x, 0) = �u0(x) = �v0(x) − g(x)y(ω), x ∈ Rd, in V′ + �L r+1 r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For all the cases given in Table 1 (excluding d = 2 with r = 1), suppose that Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1 is satisfied and lim τ→−∞ ∥uτ − �u0∥H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Then, the solution u of the system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='3) backward converges to the solution �u of the system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='25), that is, lim τ→−∞ ∥u(T + τ, τ, ϑ−τω, uτ) − �u(t, ω, �u0)∥H = 0, for all T > 0 and ω ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='26) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Let U τ(·) := u(· + τ, τ, ϑ−τω, uτ) − �u(·, ω, �u0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='3) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='25), we get dU τ dt = −µAU τ − αU τ − � B � u + gy(ϑtω) � − B � �u + gy(ϑtω) �� − β � C � u + gy(ϑtω) � − C � �u + gy(ϑtω) �� + [Pf(t + τ) − Pf ∞], (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='27) in V′ + �L r+1 r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' In view of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='27), we obtain d dt∥U τ∥2 H = −µ∥∇U τ∥2 H − α∥U τ∥2 H − � B � u + gy(ϑtω) � − B � �u + gy(ϑtω) � , u − �u � − β � C � u + gy(ϑtω) � − C � �u + gy(ϑtω) � , u − �u � + (f(t + τ) − f ∞, U τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='28) From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='5), one can rewrite −β � C � u + gy(ϑtω) � − C ��u + gy(ϑtω) � , (u + gy(ϑtω)) − (�u + gy(ϑtω)) � ≤ −β 2 ∥|u + gy(ϑtω)| r−1 2 |U τ|∥2 H − β 2 ∥|�u + gy(ϑtω)| r−1 2 |U τ|∥2 H (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='29) Applying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='4), H¨older’s and Young’s inequalities, we infer ��� B � u + gy(ϑtω) � − B � �u + gy(ϑtω) � , (u + gy(ϑtω)) − (�u + gy(ϑtω)) ��� = |b(U τ, U τ, �u + gy(ϑtω))| ≤ \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 C∥∇�u + ∇gy(ϑtω)∥2 H∥U τ∥2 H + µ 2 ∥∇U τ∥2 H, for d = 2 and r ≥ 1, µ 2 ∥∇U τ∥2 H + β 4 ∥|�u + gy(ϑtω)| r−1 2 |U τ|∥2 H + C∥U τ∥2 H, for d = 3 and r > 3, 1 2β ∥∇U τ∥2 H + β 2 ∥|�u + gy(ϑtω)||U τ|∥2 H, for d = r = 3 and 2βµ ≥ 1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='30) and |(f(t + τ) − f∞, U τ)| ≤ C∥f(t + τ) − f ∞∥2 L2(Rd) + α 2 ∥U τ∥2 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='31) ASYMPTOTIC AUTONOMY OF RANDOM ATTRACTORS FOR SCBF EQUATIONS 29 Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='28)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='31), we achieve d dt∥U τ∥2 H ≤ C × \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 ∥∇�u + gy(ϑtω)∥2 H∥U τ∥2 H + ∥f(t + τ) − f ∞∥2 L2(Rd), for d = 2 and r ≥ 1, ∥U τ∥2 H + ∥f(t + τ) − f ∞∥2 L2(Rd), for d = 3 and r > 3, ∥f(t + τ) − f ∞∥2 L2(Rd), for d = r = 3 and 2βµ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='32) Applying similar steps as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='5, we complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Increasing random absorbing sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' This subsection provides the existence of increasing D-random absorbing set for non-autonomous SCBF equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For all the cases given in Table 1 (excluding d = 2 with r = 1) and for each (τ, ω, D) ∈ R × Ω × D, there exists a time �T := �T(τ, ω, D) > 0 such that sup s≤τ sup t≥�T sup u0∈D(s−t,ϑ−tω) � ∥u(s, s − t, ϑ−sω, u0)∥2 H + α 2 � s s−t eα(ζ−s)∥u(ζ, s − t, ϑ−sω, u0)∥2 Hdζ + µ � s s−t eα(ζ−s)∥∇u(ζ, s − t, ϑ−sω, u0)∥2 Hdζ + β � s s−t eα(ζ−s)∥u(ζ, s − t, ϑ−sω, u0) + gy(ϑζ−sω)∥r+1 �Lr+1dζ � ≤ 2R sup s≤τ �K(s, ω), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='33) where R is the same as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='4) and �K(s, ω) is given by �K(s, ω) := � 0 −∞ eαζ � ∥f(ζ + s)∥2 L2(Rd) + |y(ϑζω)|2 + |y(ϑζω)|r+1 + |y(ϑζω)| 2(r+1) r−1 � dζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='34) Furthermore, for 2 < k1 < ∞, there exists a time �T∗ := �T∗(τ, ω, D, k1) > 0 such that sup s≤τ sup t≥�T∗ sup u0∈D(s−t,ϑ−tω) � s s−t eα(ζ−s)∥u(ζ, s − t, ϑ−sω, u0)∥k1 H dζ ≤ Ck1 α � � 0 −∞ e 2(k1−1)α k2 1 ζ� ∥f(ζ + s)∥2 L2(Rd) + |y(ϑζω)|2 + |y(ϑζω)|r+1 + |y(ϑζω)| 2(r+1) r−1 � dζ � k1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='35) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Let us consider the energy inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='4) for u(ζ) = u(ζ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' s − t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ϑ−sω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' u0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' that is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' d dζ ∥u(ζ)∥2 H + α∥u(ζ)∥2 H + α 2 ∥u(ζ)∥2 H + µ∥∇u(ζ)∥2 H + β∥u(ζ) + gy(ϑζ−sω)∥r+1 �Lr+1 ≤ R � ∥f(ζ)∥2 L2(Rd) + |y(ϑζ−sω)|2 + |y(ϑζ−sω)|r+1 + |y(ϑζ−sω)| 2(r+1) r−1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' In view of the variation of constants formula with respect to ζ ∈ (s − t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ξ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∥u(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' s − t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ϑ−sω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' u0)∥2 H + α 2 � ξ s−t eα(ζ−ξ)∥u(ζ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' s − t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ϑ−sω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' u0)∥2 Hdζ +µ � ξ s−t eα(ζ−ξ)∥∇u(ζ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' s − t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ϑ−sω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' u0)∥2 Hdζ + β � ξ s−t eα(ζ−ξ)∥u(ζ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' s − t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ϑ−sω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' u0) + gy(ϑζ−sω)∥r+1 �Lr+1dζ ≤ e−α(ξ−s+t)∥u0∥2 H 30 KUSH KINRA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' MANIL T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' MOHAN, AND RENHAI WANG +R � ξ−s −t eα(ζ+s−ξ) � ∥f(ζ + s)∥2 L2(Rd) + |y(ϑζω)|2 + |y(ϑζω)|r+1 + |y(ϑζω)| 2(r+1) r−1 � dζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='36) Putting ξ = s in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='36), we find ∥u(s, s − t, ϑ−sω, u0)∥2 H + α 2 � s s−t eα(ζ−s)∥u(ζ, s − t, ϑ−sω, u0)∥2 Hdζ + µ � s s−t eα(ζ−s)∥∇u(ζ, s − t, ϑ−sω, u0)∥2 Hdζ + β � s s−t eα(ζ−s)∥u(ζ, s − t, ϑ−sω, u0) + gy(ϑζ−sω)∥r+1 �Lr+1dζ ≤ e−αt∥u0∥2 H + R � 0 −∞ eαζ � ∥f(ζ + s)∥2 L2(Rd) + |y(ϑζω)|2 + |y(ϑζω)|r+1 + |y(ϑζω)| 2(r+1) r−1 � dζ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='37) for all s ≤ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Since u0 ∈ D(s−t, ϑ−tω) and D is backward tempered, the definition of backward temperedness (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='12) ensures that there exists a time �T = �T(τ, ω, D) such that for all t ≥ �T, e−αt sup s≤τ ∥u0∥2 H ≤ R � 0 −∞ eαζ � ∥f(ζ + s)∥2 L2(Rd) + |y(ϑζω)|2 + |y(ϑζω)|r+1 + |y(ϑζω)| 2(r+1) r−1 � dζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='38) Hence, Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='38) and taking supremum on s over (−∞, τ] in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='37), we arrive at (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Furthermore, the inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='35) can be obtained by using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='36) and following the similar arguments as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' □ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For all the cases given in Table 1 (excluding d = 2 with r = 1), suppose that f ∈ L2 loc(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' L2(Rd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For R and �K(s, ω), the same as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='4) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='34), respectively, we have (i) There is an increasing pullback D-random absorbing set R given by R(τ, ω) := � v ∈ H : ∥v∥2 H ≤ 4R sup s≤τ �K(s, ω) + 2∥g∥2 H|y(ω)|2 � , for all τ ∈ R and ω ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='39) Moreover, R is backward-uniformly tempered with arbitrary rate, that is, R ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (ii) There is a B-pullback random absorbing set �R given by �R(τ, ω) := � v ∈ H : ∥v∥2 H ≤ 4R �K(τ, ω) + 2∥g∥2 H|y(ω)|2� ∈ B, for all τ ∈ R and ω ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='40) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' See the proof of [73, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Backward uniform tail-estimates and backward flattening-property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' In this subsection, we prove the backward tail-estimates and backward flattening-property for the solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2) for all the cases given in Table 1 (excluding d = 2 with r = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' These estimates help us to prove the backward uniform pullback D-asymptotic compactness of the solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' We will use the cut-off function (same as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='8) to obtain these estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' The following lemma provides the backward uniform tail-estimates for the solution of the system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For all the cases given in Table 1 (excluding d = 2 with r = 1), suppose that f ∈ L2 loc(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' L2(Rd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Then, for any (τ, ω, D) ∈ R × Ω × D, the solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='1) satisfies lim k,t→+∞ sup s≤τ sup u0∈D(s−t,ϑ−tω) ∥u(s, s − t, ϑ−sω, u0)∥2 L2(Oc k) = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='41) ASYMPTOTIC AUTONOMY OF RANDOM ATTRACTORS FOR SCBF EQUATIONS 31 where Ok = {x ∈ Rd : |x| ≤ k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Let ρ be a smooth function defined in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Taking divergence to the first equation in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2), formally we obtain (see the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='8 the detailed calculations) p = (−∆)−1� ∇ · � ∇ · � (u + gy) ⊗ (u + gy) �� + β∇ · � |u + gy|r−1(u + gy) � − ∇ · f � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='42) Taking the inner product to the first equation of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2) with ρ � |x|2 k2 � u, we have 1 2 d dt � Rd ρ �|x|2 k2 � |u|2dx = µ � Rd(∆u)ρ �|x|2 k2 � udx − α � Rd ρ �|x|2 k2 � |u|2dx − b � u + gy, u + gy, ρ �|x|2 k2 � (u + gy) � + b � u + gy, u + gy, ρ �|x|2 k2 � gy � − β � Rd ρ �|x|2 k2 � |u + gy|r+1dx + β � Rd|u + gy|r−1(u + gy)ρ �|x|2 k2 � gydx − � Rd(∇p)ρ �|x|2 k2 � udx + � Rd fρ �|x|2 k2 � udx + (σ − α)y � Rd gρ �|x|2 k2 � udx + µy � Rd(∆g)ρ �|x|2 k2 � udx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='43) Let us now estimate each terms on right hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Using integration by parts, divergence free condition of u(·) and g ∈ D(A), we infer (see inequalities (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='44)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='50)) µ � Rd(∆u)ρ �|x|2 k2 � udx ≤ −µ � Rd |∇u|2ρ �|x|2 k2 � dx + C k � ∥u∥2 H + ∥∇u∥2 H � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='44) y2 b � u + gy, g, ρ �|x|2 k2 � g � ≤ C k � ∥u + gy∥2 H + |y|4∥g∥4 �L4 � ≤ C k � ∥u∥2 H + |y|2 + |y|4� , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='45) −b � u + gy, u + gy, ρ �|x|2 k2 � (u + gy) � ≤ C k ∥u + gy∥3 �L3 ≤ C k � ∥u + gy∥2 H + ∥u + gy∥r+1 �Lr+1 � ≤ C k � ∥u∥2 H + |y|2 + ∥u + gy∥r+1 �Lr+1 � , for r ≥ 2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='46) where we have used interpolation and Young’s inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Using integration by parts, divergence free condition and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='42), we obtain for r ≥ 2, − � Rd(∇p)ρ �|x|2 k2 � udx = � Rd pρ′ �|x|2 k2 � 2 k2 (x · u)dx ≤ C k � Rd ��(−∆)−1� ∇ · � ∇ · � (u + gy) ⊗ (u + gy) ����� · |u|dx + C k � Rd ��(−∆)−1� ∇ · � |u + gy|r−1(u + gy) ���� · |u|dx + C k � Rd |(−∆)−1[∇ · f]| · |u|dx =: C k [Q1(d, r) + Q2(d, r) + Q3(d, r)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='47) 32 KUSH KINRA, MANIL T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' MOHAN, AND RENHAI WANG Estimate of Q1(d, r): Using g ∈ D(A), H¨older’s inequality, Fourier transformation, Ladyzhenskaya’s and Young’s inequalities, respectively, we get for d = 2, 3, |Q1(d, r)| ≤ ��(−∆)−1� ∇ · � ∇ · � (u + gy) ⊗ (u + gy) ����� L2(Rd)∥u∥H ≤ ∥u + gy∥2 �L4∥u∥H ≤ C∥u∥2 �L4∥u∥H + C|y|2∥g∥2 �L4∥u∥H ≤ C∥u∥ 6−d 2 H ∥∇u∥ d 2 H + C|y|4 + C∥u∥2 H ≤ C � ∥∇u∥2 H + ∥u∥2 H + ∥u∥ 2(6−d) 4−d H + |y|4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='48) Estimate of Q2(d, r): Applying H¨older’s (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='48)), Gagliardo-Nirenberg’s (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='48)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' interpolation and Young’s inequalities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' we obtain |Q2(d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' r)| ≤ C × \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 ∥u + gy∥r �Lr∥u∥H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 2 and r ∈ [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∥u + gy∥r �L 6r 5 ∥u∥H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 3 and r ∈ [3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 5],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∥u + gy∥r �Lr+1∥u∥ �L 3(r+1) r+4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 3 and r ∈ (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ≤ C × \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ∥u + gy∥ (r+1)(r−2) r−1 �Lr+1 ∥u + gy∥ 2 r−1 H ∥u∥H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 2 and r ∈ [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∥u + gy∥ (r+1)(3r−5) 3(r−1) �Lr+1 ∥u + gy∥ 5−r 3(r−1) H ∥u∥H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 3 and r ∈ [3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 5],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∥u + gy∥ (r+1)(3r−5) 3(r−1) �Lr+1 ∥u∥ 2(r+1) 3(r−1) H ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 3 and r ∈ (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ≤ C � ∥u + gy∥r+1 �Lr+1 + ∥u∥r+1 H + |y|r+1� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='49) where we have used interpolation and Young’s inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Estimate of Q3(d, r): Similar to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='49), we find (for d = 2, 3) |Q3(d, r)| ≤ C∥(−∆)−1[∇ · f]∥ L d d−1 (Rd)∥u∥Ld(Rd) ≤ C � ∥f∥2 L1(Rd) + ∥u∥2 H + ∥∇u∥2 H � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='50) Finally, we estimate the remaining terms of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='43) by using H¨older’s and Young’s inequalities as follows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' yb � u + gy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ρ �|x|2 k2 � g � + βy � Rd|u + gy|r−1(u + gy)ρ �|x|2 k2 � gdx + � Rd f(x)ρ �|x|2 k2 � udx + (ℓ − α)y � Rd gρ �|x|2 k2 � udx + µy � Rd(∆g)ρ �|x|2 k2 � udx ≤ β 2 � Rd ρ �|x|2 k2 � |u + gy|r+1dx + µ 2 � Rd ρ �|x|2 k2 � |∇u|2dx + α 2 � Rd ρ �|x|2 k2 � |u|2dx + C � Rd ρ �|x|2 k2 �� |y| 2(r+1) r−1 |g| 2(r+1) r−1 + |y|r+1|g|r+1 + |f|2 + |y|2|g|2 + |y|2|∆g|2 � dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='51) Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='43)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='51), we get d dt∥u∥2 L2(Oc k) ≤ −α∥u∥2 L2(Oc k) + C k � ∥u∥2 H + ∥∇u∥2 H + ∥u + gy∥r+1 �Lr+1 + ∥u∥ 2(6−d) 4−d H + ∥u∥r+1 H + ∥f∥2 L1(Rd) + |y|2 + |y|4 + |y|r+1 � + C|y| 2(r+1) r−1 � |x|≥k |g(x)| 2(r+1) r−1 dx + C|y|r+1 � |x|≥k |g(x)|r+1dx ASYMPTOTIC AUTONOMY OF RANDOM ATTRACTORS FOR SCBF EQUATIONS 33 + C � |x|≥k |f(x)|2dx + C|y|2 � |x|≥k |g(x)|2dx + C|y|2 � |x|≥k |∆g(x)|2dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='52) Making use of the variation of constant formula to the above equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='52) on (s − t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' s) and replacing ω by ϑ−sω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' we find that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for s ≤ τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' t ≥ 0 and ω ∈ Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∥u(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' s − t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ϑ−sω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' u0)∥2 L2(Oc k) ≤ e−αt∥u0∥2 H + C k � � s s−t eα(ζ−s) � ∥u(ζ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' s − t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ϑ−sω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' u0)∥2 H + ∥∇u(ζ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' s − t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ϑ−sω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' u0)∥2 H + ∥u(ζ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' s − t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ϑ−sω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' u0) + gy(ϑζ−sω)∥r+1 �Lr+1 + ∥u(ζ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' s − t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ϑ−sω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' u0)∥ 2(6−d) 4−d H + ∥u(ζ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' s − t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ϑ−sω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' u0)∥r+1 H � dζ + � 0 −∞ eαζ � ∥f(ζ + s)∥2 L2(Rd) + |y(ϑζω)|2 + |y(ϑζω)|4 + |y(ϑζω)|r+1 � dζ � + C � 0 −∞ eαζ|y(ϑζω)| 2(r+1) r−1 dζ � |x|≥k |g(x)| 2(r+1) r−1 dx + C � 0 −∞ eαζ|y(ϑζω)|r+1dζ � |x|≥k |g(x)|r+1dx + C � 0 −∞ eαζ|y(ϑζω)|2dζ � |x|≥k |g(x)|2dx + C � 0 −∞ eαζ|y(ϑζω)|2dζ � |x|≥k |∆g(x)|2dx + C � 0 −∞ eαζ � |x|≥k |f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ζ + s)|2dxdζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='53) Now, using the definition of backward temperedness (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='12), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='9), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='8), g ∈ D(A) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='5 (both (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='33) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='35)), one can complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For all the cases given in Table 1 (excluding d = 2 with r = 1), suppose that f ∈ L2 loc(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Let (τ, ω, D) ∈ R × Ω × D and k ≥ 1 be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Then lim i,t→+∞ sup s≤τ sup u0∈D(s−t,ϑ−tω) ∥(I − Pi)¯u(s, s − t, ϑ−sω, ¯u0,2)∥2 L2(O√ 2k) = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='54) where ¯u0,2 = (I − Pi)(̺ku0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' The first equation of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2) can be rewritten as (multiplying by ̺k): d¯u dt − µ∆¯u + ̺k � (u + gy) · ∇ � (u + gy) + α¯u + ̺k|u + gy|r−1(u + gy) + ̺k∇p = −µu∆̺k − 2µ∇̺k · ∇u + ̺kf + (σ − α)̺kgy + µy̺k∆g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='55) Applying the projection (I − Pi) and taking the inner product with ¯ui,2 in L2(O√ 2k) to the equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='55), we get 1 2 d dt∥¯ui,2∥2 L2(O√ 2k) + µ∥∇¯ui,2∥2 L2(O√ 2k) + α∥¯ui,2∥2 L2(O√ 2k) + β∥|u + gy| r−1 2 (¯ui,2 + ¯gi,2y)∥2 L2(O√ 2k) = − 2 � q,m=1 � O√ 2k (I − Pi) � (uq + gqy)∂(um + gmy) ∂xq {̺k(x)}2(um + gmy) � dx � �� � :=L1 34 KUSH KINRA, MANIL T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' MOHAN, AND RENHAI WANG + y 2 � q,m=1 � O√ 2k (I − Pi) � (uq + gqy)∂(um + gmy) ∂xq {̺k(x)}2gm � dx � �� � :=L2 + y � O√ 2k � |u + gy|r−1(¯ui,2 + ¯gi,2y)¯gi,2 � dx � �� � :=L3 − � ̺k(x)∇p, ¯ui,2 � � �� � :=L4 − � µ � u∆̺k + 2∇̺k · ∇u, ¯ui,2 � − � ̺kf, ¯ui,2 � − (σ − α)y � ̺kg, ¯ui,2 � − µy � ̺k∆g, ¯ui,2 �� � �� � :=L5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='56) Next, we estimate each terms on the RHS of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='56) as follows: Estimate of L1: Using integration by parts, divergence free condition of u(·), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='54) (WLOG we assume that λi ≥ 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' H¨older’s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Ladyzhenskaya’s (for d = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 3) and Young’s inequalities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' we find |L1| = ����� � O√ 2k (I − Pi) � ρ′ �|x|2 k2 � x k2 · {¯u + ¯gy}|u + gy|2 � dx ����� ≤ C∥¯ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2 + ¯gi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2y∥L4(O√ 2k)∥u + gy∥�L4∥u + gy∥H ≤ C∥¯ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2 + ¯gi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2y∥ 4−d 4 L2(O√ 2k)∥∇(¯ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2 + ¯gi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2y)∥ d 4 L2(O√ 2k)∥∇(u + gy)∥ d 4 H∥u + gy∥ 8−d 4 H ≤ Cλ − 4−d 8 i+1 ∥u + gy∥ 4+d 4 V ∥u + gy∥ 8−d 4 H ≤ Cλ − 4−d 8 i+1 � ∥u + gy∥2 V + ∥u + gy∥ 2(8−d) 4−d H � ≤ Cλ − 4−d 8 i+1 � ∥u∥2 H + ∥∇u∥2 H + ∥u∥ 2(8−d) 4−d H + |y|2 + |y| 2(8−d) 4−d � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='57) Estimate of L2 and L3: Using g ∈ D(A), H¨older’s, Agmon’s, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='54),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' interpolation and Young’s inequalities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' we get for d = 2 with r ≥ 2 and d = 3 with r ≥ 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' |L2 + L3| ≤ |y|∥¯gi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2∥L∞(O√ 2k) � ∥u + gy∥H∥∇(u + gy)∥H + ∥u + gy∥r �Lr � ≤ C∥¯gi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2∥ 4−d 4 L2(O√ 2k)∥¯gi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2∥ d 4 H2(O√ 2k) � |y|∥u + gy∥H∥∇(u + gy)∥H + |y|∥u + gy∥ 2 r−1 H ∥u + gy∥ (r+1)(r−2) r−1 �Lr+1 � ≤ Cλ − 4−d 8 i+1 ∥∇¯gi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2∥ 4−d 4 L2(O√ 2k)∥¯gi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2∥ d 4 H2(O√ 2k) � ∥u + gy∥4 H + ∥∇(u + gy)∥2 H + ∥u + gy∥r+1 �Lr+1 + |y|4 + |y|2(r−1) � ≤ Cλ − 4−d 8 i+1 � ∥∇u∥2 H + ∥u∥4 H + ∥u + gy∥r+1 �Lr+1 + |y|2 + |y|4 + |y|2(r−1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='58) Estimate of L4: Using integration by parts, divergence free condition for u(·) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='42), we obtain |L4| = ����� � O√ 2k (I − Pi) � pρ′ �|x|2 k2 � 4 k2 (x · ¯u) � dx ����� ASYMPTOTIC AUTONOMY OF RANDOM ATTRACTORS FOR SCBF EQUATIONS 35 ≤ C � O√ 2k ��(−∆)−1� ∇ · � ∇ · � (u + gy) ⊗ (u + gy) ����� · |¯ui,2|dx + C � O√ 2k ��(−∆)−1� ∇ · � |u + gy|r−1(u + gy) ���� · |¯ui,2|dx + C � O√ 2k |(−∆)−1[∇ · f]| · |¯ui,2|dx =: C � �Q1(d, r) + �Q2(d, r) + �Q3(d, r) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='59) Estimate of �Q1(d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' r): Using H¨older’s inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Fourier transformation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Ladyzhenskaya’s and Young’s in- equalities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' we get for d = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 3 | �Q1(d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' r)| ≤ ��(−∆)−1� ∇ · � ∇ · � (u + gy) ⊗ (u + gy) ����� L2(Rd)∥¯ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2∥L2(O√ 2k) ≤ ∥u + gy∥2 �L4∥¯ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2∥L2(O√ 2k) ≤ Cλ − 4−d 8 i+1 ∥∇(u + gy)∥ d 2 H∥u + gy∥ 4−d 2 H ∥∇¯ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2∥ 4−d 4 L2(O√ 2k)∥¯ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2∥ d 4 L2(O√ 2k) ≤ Cλ − 4−d 8 i+1 � ∥u + gy∥2 V + ∥u + gy∥ 2(4−d) 2−d H + ∥u∥2 V + ∥u∥2 H � ≤ Cλ − 4−d 8 i+1 � ∥u∥2 H + ∥∇u∥2 H + ∥u∥ 2(4−d) 2−d H + |y|2 + |y| 2(4−d) 2−d � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='60) Estimate of �Q2(d, r): Applying H¨older’s (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='62)), Gagliardo-Nirenberg’s (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='62)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' interpolation and Young’s inequalities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' we find | �Q2(d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' r)| ≤ C × \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ∥u + gy∥r �Lr∥¯ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2∥L2(O√ 2k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 2 and r ∈ [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∥u + gy∥r �L 6r 5 ∥¯ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2∥L2(O√ 2k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 3 and r ∈ [3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 5],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∥u + gy∥r �Lr+1∥¯ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2∥ L 3(r+1) r+4 (O√ 2k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 3 and r ∈ (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ≤ C × \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ∥u + gy∥ (r+1)(r−2) r−1 �Lr+1 ∥u + gy∥ 2 (r−1) H ∥¯ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2∥L2(O√ 2k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 2 and r ∈ [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∥u + gy∥ (r+1)(3r−5) 3(r−1) �Lr+1 ∥u + gy∥ 5−r 3(r−1) H ∥¯ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2∥L2(O√ 2k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 3 and r ∈ [3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 5],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∥u + gy∥r �Lr+1∥u∥ r−5 3(r−1) �Lr+1 ∥¯ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2∥ 2(r+1) 3(r−1) L2(O√ 2k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 3 and r ∈ (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ≤ C × \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 λ − 1 2(r−1) i+1 ∥u + gy∥ (r+1)(r−2) r−1 �Lr+1 ∥u + gy∥ 2 r−1 H ∥u∥ r−2 r−1 H ∥∇¯ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2∥ 1 r−1 L2(O√ 2k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 2 and r ∈ [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' λ − 1 12 i+1 ∥u + gy∥ (r+1)(3r−5) 3(r−1) �Lr+1 ∥u + gy∥ 5−r 3(r−1) H ∥u∥ 5 6 H∥∇¯ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2∥ 1 6 L2(O√ 2k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 3 and r ∈ [3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 5],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' λ − r+1 3r(r−1) i+1 ∥u + gy∥r �Lr+1∥u∥ r−5 3(r−1) �Lr+1 ∥u∥ 2(r+1) 3r H ∥∇¯ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2∥ 2(r+1) 3r(r−1) L2(O√ 2k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' for d = 3 and r ∈ (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' ∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 36 KUSH KINRA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' MANIL T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' MOHAN, AND RENHAI WANG ≤ C × \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 λ − 1 2(r−1) i+1 � ∥u + gy∥r+1 �Lr+1 + ∥u + gy∥4(r−1) H + ∥u∥2(r−1) H + ∥u∥2 V � , for d = 2 and r ∈ [2, ∞), λ − 1 12 i+1 � ∥u + gy∥r+1 �Lr+1 + ∥u + gy∥2 H + ∥u∥10 H + ∥u∥2 V � , for d = 3 and r ∈ [3, 5], λ − r+1 3r(r−1) i+1 � ∥u + gy∥r+1 �Lr+1 + ∥u∥r+1 �Lr+1 + ∥u∥r+1 H + ∥u∥2 V � , for d = 3 and r ∈ (5, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='61) Estimate of �Q3(d, r): Similar to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='63), we find (for d = 2, 3) | �Q3(d, r)| ≤ C∥(−∆)−1[∇ · f]∥ L d d−1 (Rd)∥¯ui,2∥Ld(O√ 2k) ≤ C∥f∥L1(Rd)∥¯ui,2∥ 4−d 2 L2(O√ 2k)∥∇¯ui,2∥ d−2 2 L2(O√ 2k) ≤ Cλ − 4−d 4 i+1 ∥f∥L1(Rd)∥∇¯ui,2∥L2(O√ 2k) ≤ µ 4 ∥¯ui,2∥2 L2(O√ 2k) + Cλ − 4−d 2 i+1 ∥f∥2 L1(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='62) Estimate of L5: Applying H¨older’s and Young’s inequalities, we deduce |L5| ≤ C � ∥u∥H + ∥∇u∥H + ∥f∥L2(Rd) + |y| � ∥¯ui,2∥L2(O√ 2k) ≤ Cλ−1/2 i+1 � ∥u∥H + ∥∇u∥H + ∥f∥L2(Rd) + |y| � ∥∇¯ui,2∥L2(O√ 2k) ≤ µ 4∥∇¯ui,2∥L2(O√ 2k) + Cλ−1 i+1 � ∥u∥2 H + ∥∇u∥2 H + ∥f∥2 L2(Rd) + |y|2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='63) Now, combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='56)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='63), applying the variation of constant formula, using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='5 (both (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='33) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='35)) and passing limit i → ∞, (λi+1 → 0 as i → ∞), we demonstrate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='54), as desired (see the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='9), which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' This subsection is devoted to the proof of main result of this section, that is, the existence of pullback D-random attractors and their asymptotic autonomy for the solution of the system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='6) with S(v) = g ∈ D(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For all the cases given in Table 1 (excluding d = 2 with r = 1), the existence of pullback D-random attractors for non-autonomous SCBF equations driven by additive noise on the whole space is established in [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' For all the cases given in Table 1 (excluding d = 2 with r = 1), as the existence of a unique pullback random attractor is known for each τ, one can obtain the existence of a unique random attractor for an autonomous SCBF equations driven by additive noise on the whole space (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' [38]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' In view of Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='4 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='6, and Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='7 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='8, the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='3 can be obtained by applying similar arguments as in the proof of [73, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='6] (Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='5 in [73]) and [9, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Acknowledgments: The first author would like to thank the Council of Scientific & Industrial Research (CSIR), India for financial assistance (File No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' 09/143(0938)/2019-EMR-I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Mohan would like to ASYMPTOTIC AUTONOMY OF RANDOM ATTRACTORS FOR SCBF EQUATIONS 37 thank the Department of Science and Technology (DST), Govt of India for Innovation in Science Pursuit for Inspired Research (INSPIRE) Faculty Award (IFA17-MA110).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Renhai Wang was supported by China Postdoctoral Science Foundation under grant numbers 2020TQ0053 and 2020M680456.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Declarations: Ethical Approval: Not applicable Competing interests: The authors declare no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Authors’ contributions: All authors have contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Funding: CSIR, India, 09/143(0938)/2019-EMR-I (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Kinra), DST, India, IFA17-MA110 (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Mohan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Availability of data and materials: 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INDIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Email address, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Mohan: maniltmohan@ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='iitr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='in, maniltmohan@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='com (Renhai Wang) School of Mathematics and Statistics, Southwest University, Chongqing 400715, CHINA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Email address, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content=' Wang: rwang-math@outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} +page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfYvfO/content/2301.00211v1.pdf'} diff --git a/odAzT4oBgHgl3EQfOfsX/vector_store/index.pkl b/odAzT4oBgHgl3EQfOfsX/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..968a08c0ea70afce5d40e69b10af0db4787ebdd0 --- /dev/null +++ b/odAzT4oBgHgl3EQfOfsX/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:66b68a7ab4d5574e70b35519a5b0eeac5481ad73b8259b03d0165a34a8ead002 +size 986788 diff --git a/odE2T4oBgHgl3EQfKAYw/content/tmp_files/2301.03697v1.pdf.txt b/odE2T4oBgHgl3EQfKAYw/content/tmp_files/2301.03697v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e4479f0be5d06ea147eb9186c77cd2e490648441 --- /dev/null +++ b/odE2T4oBgHgl3EQfKAYw/content/tmp_files/2301.03697v1.pdf.txt @@ -0,0 +1,863 @@ +Relation Between Photoionization Cross Sections and Attosecond Time Delays +Jia-Bao Ji,1, ∗ Kiyoshi Ueda,1, 2, † Meng Han,1 and Hans Jakob W¨orner1, ‡ +1Laboratorium f¨ur Physikalische Chemie, ETH Z¨urich, 8093 Z¨urich, Switzerland +2Department of Chemistry, Tohoku University, Sendai, 980-8578, Japan +(Dated: January 11, 2023) +By analyzing the reported theoretical and experimental results on photoionization, we show that +for an (anti)resonance, there is a Kramers-Kronig-like relation between the cross section and the +time delay. We derive the analytical time-delay formulae for the (anti)resonance whose cross section +is described by the Lorentz or Fano ansatz with consideration of the interference of the resonant and +non-resonant channels. We further demonstrate that the relation is also valid for an energetically- +confined cross-section feature with multiple maxima and minima. The validity, however, depends +on the topology of the transition amplitude on the complex plane, which causes the discordance +of various theoretical approaches on the Ar 3s Cooper minimum. Our work bridges two kinds of +measurements and highlights the roll of the analyticity of the ionization amplitude. +Photoionization is one of the fundamental processes that reveal the electronic behavior in atoms and molecules. +For a photoelectron emitted with kinetic energy Ek, one can write the angle-differential dipole transition amplitude +as D(E, ˆk), where E is the photon energy, and ˆk is the emission angle, and the angle-integrated amplitude D(E), +then the photoionization cross section can be expressed as +σ(E) = 4π2 +Ec |D(E)|2 ∝ Im{d(E)}, +(1) +where c is the speed of light, and d(E) is the dipole response function in the frequency domain regarding the atom +(molecule) as the medium of light. The atomic units are used through this Letter unless otherwise stated. The +differential photoionization cross section at emission angle ˆk can be described as σ(E, ˆk) = +π +Ec|D(E, ˆk)| +2 = σ(E) +4π {1 + +β(E)P2[cos(ˆk · ˆe)]} with ˆe being the light polarization vector, and P2 is the second Legendre polynomial. Conventional +photoionization studies have been focusing on the measurements of cross sections σ(E) and asymmetry parameters +β(E) [1]. More recently, the time delays of photoionization of atoms and molecules have been measured by techniques +such as the reconstruction of attosecond beating by interference of two-photon transitions (RABBITT) technique +[2–4] and the attosecond streaking [5, 6], with the Wigner time delay [7, 8] defined as +τ(E) = ∂arg{D(E)} +∂E += ∂Im{ln[D(E)]} +∂E +. +(2) +These time delays are typically at the order of attosecond (as, 1 as = 10−18 s), which corresponds to the natural +time scale of electronic motion in an atom or a molecule. Both theoretical and experimental studies indicate that a +structured continuum, or an (anti)resonance, such as a Fano resonance [9–11], a shape resonance [12–15], a Cooper +minimum [16], and a two-center interference [17], usually leads to both the modulation of cross section and a significant +time delay of ±102 as. In this Letter, we present the quantitative relation between the cross section and the time +delay in such an (anti)resonance, which relates the two aspects of photoionization. +Photoionization can be regarded as a half-scattering process [18], and its transition amplitude can be presented by +the S-matrix or the related R-matrix [19]. If the S-matrix is diagonalized according to the angular momentum l as +S(k) = +� +l,m +��l, m +� +e2iφl� +l, m +��, +(3) +where 2φl is the l-th phase shift, then the photoionization amplitude corresponds to +S(k) − 1= +� +l,m +��l, m +� +(e2iφl − 1) +� +l, m +�� += 2i +� +l,m +��l, m +� +eiφl sin φl +� +l, m +��, +(4) +where the eigenvalues no longer have moduli of 1, and the phases are halved. Equation (2) yields τl(E) = ∂φl/∂E, +which is the original time delay proposed by Wigner [7]. The experimental time delay, on the other hand, is usually +expressed in the ˆk-space with interference of different l’s. Nonetheless, Eq. (2) can be interpreted as the “generalized +arXiv:2301.03697v1 [physics.atom-ph] 9 Jan 2023 + +2 +Wigner delay”, which deals with the off-diagonal terms following Eisenbud’s formula [20–22]. The analyticity of the S- +matrix based on causality has been intensely studied [23–30] and the scattering amplitude was shown to be analytical +on the upper-half complex k-plane for the physical region of the reaction, where k is the incident momentum. This +is the foundation of the complex-scaling method for computation [31–33]. On the other hand, the analyticity can +be also expressed regarding energy, and by applying the energy-time uncertainty principle, a time-domain picture +of the scattering processes arises [34–36], where the connection of this “microscopic” time and the attosecond time +delay is discussed in the next section. The analycity leads to the Kramers-Kronig (KK) relations [37–39] that allow +one to construct an energy-dependent complex function f(E) using only its real or imaginary part. For example, +the energy-domain response function d(E) can be obtained from its imaginary part using Eq. (1), and hence the +time-domain response function ˜d(t) via Fourier transform [40–42]. In this Letter we show that using the analyticity +of the scattering amplitude, Im{ln[D(E)]} is similarly connected to its real part, and the Wigner time delay can be +retrieved via Eq. (2). The KK relations require that the absorption variation vanishes beyond a certain frequency +region, which is true for an (anti)resonance where the cross section below and above the resonance region (in the +vicinity of the resonant energy Er) can be approximated as constant, and the variation of E in Eq. (1) is negligible +compared to the drastic modulation of D(E), so Eq. (1) can be written as (see Appendix for details) +Re{ln[D(E)]} ≈ 1 +2 ln [σ(E)] + const. +(5) +Therefore, the trajectory of (Re{D(E)}, Im{D(E)}) from E ≪ Er to E ≫ Er is approximately a closed contour. +Using the KK relations of the amplitude and the phase of a complex function, it has been shown [43–47] that the +phase can be retrieved (with freedom of a common phase shift) from the absolute value when ln[D(E)] vanishes faster +than 1 +E and D(E) fulfills the minimum-phase condition, namely, that all the poles and zeros are located on the same +half-plane. +In the interaction picture ˆH(∆t) = ˆH0 + ˆH1(∆t), where ˆH0 is the field-free Hamiltonian, and ˆH1(∆t) is the +interaction term as a function of retardation, the interaction is turned on at ∆t = 0, which triggers a transition at +∆t ≥ 0. This retardation is the “microscopic” time introduced by Branson, and we have ignored the relativistic effect +of the virtual-particle “cloud” [34], whose temporal feature is much shorter than the electronic response in atoms +(molecules), as analogously shown in Ref. [36]. Thus, in the energy (frequency) domain, we have +D(E) = +� +∞ +0 +˜D(∆t)eiE∆td∆t. +(6) +This ensures D(E) has no poles on the upper-half of the complex plane. The minimum-phase condition thus requires +no zeros on the upper half-plane, which means that the transition amplitude is invertible. It also implies that the origin +is not enclosed in the trajectory (the winding number is 0), and the total phase change throughout the (anti)resonant +region is 0 instead of ±2π [47]. We shall come back to this point in the discussion of the Cooper minimum. From Eq. +(6), we can define the complex “average retardation”, with the same expression derived by Pollak and Miller [48], as +T(E) = +� +∞ +0 +∆t ˜D(∆t)eiE∆td∆t +� +∞ +0 +˜D(∆t)eiE∆td∆t += −i∂ ln[D(E)] +∂E +. +(7) +The Wigner time delay τ(E) is connected to T(E) and σ(E) using Eqs. (2) and (5) and the KK relations by +τ(E)= Re{T(E)} = −H{Im{T(E)}} += 1 +2 +∂ +∂E H{ln[σ(E)]} = 1 +2H +� +1 +σ(E) +∂σ +∂E +� +, +(8) +where H is the Hilbert transform: +H{f(E)} = +1 +πE ∗ f(E) = 1 +π P +� +∞ +−∞ +f(E′) +E − E′ dE′, +(9) +which is linear and commutative to the derivative. Here ∗ represents the convolution, and P refers to the Cauchy +principal value. The boundary condition at infinity hence becomes +lim +(E−Er)→∞ +|E − Er| +σ(E) +∂σ +∂E = 0. +(10) + +3 +The cross section of an (anti)resonance can be expressed by the Lorentzian lineshape, e.g., the Breit-Wigner +resonance formula[49, 50]: +σL(ϵ) = σa +1 +ϵ2 + 1 + σb +(11) +or the Fano lineshape [51, 52]: +σF(ϵ) = σa +(ϵ + q)2 +ϵ2 + 1 + σb, +(12) +where ϵ = (E − Er)/( Γ +2 ) is the relative energy, and σa and σb are the coefficients of resonant and non-resonant +pathways, respectively. We assume that they contribute to the overall transition amplitude coherently, which fits +in the Feshbach picture [53] and was recently demonstrated in the x-ray regime [54]. The Lorentzian lineshape is +symmetric, while the Fano lineshape is asymmetric, with q being the asymmetric parameter. Γ describes the peak +width, and from Fano’s picture, the resonant state has a lifetime of 1/Γ. Let r = σb/σa, and the cross section for +both Lorentzian and Fano linehsapes be expressed as +σ(ϵ) = S(ϵ + Q)2 + γ2 +ϵ2 + 1 +. +(13) +For the Lorentzian lineshape, Q = 0, γ2 = 1 + 1 +r, and S = σb, while for the Fano lineshape, Q = +q +r+1, γ2 = r(r+q2+1) +(r+1)2 +, +and S = σa + σb. The detailed proof can be found in the Appendix. Using +∂ +∂E = 2 +Γ +∂ +∂ϵ, we have +1 +σ(E) +∂σ +∂E = 2 +Γ +� +�� +2 +γ +� +ϵ+Q +γ +� +� +ϵ+Q +γ +�2 ++ 1 +− +2ϵ +ϵ2 + 1 +� +�� . +(14) +It is easy to verify that the boundary condition Eq. (10) is fulfilled, and Eq. (8) yields +τ(E) = 2 +Γ +� +��− +1 +|γ| +� +ϵ+Q +|γ| +�2 ++ 1 ++ +1 +ϵ2 + 1 +� +�� , +(15) +where we used the property of the Hilbert transform that H{f(aE)}(E) = sgn(a)[H{f(E)}(aE)]. The time delay +can be decomposed as two Lorentzian lineshapes with widths of γΓ and Γ and opposite signs, centered at (Er − QΓ +2 ) +and Er, respectively. For the Fano lineshape with r → 0, |γ| → 0 the first time-delay peak approaches −πδ(ϵ + Q). +On the other hand, for the Lorentzian lineshape with r → 0, |γ| → ∞ the first time-delay peak vanishes, so the time +delay is also a Lorentzian lineshape with τ(Er) = 2/Γ, as discussed in Ref. [8]. +Since (Q + iγ) can be regarded as the “complex asymmetric parameter” [54], a more insightful approach is to +investigate the function [9, 55] +D(ϵ) = +√ +Sϵ + Q + iγ +ϵ + i += +√ +S +� +1 + Q + i(γ − 1) +ϵ + i +� +. +(16) +It fulfills |D(ϵ)|2 = σ(ϵ) and it is minimum-phase when γ > 0, so the phase of the transition amplitude is the same as +the phase of D(ϵ) with the freedom of a common phase shift: +arg {D(ϵ)} = arg {D(ϵ)} + φ0 += cot−1 +�ϵ + Q +γ +� +− cot−1(ϵ) + φ0, +(17) +and thus the Wigner time delay will be the same, as expressed in Eq. (15). For the Lorentzian lineshape with Q = 0, +the maximal phase jump is (see Appendix for details) +∆φmax = φ(√γ) − φ(−√γ) = 4 tan−1(√γ) − π, +(18) +which approaches +π and −π for γ → +∞ (r → 0) and γ → 0 (r → −1), respectively. The trajectory of D(ϵ) +from ϵ → −∞ to ϵ → +∞ is a counter-clockwise rotating (due to the causality) closed circle [9, 30] with radius of + +4 +FIG. 1. The transition amplitudes for the Lorentzian and Fano resonances. The cross sections (solid) and the phases (dashed) +(a, b), the derivative of the logarithm of cross sections (solid) and the time delays (c, d), and transition-amplitude trajectories +(e, f) for the Lorentzian lineshape (a, c, e) and the Fano lineshape with q = 2 (b, d, f), where σa = 1 and σb → 0+ (blue), +σb = 0.1 (orange). The magenta dots in (e) and (f) indicate the beginning of the trajectories at ϵ → −∞. +√ +S +2 +� +(γ − 1)2 + Q2 centered at +�√ +S γ+1 +2 , − +√ +S Q +2 +� +. It can be easily verified that the origin is not enclosed when +γ > 0. Take the Fano resonance as an example, at ϵ → ±∞, the transition amplitudes are dominated by the direct +ionization channel, which is approximately constant in this region. As r → 0 (“pure” Fano lineshape), the trajectory +approaches tangential to the origin, where a (−π)-phase jump occurs within an infinitely small energy interval, which +corresponds to the −πδ(ϵ+Q) peak. For the “pure” Lorentzian transition amplitude D(ϵ) ∝ +1 +ϵ+i [30], there is a phase +offset of π between ϵ → +∞ and ϵ → −∞, which does not fulfill the minimum-phase condition since it encloses “half” +the origin. Its phase retrieved by the logarithm Hilbert transform (LHT) method starts and ends at the same value +at infinities, which corresponds to shifting the circle by an infinitely small amount away from the origin, and thus +subtracts an infinitely broad Lorentzian lineshape according to Eq. (15), where Q = 0 and γ → +∞. Its behavior +near ϵ = 0, which is of most interest, is largely unchanged. Examples of Lorentzian and Fano lineshapes are plotted +in Fig. 1, where the increase of r results in the decrease of the extreme(s) of the time delay. +Recently, Kheifets and Catsamas [56] proposed an approach for shape-resonance analysis by utilising the relation +σl(E) ∝ sin2 φl from Eq. (4), which is applicable if only one l is dominating. Let fl(E) = tan[φl(E) + π/2] = +− cot[φl(E)] be a real function, then +sin2[φl(E)] = +1 +fl(E)2 + 1 += +��� +1 +fl(E) ± i +��� +2 +. +(19) +Therefore, assuming fl(E) to be analytical, +H +�1 +2 ln[σl(E)] +� += − arg{fl(E) ± i} += ∓ cot−1[fl(E)] = ±φl(E), +(20) +if one of the two branches (fl(E) ± i) has no poles or zeros on the upper half-plane of E. This is the condition under +which Kheifets’s approach is equivalent to the method introduced in this Letter. For example, let fl(E) = E/( Γl +2 ), +where Γl > 0, (fl(E) + i) fulfills the condition and leads to the “pure” Breit-Wigner resonance. +In real experiments, the measurable energy range is finite, so the boundary condition is not strictly fulfilled, +which leads to a deviation of the retrieved time delay from its physical counterpart. However, when comparing two +photoionization processes, the relative time delay, which is usually the experimental observable [57], can be obtained + +(a) 101 +(b)101 +7 +100 +0 +E +10-1 +0 +9 +6 +0 +10-1 +-2 +-1 +10-2 +10-3 +-10 +10 +0 +-10 +10 +E +(d) +c +0.5 +1.0 +dln(V)/de +3p/ +2 +L +dln(V)/ +3p/ +0.5 +0.0 +/Φp +0 +/pp +0.0 +2 +-1 +-0.5 +-10 +0 +10 +-10 +10 +E +(e) +0.5 +(f) +0 +- +[()g}ul +[()]ul +0.0 +-1 +-2 +-0.5 +0 +1 +0 +2 +Re[の(e)) +Re[(e))5 +FIG. 2. Applications of the LHT formalism to Xe and Xe@C60. (a) The calculated cross-section ratio (green dashed, right +vertical axis) using the RPAE method and the time delay (blue, left vertical axis) [59] and the time delay retrieved from the +cross-section using the LHT (red, left vertical axis). The filled area indicates the difference between first taking the energy +derivative then perform the Hilbert transform and first performing the Hilbert transform then taking the energy derivative, +which are the same when the boundary condition is strictly fulfilled. (b) The transition-amplitude trajectory constructed by +DXe@C60/DXe(free) = exp[ 1 +2 ln(r) + i +2H{ln(r)}], where r = σXe@C60/σXe(free) is the cross-section ratio. +FIG. 3. Comparison of the trajectories with different topologies. Cross sections (a), phases (b), time delays (c), and trajectories +(d, the magenta dot indicates ϵ → −∞) of the anti-Lorentzian lineshapes with S = 1 , Q = 0 and γ = 0+ (blue, dashed), 0.1 +(orange, solid), or −0.1 (green, dotted), according to Eq. (16). +from their cross-section ratio, which converges faster than the individual time delays: +∆τ(E) = τ1(E) − τ2(E) = ∂ +∂E arg +�D1 +D2 +� += 1 +2 +∂ +∂E H +� +ln +�σ1 +σ2 +�� += 1 +2H +�σ2 +σ1 +∂(σ1/σ2) +∂E +� +. +(21) +Note that the amplitude-phase relation can be extended beyond “simple” resonances with Lorentzian or Fano line- +shapes. For example, the cross sections and time delays of free xenon and xenon in a C60 cage in the region of the +giant resonance have been computed using the random-phase approximation (RPA), relativistic RPA (RRPA) [58], +RPA with exchange (RPAE), and time-dependent Schr¨odinger equation (TDSE) methods [59]. As shown in Fig. 2, +the cross-section ratio shows multiple peaks in the giant-resonance region, which reflects the influence of the C60 +shell, and the time delay retrieved by the LHT, which does not require the phase of the transition amplitude, almost +quantitatively agrees with the reported value obtained from the phase. Positive (negative) time delays are typically +associated with local maxima (minima) of the cross-section ratio. The reconstructed trajectory of DXe@C60/DXe(free) +has a multiple-spiral structure, which manifests the validity of the LHT method for complex resonances. +Now we discuss a specific type of antiresonance, the Cooper minimum (CM), which has a distinctly different origin +compared to Fano or shape resonance [16, 60, 61]. Nevertheless, the cross section near the minimum is approximately +parabolic, which can be locally fitted by a negative-Lorentzian curve, which corresponds to r = σb/σa → −1, and +therefore Q = 0, γ → 0 in Eq. (13). We note that using Q ̸= 0 one can describe the asymmetry of the CM (see +Appendix for details). Although ±γ give the same cross section, they have different topological structures on the +complex plane. Only the transition amplitude corresponding to the positive γ is minimum-phase, which leads to a +negative time delay, while the negative γ gives a positive time delay, as plotted in Fig. 3. The latter has a zero at + +(a) +(b) +2.5 +1 +150 - + () +0.5 +(as) +2.0 +energy ( +100 + TXe(free) +120 +0.0 +50 +1.5 +Xe@C60 / +100 +photon +8 +0 +? +TXe@C +6 +-0.5- +Im +80 +-50 +- +0.5 +0.0 +0.5 +1.0 +1.5 +75 +100 +125 +150 +Re[xe@Co /xe(free) ] +photon energy (eV)0.2 +(b) +(a) +5 +0.1 +6 +0 +0.0 +-0.5 +0.0 +0.5 +-5 +0 +5 +3 +3 +()gjul +(c) +10 +0.5 +p/Φp +0 +0.0 +-10 +-0.5 +-0.5 +0.0 +0.5 +0 +1 +Re{②(e) ) +E6 +FIG. 4. Experimental and theoretical relative time delays between argon 3s and 3p [63, 64], compared with the time delay +retrieved from the experimental cross section [66, 67], interpolated and smoothed to fit the finite-differential energy resolution, +where the cross sections are compared in the inset panel. The filled area is the same as in Fig. 2. +(−Q+i|γ|), and its transition amplitude is denoted as ˘D(ϵ). The LHT-retrieved transition amplitude can be expressed +as +D(ϵ) = ϵ + Q + i|γ| +ϵ + Q − i|γ| +˘D(ϵ)eiφ0, +(22) +which gives rise to a Lorentzian lineshape that should be added to the LHT-retrieved time delay: +∂ arg{ ˘D(ϵ)} +∂ϵ += ∂ arg{D(ϵ)} +∂ϵ ++ +2 +|γ| +� +ϵ+Q +|γ| +�2 ++ 1 +. +(23) +Therefore, the first term of Eq. (15) for ˘D(ϵ) will become positive, and in the limit of γ → 0−, it becomes a +πδ(ϵ+Q) +peak. For example, it is known that the calculated time delay of argon 3s-CM around 42 eV by RPAE is positive while +it is negative by the time-dependent local density approximation (TDLDA), which is attributed to the correlation +effect [62]. The former has winding number of 1 while the latter has 0. A recent experimental study by Alexandridi +et al using RABBITT [63] shows negative (τ A +3s − τ A +3p) value at the argon 3s-CM, which is opposite to the theoretical +prediction by the two-photon-two-color RPAE (2P2C-RPAE) but is in good agreement with the TDLDA [64] and +the continuum-continuum (cc) delay [65]: ∆τ A ≈ ∆τ W + ∆τ cc, where τ W is the one-photon Wigner time delay, and +τ A is the experimentally measured two-photon atomic time delay. The LHT of the previously reported cross sections +[66, 67] yields very good agreement with the TDLDA and the experiment (Fig. 4). The fine cross-section structure +around 30 ∼ 37 eV corresponds to time-delay fluctuations that are unresolvable by the finite-difference method. The +negative time delay indicates that the trajectory of the argon 3s-CM, taking all contributing ionization pathways +into account, has the winding number of 0, which is the same as the argon 3p-CM around 50 eV, where all methods +reasonably agree with the experiment [16]. We note that there has been theoretical predictions that the 3p-emission +of Ar@C60 near the 3p-CM will have two branches with different topologies, depending on the symmetry of wave +function mixing [68]. +In conclusion, we have introduced a Kramers-Kronig-like relation between photoionization cross section and at- +tosecond time delays, which relies on the general properties of coherence and causality of the transition amplitude. +We have derived a unified time-delay formula of (anti)Lorentzian and Fano lineshapes with constant and coherent +background, where both Breit-Wigner and Fano resonances correspond to a pair of Lorentzian time-delay peaks with +opposite signs. By analysing the cross-section ratio and relative time delays between Xe@C60 and free Xe, we have +established that this relation holds for more complicated resonances. The LHT further requires the topological struc- +ture that the trajectory has winding number of 0, which is particularly relevant for antiresonances where the minimal +cross section approaches zero. The experimental result of argon 3s- and 3p-Cooper minima suggests that they both +have winding number of 0 and that the time delay can be directly retrieved by the LHT, which quantitatively agrees +with the RABBITT measurement and the TDLDA calculation. We note that our method is not restricted to atomic +one-photon processes, since similar resonances have been found in molecular systems [12, 14, 15, 69] and in two-photon +transitions [13, 55, 70]. Our result bridges two kinds of experiments in atomic and molecular physics and broadens +the understanding of attosecond processes. + +400 +LHT (smooth) + cc +TDLDA + CC +300 +2P2C-RPAE +CC +200 +exp. Saclay +exp. Lund +100 +(as) +A3 +A3 +T +-100 + cross section (arb. unit) +smooth +TDLDA +-200 +exp. +100 +-300 +Ar 3s ( +30 +35 +40 +45 +50 +55 +photon energy (eV) +-400 +35 +40 +45 +50 +55 +60 +65 +70 +photon energy (eV)7 +J.-B.Ji acknowledges the funding from the ETH grant 41-20-2. M.H.’s work was funded by the European Union’s +Horizon 2020 research and innovation programme under Marie Sk�lodowska-Curie agreement grant No. 801459, FP- +RESOMUS. J.-B.Ji thanks Prof. Dr. J.O. Richardson (ETH Z¨urich) for discussions. +Appendix +From Eq (1) we have +|D(E)| = +� +Ec +4π2 σ(E). +(24) +Approximating E as constant near the resonant region, we have Eq. (5). +For the Lorentzian peak, +σ ∝ +1 +ϵ2 + 1 + r = rϵ2 + r + 1 +ϵ2 + 1 +. +(25) +Compared with Eq. (13), we have Q = 0 and γ2 = r+1 +r . For the Fano peak, +σ ∝ (ϵ + q)2 +ϵ2 + 1 + r = (r + 1)ϵ2 + 2qϵ + q2 + r +ϵ2 + 1 +. +(26) +Compared with Eq. (13), we have 2Q = +2q +r+1 and Q2 + γ2 = q2+r +r+1 , which leads to the expressions in the main text. +From Eq. (13) we have +1 +σ +∂σ +∂ϵ = 2(ϵ + Q)(ϵ2 + 1) − [(ϵ + Q)2 + γ2] · 2ϵ +[(ϵ + Q)2 + γ2](ϵ2 + 1) += +2 (ϵ + Q) +(ϵ + Q)2 + γ2 − +2ϵ +ϵ2 + 1, +(27) +which yields Eq. (14). +For D(ϵ) in Eq. (16) with S = 1, we have +Re{D(ϵ)}= ϵ2 + Qϵ + γ +ϵ2 + 1 +, +Im{D(ϵ)}= (γ − 1)ϵ − Q +ϵ2 + 1 +. +(28) +It can be verified that +� +Re{D(ϵ)} − γ + 1 +2 +�2 ++ +� +Im{D(ϵ)} + Q +2 +�2 += 1 +4 +� +(γ − 1)2 + Q2� +. +(29) +The extrema of arg{D(ϵ)} are reached when the time delay crosses zero. For the Lorentzian peak with Q = 0, the +zeros of Eq. (15) are at ϵ = ±√γ, so the corresponding phases are +φ(√γ)= tan−1(√γ) − tan−1( 1 +√γ ) + φ0, +φ(−√γ)= tan−1(−√γ) − tan−1(− 1 +√γ ) + φ0, +(30) +and the phase difference is expressed as Eq. (18), which is positive when γ > 1 (maximum in cross section) and is +negative when γ < 1 (minimum in cross section). +For a CM at Em, we can write the cross section near the CM: +σ(E) =a + b(E − Em)2 + c(E − Em)3 ++O +� +(E − Em)4� +. +(31) + +8 +Expanding Eq. (12) at its minimum, we have +σ(E) = σb + σa(q2 + 1) +� +ξ2 + 2qξ3 + O(ξ4) +� +, +(32) +where ξ = +ϵ+q +q2+1, and σb and (σa + σb) are the cross sections at the CM and away from the CM region, respectively. +If the cross sections are uneven at the both ends of the region, we divide the cross section by a slow-varying function +whose phase is known from an analytical model or approximated as constant, so that the boundary condition for the +LHT is fulfilled. Hence we have q = +c +2b, ϵ = +� +b +σa (q2 + 1)(E −Em)−q, Γ = 2/ +� +b +σa (q2 + 1), and Er = Em + qΓ +2 . Using +S = σa + σb, r = σb +σa , Q = +q +r+1, and γ = +� +r(r+q2+1) +(r+1)2 +, Eqs. (13) and (16) are obtained. 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Mart´ın, Physical Review Letters 113, 263001 (2014). + diff --git a/odE2T4oBgHgl3EQfKAYw/content/tmp_files/load_file.txt b/odE2T4oBgHgl3EQfKAYw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6bad73b3bbb309e0122c5c77265a4ea632c92ea1 --- /dev/null +++ b/odE2T4oBgHgl3EQfKAYw/content/tmp_files/load_file.txt @@ -0,0 +1,674 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf,len=673 +page_content='Relation Between Photoionization Cross Sections and Attosecond Time Delays Jia-Bao Ji,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' ∗ Kiyoshi Ueda,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' † Meng Han,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='1 and Hans Jakob W¨orner1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' ‡ 1Laboratorium f¨ur Physikalische Chemie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' ETH Z¨urich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' 8093 Z¨urich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Switzerland 2Department of Chemistry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Tohoku University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Sendai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' 980-8578,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Japan (Dated: January 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' 2023) By analyzing the reported theoretical and experimental results on photoionization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' we show that for an (anti)resonance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' there is a Kramers-Kronig-like relation between the cross section and the time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' We derive the analytical time-delay formulae for the (anti)resonance whose cross section is described by the Lorentz or Fano ansatz with consideration of the interference of the resonant and non-resonant channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' We further demonstrate that the relation is also valid for an energetically- confined cross-section feature with multiple maxima and minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' The validity, however, depends on the topology of the transition amplitude on the complex plane, which causes the discordance of various theoretical approaches on the Ar 3s Cooper minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Our work bridges two kinds of measurements and highlights the roll of the analyticity of the ionization amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Photoionization is one of the fundamental processes that reveal the electronic behavior in atoms and molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' For a photoelectron emitted with kinetic energy Ek, one can write the angle-differential dipole transition amplitude as D(E, ˆk), where E is the photon energy, and ˆk is the emission angle, and the angle-integrated amplitude D(E), then the photoionization cross section can be expressed as σ(E) = 4π2 Ec |D(E)|2 ∝ Im{d(E)}, (1) where c is the speed of light, and d(E) is the dipole response function in the frequency domain regarding the atom (molecule) as the medium of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' The atomic units are used through this Letter unless otherwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' The differential photoionization cross section at emission angle ˆk can be described as σ(E, ˆk) = π Ec|D(E, ˆk)| 2 = σ(E) 4π {1 + β(E)P2[cos(ˆk · ˆe)]} with ˆe being the light polarization vector, and P2 is the second Legendre polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Conventional photoionization studies have been focusing on the measurements of cross sections σ(E) and asymmetry parameters β(E) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' More recently, the time delays of photoionization of atoms and molecules have been measured by techniques such as the reconstruction of attosecond beating by interference of two-photon transitions (RABBITT) technique [2–4] and the attosecond streaking [5, 6], with the Wigner time delay [7, 8] defined as τ(E) = ∂arg{D(E)} ∂E = ∂Im{ln[D(E)]} ∂E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (2) These time delays are typically at the order of attosecond (as, 1 as = 10−18 s), which corresponds to the natural time scale of electronic motion in an atom or a molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Both theoretical and experimental studies indicate that a structured continuum, or an (anti)resonance, such as a Fano resonance [9–11], a shape resonance [12–15], a Cooper minimum [16], and a two-center interference [17], usually leads to both the modulation of cross section and a significant time delay of ±102 as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' In this Letter, we present the quantitative relation between the cross section and the time delay in such an (anti)resonance, which relates the two aspects of photoionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Photoionization can be regarded as a half-scattering process [18], and its transition amplitude can be presented by the S-matrix or the related R-matrix [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' If the S-matrix is diagonalized according to the angular momentum l as S(k) = � l,m ��l, m � e2iφl� l, m ��, (3) where 2φl is the l-th phase shift, then the photoionization amplitude corresponds to S(k) − 1= � l,m ��l, m � (e2iφl − 1) � l, m �� = 2i � l,m ��l, m � eiφl sin φl � l, m ��, (4) where the eigenvalues no longer have moduli of 1, and the phases are halved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Equation (2) yields τl(E) = ∂φl/∂E, which is the original time delay proposed by Wigner [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' The experimental time delay, on the other hand, is usually expressed in the ˆk-space with interference of different l’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Nonetheless, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (2) can be interpreted as the “generalized arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='03697v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='atom-ph] 9 Jan 2023 2 Wigner delay”, which deals with the off-diagonal terms following Eisenbud’s formula [20–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' The analyticity of the S- matrix based on causality has been intensely studied [23–30] and the scattering amplitude was shown to be analytical on the upper-half complex k-plane for the physical region of the reaction, where k is the incident momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' This is the foundation of the complex-scaling method for computation [31–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' On the other hand, the analyticity can be also expressed regarding energy, and by applying the energy-time uncertainty principle, a time-domain picture of the scattering processes arises [34–36], where the connection of this “microscopic” time and the attosecond time delay is discussed in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' The analycity leads to the Kramers-Kronig (KK) relations [37–39] that allow one to construct an energy-dependent complex function f(E) using only its real or imaginary part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' For example, the energy-domain response function d(E) can be obtained from its imaginary part using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (1), and hence the time-domain response function ˜d(t) via Fourier transform [40–42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' In this Letter we show that using the analyticity of the scattering amplitude, Im{ln[D(E)]} is similarly connected to its real part, and the Wigner time delay can be retrieved via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' The KK relations require that the absorption variation vanishes beyond a certain frequency region, which is true for an (anti)resonance where the cross section below and above the resonance region (in the vicinity of the resonant energy Er) can be approximated as constant, and the variation of E in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (1) is negligible compared to the drastic modulation of D(E), so Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (1) can be written as (see Appendix for details) Re{ln[D(E)]} ≈ 1 2 ln [σ(E)] + const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (5) Therefore, the trajectory of (Re{D(E)}, Im{D(E)}) from E ≪ Er to E ≫ Er is approximately a closed contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Using the KK relations of the amplitude and the phase of a complex function, it has been shown [43–47] that the phase can be retrieved (with freedom of a common phase shift) from the absolute value when ln[D(E)] vanishes faster than 1 E and D(E) fulfills the minimum-phase condition, namely, that all the poles and zeros are located on the same half-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' In the interaction picture ˆH(∆t) = ˆH0 + ˆH1(∆t), where ˆH0 is the field-free Hamiltonian, and ˆH1(∆t) is the interaction term as a function of retardation, the interaction is turned on at ∆t = 0, which triggers a transition at ∆t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' This retardation is the “microscopic” time introduced by Branson, and we have ignored the relativistic effect of the virtual-particle “cloud” [34], whose temporal feature is much shorter than the electronic response in atoms (molecules), as analogously shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Thus, in the energy (frequency) domain, we have D(E) = � +∞ 0 ˜D(∆t)eiE∆td∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (6) This ensures D(E) has no poles on the upper-half of the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' The minimum-phase condition thus requires no zeros on the upper half-plane, which means that the transition amplitude is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' It also implies that the origin is not enclosed in the trajectory (the winding number is 0), and the total phase change throughout the (anti)resonant region is 0 instead of ±2π [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' We shall come back to this point in the discussion of the Cooper minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (6), we can define the complex “average retardation”, with the same expression derived by Pollak and Miller [48], as T(E) = � +∞ 0 ∆t ˜D(∆t)eiE∆td∆t � +∞ 0 ˜D(∆t)eiE∆td∆t = −i∂ ln[D(E)] ∂E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (7) The Wigner time delay τ(E) is connected to T(E) and σ(E) using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (2) and (5) and the KK relations by τ(E)= Re{T(E)} = −H{Im{T(E)}} = 1 2 ∂ ∂E H{ln[σ(E)]} = 1 2H � 1 σ(E) ∂σ ∂E � , (8) where H is the Hilbert transform: H{f(E)} = 1 πE ∗ f(E) = 1 π P � +∞ −∞ f(E′) E − E′ dE′, (9) which is linear and commutative to the derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Here ∗ represents the convolution, and P refers to the Cauchy principal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' The boundary condition at infinity hence becomes lim (E−Er)→∞ |E − Er| σ(E) ∂σ ∂E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (10) 3 The cross section of an (anti)resonance can be expressed by the Lorentzian lineshape, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=', the Breit-Wigner resonance formula[49, 50]: σL(ϵ) = σa 1 ϵ2 + 1 + σb (11) or the Fano lineshape [51, 52]: σF(ϵ) = σa (ϵ + q)2 ϵ2 + 1 + σb, (12) where ϵ = (E − Er)/( Γ 2 ) is the relative energy, and σa and σb are the coefficients of resonant and non-resonant pathways, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' We assume that they contribute to the overall transition amplitude coherently, which fits in the Feshbach picture [53] and was recently demonstrated in the x-ray regime [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' The Lorentzian lineshape is symmetric, while the Fano lineshape is asymmetric, with q being the asymmetric parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Γ describes the peak width, and from Fano’s picture, the resonant state has a lifetime of 1/Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Let r = σb/σa, and the cross section for both Lorentzian and Fano linehsapes be expressed as σ(ϵ) = S(ϵ + Q)2 + γ2 ϵ2 + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (13) For the Lorentzian lineshape, Q = 0, γ2 = 1 + 1 r, and S = σb, while for the Fano lineshape, Q = q r+1, γ2 = r(r+q2+1) (r+1)2 , and S = σa + σb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' The detailed proof can be found in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Using ∂ ∂E = 2 Γ ∂ ∂ϵ, we have 1 σ(E) ∂σ ∂E = 2 Γ � �� 2 γ � ϵ+Q γ � � ϵ+Q γ �2 + 1 − 2ϵ ϵ2 + 1 � �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (14) It is easy to verify that the boundary condition Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (10) is fulfilled, and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (8) yields τ(E) = 2 Γ � ��− 1 |γ| � ϵ+Q |γ| �2 + 1 + 1 ϵ2 + 1 � �� , (15) where we used the property of the Hilbert transform that H{f(aE)}(E) = sgn(a)[H{f(E)}(aE)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' The time delay can be decomposed as two Lorentzian lineshapes with widths of γΓ and Γ and opposite signs, centered at (Er − QΓ 2 ) and Er, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' For the Fano lineshape with r → 0, |γ| → 0 the first time-delay peak approaches −πδ(ϵ + Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' On the other hand, for the Lorentzian lineshape with r → 0, |γ| → ∞ the first time-delay peak vanishes, so the time delay is also a Lorentzian lineshape with τ(Er) = 2/Γ, as discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Since (Q + iγ) can be regarded as the “complex asymmetric parameter” [54], a more insightful approach is to investigate the function [9, 55] D(ϵ) = √ Sϵ + Q + iγ ϵ + i = √ S � 1 + Q + i(γ − 1) ϵ + i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (16) It fulfills |D(ϵ)|2 = σ(ϵ) and it is minimum-phase when γ > 0, so the phase of the transition amplitude is the same as the phase of D(ϵ) with the freedom of a common phase shift: arg {D(ϵ)} = arg {D(ϵ)} + φ0 = cot−1 �ϵ + Q γ � − cot−1(ϵ) + φ0, (17) and thus the Wigner time delay will be the same, as expressed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' For the Lorentzian lineshape with Q = 0, the maximal phase jump is (see Appendix for details) ∆φmax = φ(√γ) − φ(−√γ) = 4 tan−1(√γ) − π, (18) which approaches +π and −π for γ → +∞ (r → 0) and γ → 0 (r → −1), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' The trajectory of D(ϵ) from ϵ → −∞ to ϵ → +∞ is a counter-clockwise rotating (due to the causality) closed circle [9, 30] with radius of 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' The transition amplitudes for the Lorentzian and Fano resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' The cross sections (solid) and the phases (dashed) (a, b), the derivative of the logarithm of cross sections (solid) and the time delays (c, d), and transition-amplitude trajectories (e, f) for the Lorentzian lineshape (a, c, e) and the Fano lineshape with q = 2 (b, d, f), where σa = 1 and σb → 0+ (blue), σb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='1 (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' The magenta dots in (e) and (f) indicate the beginning of the trajectories at ϵ → −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' √ S 2 � (γ − 1)2 + Q2 centered at �√ S γ+1 2 , − √ S Q 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' It can be easily verified that the origin is not enclosed when γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Take the Fano resonance as an example, at ϵ → ±∞, the transition amplitudes are dominated by the direct ionization channel, which is approximately constant in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' As r → 0 (“pure” Fano lineshape), the trajectory approaches tangential to the origin, where a (−π)-phase jump occurs within an infinitely small energy interval, which corresponds to the −πδ(ϵ+Q) peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' For the “pure” Lorentzian transition amplitude D(ϵ) ∝ 1 ϵ+i [30], there is a phase offset of π between ϵ → +∞ and ϵ → −∞, which does not fulfill the minimum-phase condition since it encloses “half” the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Its phase retrieved by the logarithm Hilbert transform (LHT) method starts and ends at the same value at infinities, which corresponds to shifting the circle by an infinitely small amount away from the origin, and thus subtracts an infinitely broad Lorentzian lineshape according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (15), where Q = 0 and γ → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Its behavior near ϵ = 0, which is of most interest, is largely unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Examples of Lorentzian and Fano lineshapes are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' 1, where the increase of r results in the decrease of the extreme(s) of the time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Recently, Kheifets and Catsamas [56] proposed an approach for shape-resonance analysis by utilising the relation σl(E) ∝ sin2 φl from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (4), which is applicable if only one l is dominating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Let fl(E) = tan[φl(E) + π/2] = − cot[φl(E)] be a real function, then sin2[φl(E)] = 1 fl(E)2 + 1 = ��� 1 fl(E) ± i ��� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (19) Therefore, assuming fl(E) to be analytical, H �1 2 ln[σl(E)] � = − arg{fl(E) ± i} = ∓ cot−1[fl(E)] = ±φl(E), (20) if one of the two branches (fl(E) ± i) has no poles or zeros on the upper half-plane of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' This is the condition under which Kheifets’s approach is equivalent to the method introduced in this Letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' For example, let fl(E) = E/( Γl 2 ), where Γl > 0, (fl(E) + i) fulfills the condition and leads to the “pure” Breit-Wigner resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' In real experiments, the measurable energy range is finite, so the boundary condition is not strictly fulfilled, which leads to a deviation of the retrieved time delay from its physical counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' However, when comparing two photoionization processes, the relative time delay, which is usually the experimental observable [57], can be obtained (a) 101 (b)101 7 100 0 E 10-1 0 9 6 0 10-1 2 1 10-2 10-3 10 10 0 10 10 E (d) c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='0 dln(V)/de 3p/ 2 L dln(V)/ 3p/ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='0 /Φp 0 /pp 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='0 2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='5 10 0 10 10 10 E (e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='5 (f) 0 [()g}ul [()]ul 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='0 1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='5 0 1 0 2 Re[の(e)) Re[(e))5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Applications of the LHT formalism to Xe and Xe@C60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (a) The calculated cross-section ratio (green dashed, right vertical axis) using the RPAE method and the time delay (blue, left vertical axis) [59] and the time delay retrieved from the cross-section using the LHT (red, left vertical axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' The filled area indicates the difference between first taking the energy derivative then perform the Hilbert transform and first performing the Hilbert transform then taking the energy derivative, which are the same when the boundary condition is strictly fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (b) The transition-amplitude trajectory constructed by DXe@C60/DXe(free) = exp[ 1 2 ln(r) + i 2H{ln(r)}], where r = σXe@C60/σXe(free) is the cross-section ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Comparison of the trajectories with different topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Cross sections (a), phases (b), time delays (c), and trajectories (d, the magenta dot indicates ϵ → −∞) of the anti-Lorentzian lineshapes with S = 1 , Q = 0 and γ = 0+ (blue, dashed), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='1 (orange, solid), or −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='1 (green, dotted), according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' from their cross-section ratio, which converges faster than the individual time delays: ∆τ(E) = τ1(E) − τ2(E) = ∂ ∂E arg �D1 D2 � = 1 2 ∂ ∂E H � ln �σ1 σ2 �� = 1 2H �σ2 σ1 ∂(σ1/σ2) ∂E � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (21) Note that the amplitude-phase relation can be extended beyond “simple” resonances with Lorentzian or Fano line- shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' For example, the cross sections and time delays of free xenon and xenon in a C60 cage in the region of the giant resonance have been computed using the random-phase approximation (RPA), relativistic RPA (RRPA) [58], RPA with exchange (RPAE), and time-dependent Schr¨odinger equation (TDSE) methods [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' 2, the cross-section ratio shows multiple peaks in the giant-resonance region, which reflects the influence of the C60 shell, and the time delay retrieved by the LHT, which does not require the phase of the transition amplitude, almost quantitatively agrees with the reported value obtained from the phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Positive (negative) time delays are typically associated with local maxima (minima) of the cross-section ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' The reconstructed trajectory of DXe@C60/DXe(free) has a multiple-spiral structure, which manifests the validity of the LHT method for complex resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Now we discuss a specific type of antiresonance, the Cooper minimum (CM), which has a distinctly different origin compared to Fano or shape resonance [16, 60, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Nevertheless, the cross section near the minimum is approximately parabolic, which can be locally fitted by a negative-Lorentzian curve, which corresponds to r = σb/σa → −1, and therefore Q = 0, γ → 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' We note that using Q ̸= 0 one can describe the asymmetry of the CM (see Appendix for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Although ±γ give the same cross section, they have different topological structures on the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Only the transition amplitude corresponding to the positive γ is minimum-phase, which leads to a negative time delay, while the negative γ gives a positive time delay, as plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' The latter has a zero at (a) (b) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='5 1 150 - () 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='5 (as) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='0 energy ( 100 TXe(free) 120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='0 50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='5 Xe@C60 / 100 photon 8 0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' TXe@C 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='5- Im 80 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='5 75 100 125 150 Re[xe@Co /xe(free) ] photon energy (eV)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='2 (b) (a) 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='1 6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='5 5 0 5 3 3 ()gjul (c) 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='5 p/Φp 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='0 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='5 0 1 Re{②(e) ) E6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Experimental and theoretical relative time delays between argon 3s and 3p [63, 64], compared with the time delay retrieved from the experimental cross section [66, 67], interpolated and smoothed to fit the finite-differential energy resolution, where the cross sections are compared in the inset panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' The filled area is the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (−Q+i|γ|), and its transition amplitude is denoted as ˘D(ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' The LHT-retrieved transition amplitude can be expressed as D(ϵ) = ϵ + Q + i|γ| ϵ + Q − i|γ| ˘D(ϵ)eiφ0, (22) which gives rise to a Lorentzian lineshape that should be added to the LHT-retrieved time delay: ∂ arg{ ˘D(ϵ)} ∂ϵ = ∂ arg{D(ϵ)} ∂ϵ + 2 |γ| � ϵ+Q |γ| �2 + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (23) Therefore, the first term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (15) for ˘D(ϵ) will become positive, and in the limit of γ → 0−, it becomes a +πδ(ϵ+Q) peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' For example, it is known that the calculated time delay of argon 3s-CM around 42 eV by RPAE is positive while it is negative by the time-dependent local density approximation (TDLDA), which is attributed to the correlation effect [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' The former has winding number of 1 while the latter has 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' A recent experimental study by Alexandridi et al using RABBITT [63] shows negative (τ A 3s − τ A 3p) value at the argon 3s-CM, which is opposite to the theoretical prediction by the two-photon-two-color RPAE (2P2C-RPAE) but is in good agreement with the TDLDA [64] and the continuum-continuum (cc) delay [65]: ∆τ A ≈ ∆τ W + ∆τ cc, where τ W is the one-photon Wigner time delay, and τ A is the experimentally measured two-photon atomic time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' The LHT of the previously reported cross sections [66, 67] yields very good agreement with the TDLDA and the experiment (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' The fine cross-section structure around 30 ∼ 37 eV corresponds to time-delay fluctuations that are unresolvable by the finite-difference method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' The negative time delay indicates that the trajectory of the argon 3s-CM, taking all contributing ionization pathways into account, has the winding number of 0, which is the same as the argon 3p-CM around 50 eV, where all methods reasonably agree with the experiment [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' We note that there has been theoretical predictions that the 3p-emission of Ar@C60 near the 3p-CM will have two branches with different topologies, depending on the symmetry of wave function mixing [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' In conclusion, we have introduced a Kramers-Kronig-like relation between photoionization cross section and at- tosecond time delays, which relies on the general properties of coherence and causality of the transition amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' We have derived a unified time-delay formula of (anti)Lorentzian and Fano lineshapes with constant and coherent background, where both Breit-Wigner and Fano resonances correspond to a pair of Lorentzian time-delay peaks with opposite signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' By analysing the cross-section ratio and relative time delays between Xe@C60 and free Xe, we have established that this relation holds for more complicated resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' The LHT further requires the topological struc- ture that the trajectory has winding number of 0, which is particularly relevant for antiresonances where the minimal cross section approaches zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' The experimental result of argon 3s- and 3p-Cooper minima suggests that they both have winding number of 0 and that the time delay can be directly retrieved by the LHT, which quantitatively agrees with the RABBITT measurement and the TDLDA calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' We note that our method is not restricted to atomic one-photon processes, since similar resonances have been found in molecular systems [12, 14, 15, 69] and in two-photon transitions [13, 55, 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Our result bridges two kinds of experiments in atomic and molecular physics and broadens the understanding of attosecond processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' 400 LHT (smooth) + cc TDLDA + CC 300 2P2C-RPAE CC 200 exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Saclay exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Lund 100 (as) A3 A3 T 100 cross section (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' unit) smooth TDLDA 200 exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' 100 300 Ar 3s ( 30 35 40 45 50 55 photon energy (eV) 400 35 40 45 50 55 60 65 70 photon energy (eV)7 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='Ji acknowledges the funding from the ETH grant 41-20-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='H.’s work was funded by the European Union’s Horizon 2020 research and innovation programme under Marie Sk�lodowska-Curie agreement grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' 801459, FP- RESOMUS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='Ji thanks Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Richardson (ETH Z¨urich) for discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Appendix From Eq (1) we have |D(E)| = � Ec 4π2 σ(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (24) Approximating E as constant near the resonant region, we have Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' For the Lorentzian peak, σ ∝ 1 ϵ2 + 1 + r = rϵ2 + r + 1 ϵ2 + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (25) Compared with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (13), we have Q = 0 and γ2 = r+1 r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' For the Fano peak, σ ∝ (ϵ + q)2 ϵ2 + 1 + r = (r + 1)ϵ2 + 2qϵ + q2 + r ϵ2 + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (26) Compared with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (13), we have 2Q = 2q r+1 and Q2 + γ2 = q2+r r+1 , which leads to the expressions in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (13) we have 1 σ ∂σ ∂ϵ = 2(ϵ + Q)(ϵ2 + 1) − [(ϵ + Q)2 + γ2] · 2ϵ [(ϵ + Q)2 + γ2](ϵ2 + 1) = 2 (ϵ + Q) (ϵ + Q)2 + γ2 − 2ϵ ϵ2 + 1, (27) which yields Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' For D(ϵ) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (16) with S = 1, we have Re{D(ϵ)}= ϵ2 + Qϵ + γ ϵ2 + 1 , Im{D(ϵ)}= (γ − 1)ϵ − Q ϵ2 + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (28) It can be verified that � Re{D(ϵ)} − γ + 1 2 �2 + � Im{D(ϵ)} + Q 2 �2 = 1 4 � (γ − 1)2 + Q2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (29) The extrema of arg{D(ϵ)} are reached when the time delay crosses zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' For the Lorentzian peak with Q = 0, the zeros of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (15) are at ϵ = ±√γ, so the corresponding phases are φ(√γ)= tan−1(√γ) − tan−1( 1 √γ ) + φ0, φ(−√γ)= tan−1(−√γ) − tan−1(− 1 √γ ) + φ0, (30) and the phase difference is expressed as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (18), which is positive when γ > 1 (maximum in cross section) and is negative when γ < 1 (minimum in cross section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' For a CM at Em, we can write the cross section near the CM: σ(E) =a + b(E − Em)2 + c(E − Em)3 +O � (E − Em)4� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (31) 8 Expanding Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (12) at its minimum, we have σ(E) = σb + σa(q2 + 1) � ξ2 + 2qξ3 + O(ξ4) � , (32) where ξ = ϵ+q q2+1, and σb and (σa + σb) are the cross sections at the CM and away from the CM region, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' If the cross sections are uneven at the both ends of the region, we divide the cross section by a slow-varying function whose phase is known from an analytical model or approximated as constant, so that the boundary condition for the LHT is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Hence we have q = c 2b, ϵ = � b σa (q2 + 1)(E −Em)−q, Γ = 2/ � b σa (q2 + 1), and Er = Em + qΓ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' Using S = σa + σb, r = σb σa , Q = q r+1, and γ = � r(r+q2+1) (r+1)2 , Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (13) and (16) are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' The time delay near the CM for the zero-winding-number trajectory is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (15) and is for the one-winding-trajectory modified according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content=' ∗ jiabao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='ji@phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} +page_content='ethz.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE2T4oBgHgl3EQfKAYw/content/2301.03697v1.pdf'} diff --git a/odE5T4oBgHgl3EQfkA-1/content/2301.05660v1.pdf b/odE5T4oBgHgl3EQfkA-1/content/2301.05660v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..d4064af340379531ad4f9777f3b41aab5c3e3fac --- /dev/null +++ b/odE5T4oBgHgl3EQfkA-1/content/2301.05660v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7d8b0936eaa2250b35265d43a37bc17c3ed50d3ff0303afd1271c14dbac97a66 +size 391397 diff --git a/odE5T4oBgHgl3EQfkA-1/vector_store/index.faiss b/odE5T4oBgHgl3EQfkA-1/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..40f722128abe041bad17f858f4a29568884c3aca --- /dev/null +++ b/odE5T4oBgHgl3EQfkA-1/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:edb12efa23de5e5e3374d1ace1c4c7cd57c3f45a09e92b3105a5712b53d3e39c +size 2949165 diff --git 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+ematics and Physics. The sets of configurations of the 6-vertex model +can be described as the sets of paths in multigraphs. In this article the +transition probability of the simple random walk on the multigraphs is +computed. The unexpected point of the results is the use of continuous +fractions to compute the transition probability. +Keywords: Random walk, Markov Chain AMS classification (2000): 05C81, +60F05. +1 +Introduction +In this article we are interested in a simple random walk Y on particular +(multi)graphs GK indexed by an integer K. The set of vertices of GK is +VK +def += {−1, 1}K and the set EK of edges is defined so that the set of paths +of length n is isomorphic with the set of configurations of the so-called 6- +vertex model, on a rectangle K×n in the particular case of the ice-model [6]. +Please note that the uniform distribution is usually considered in the 6- +vertex model, with various admissible boundary conditions on the rectangle +when n, K → ∞ of a rectangular lattice. In this paper we endow the sets +of configurations with the distribution of Y0, . . . , Yn which is actually easier +to study that the uniform distribution on the set of paths of length n. In +the 6-vertex model the height function h which is a map from the rectangles +K × n to the set of integers Z has physical meaning, and more precisely +∗Institut de Math´ematiques de Toulouse; UMR 5219, Universit´e de Toulouse; CNRS, +UT3 F-31062 Toulouse Cedex 9, France. First-Name.Name@math.univ-toulouse.fr +1 + +the decay of the variance of the difference of height function between two +distant points on the rectangle is of special importance. This question is +also related to random graph homomorphisms [1]. See [3] where the decay +is shown to be of logarithmic order for periodic boundary conditions on +the rectangle. For the Markov chain Y the height fonction is related to an +additive functional Z of the Y, which collects a particular vector field defined +on the edges of GK. In a previous paper [2] it is shown that the variance +of the height function decays like +2 +(2+K)n. Still for finite K the variance is +computed in [4] for periodic boundary condition in the variable between 1 +and K, and in [5] for other constraints. +The easiest way to describe the simple random walk on GK in the sta- +tionary regime is to state that the distribution of the pair (YK(0), YK(1)) +is the uniform distribution on the edges EK. One aim of this article is to +provide a formula for the transition probability of the simple random walk +that starts from a given vertex. +Surprisingly enough the formula in the +Theorem 3.1 uses the continuous fraction associated to the length of the +constancy blocks of the digits of the given vertex. The authors see two use- +ful consequences of this result. The first one is the fact that the transition +probability of the simple random walk will considerably make easier and +faster simulations of the walk. See [7] for a survey of previous simulation +methods, which are both memories and computationally intensive due the +difficulty to describe easily the neighbors of a given vertex in GK for large +K. Another interesting consequence of Theorem 3.1 is that it can be ex- +tended to the case K = ∞. This in turns gives a sense to the decay of the +variance of the height function in G∞. The definitions and the models are +given in section 2. In section 3 the main theorems are written and they are +proved in the following sections. +2 +The model +Let +� +Z(1) +t +, . . . , Z(K+1) +t +� +t∈N ∈ ZK+1 denote the heights of K + 1 simple ran- +dom walks on Z, conditioned on satisfying +∀t ∈ N, ∀i ∈ [1, K] , +���Z(i+1) +t +− Z(i) +t +��� = 1. +(1) +More precisely, the random walk is a Markov chain on the state space of +K-step walks in Z +SK = {(z(1), . . . , z(K+1)) ∈ ZK+1, +∀i ∈ [1, K] , |z(i+1) − z(i)| = 1} +(2) +2 + +where the next step from z0 ∈ SK is selected uniformly among the z1s that +belong to SK such that ∀i ∈ [1, K + 1], z(i) +1 − z(i) +0 +∈ {−1, 1}. In other words, +we consider K + 1 simple random walks on the lattice Z coupled under a +shape condition. One can associate to a path of length n (Z(i) +t )0≤t≤n−1 a +height function h on a rectangle ((t, i))0≤t≤n−1, 1≤i≤K+1 by h(t, i) = Z(i) +t , +which makes the link with the height function of the 6-vertex model. +In [2] an equivalent depiction is provided as a simple random walk on a +(multi)-graph. +Definition 2.1. Let VK +def += {−1, 1}K, E+ +K, E− +K ∈ VK×VK will be respectively +set of postive, negative edges on VK. +A pair (a, b) ∈ VK × VK such that a ̸= b belongs to E+ +K if non vanishing +coordinates of the vector (b − a) ∈ {−2, 0, 2}K have alternate signs with the +first sign negative. For every vertex a ∈ VK there is an edge from a to a +E+ +K denoted by (a, a)+. +In a similar manner there is a pair (a, b) ∈ VK × VK such that a ̸= b +belongs to E− +K if non vanishing coordinates of the vector (b−a) ∈ {−2, 0, 2}K +have alternate signs with the first sign positive. For every vertex a ∈ VK +theres is an edge from a to a E− +K denoted by (a, a)−. +For every K ∈ N the (multi)graph GK +def += (VK, EK), where EK +def += +E+ +K ∪ E− +K. +The study of (Z(1) +t +, ..., Z(K+1) +t +) can be split in the study of the first +coordinate Z(1) +t +and of the increments +YK(t) +def += +� +Z(2) +t +− Z(1) +t +, . . . , Z(K+1) +t +− Z(K) +t +� +, +which is always an element of VK (because of (1)). +Let us remark that the uniform distribution on the set of edges EK is +the same as the distribution of (YK(0), YK(1)) if the Markov process YK +is stationary. We have another useful characterization of this distribution +given by the following Lemma. +Lemma 2.1. Let ǫ be a random variable such that P(ǫ = −1) = P(ǫ = +1) = 1 +2), let (αk)k≥1 be an i.i.d. sequence of Bernoulli random variables +with parameter 1 +3 and let (βk)k≥1 be an i.i.d. sequence of random variables +such that P(βk = −1) = P(βk = 1) = 1 +2. The previous random variables are +mutually independent. Let us define ∀k ≥ 2 : +γk +def += ǫ(−1) +�k−1 +i=1 αi, +3 + +with the convention that γ1 = ǫ. +Let us also define k ≥ 1 : +Ak += +(1 − αk)βk + αkγk +(3) +Bk += +(1 − αk)βk − αkγk. +(4) +The distribution of the pair ((Ak)1≤k≤K, (Bk)1≤k≤K) is the uniform dis- +tribution on EK. If we denote by (A, B)K the edge defined by : +(A, B)K +def += +� ((Ak)1≤k≤K, (Bk)1≤k≤K) +if (Ak)1≤k≤K ̸= (Bk)1≤k≤K +((Ak)1≤k≤K, (Bk)1≤k≤K)ǫ +if (Ak)1≤k≤K = (Bk)1≤k≤K +then (A, B)K +L= (YK(0), YK(1)) if YK(0) ̸= YK(1) and +(A, A)ǫ +K +L= (YK(0), YK(0))Z(1) +1 +−Z(1) +0 +if +YK(0) = YK(1). +Moreover (Ak)1≤k≤K +L= YK(0) and (Bk)1≤k≤K +L= YK(0). +Proof. The proof is by induction and can be found in [7]. Let us first prove +that (A, B)K ∈ EK. We remark that Bk −Ak = −2αkγk, then it is vanishing +if αk = 0, and the alternating rule sign is fulfilled because every time αk = 1, +γk has a different sign from γk−1. Let us denote by DK the cardinal of EK. +A simple computation yields D1 = 6 and by induction DK = 2 × 3K. It +is also obvious to check P((A, B)1 = e) = 1 +6 for every edge in E1. Let us +assume that P((A, B)K = e) = +1 +DK is true for every edge in EK Let us +consider u′, v′ ∈ VK and denote by u′ ± 1 the vertex in VK+1 obtained by +concatenating ±1 on the right of u′. Then for +P((A, B)K+1 = (u′1, v′1)) = P((A, B)K = (u′, v′) ∩ αK+1 = 0 ∩ βK+1 = 1) += +1 +DK +2 +3 +1 +2 += +1 +DK+1 +. +The same holds for (u′ −1, v′ −1). If the concatenated digit to u′ is different +from the one concatenated to v′ and u′ ̸= v′ there is only one possible choice +which leads to a non vanishing probability depending on the last digit that +differs between u′ and v′. Then for this choice +P((A, B)K+1 = (u′ ± 1, v′ ∓ 1)) = P((A, B)K = (u′, v′) ∩ αK+1 = 1) += +1 +DK +1 +3 += +1 +DK+1 +. +4 + +The proof is complete when we consider the case u′ = v′ and in this case we +get also P((A, B)K+1 = (u′ ± 1, v′ ∓ 1)) = +1 +DK+1 thanks to the distribution +of ǫ. +Remark 2.1. One important consequence of the previous result is the fact +that the graph GK is defined for K = ∞. Let us be more precise. When K = +∞, V∞ +def += {−1, 1}N, and E+ +∞, E− +∞ are defined with the same alternating +rules as for finite K. The Lemma 2.1 is still true when K infinite. +3 +Results +The aim of this article is to compute the transition probability of the sta- +tionary Markov chain associated with the simple random walks on the +(multi)graphs GK for K ∈ N ∪ ∞. Hence we will compute the conditional +probability that (Bk)k≤K takes a particular value in VK once the sequence +(Ak)k≤K is given. +Please note that when K = ∞, the law of large number implies for both +sequences (Ak)k∈N, (Bk)k∈N that there are not constant for k big enough +almost surely. +Let us assume that the sequence (ak)k∈N ∈ {−1, 1}N starts with a1 = 1. +Let us fix the consecutive times where a is constant and denote by im = +(−1)m+1. (If a1 = −1, then im = (−1)m). By convention we set S0 = 0, +and for m ≥ 1, we assume that the m-th block of constancy of a starts with +Sm−1 + 1 and stops with Sm. +Let us denote for m ≥ 1, the event +A(m) = {a ∈ {−1, 1}N such that aSm−1+1 = im, . . . , aSm = im} +(5) +of sequences which are equal to a on the m-th block of constancy of a. +Remark 3.1. In the following we are conditioning the distribution of B with +respect of events of the form {A = a}, where a is a deterministic sequence. +Once a is given, so is the sequence S and the conditioning with respect of +A(m) actually means with respect of the event {ASm−1+1 = im, . . . , ASn = +im}. We will use the abuse of notation P(.|a), P(.|A(m)) in the sequel. +Hence {a} = {(ak)k≤K} = ∩N +m=1A(m) where N is the number of blocks +of constancy of a. By definition of Mm the length of the m-th block of +constancy of a is equal to +Mm = Sm − Sm−1. +(6) +5 + +Let for 1 ≤ m ≤ n ≤ N +xn +m +def += +1 +Mm + +1 +Mm+1+ +1 +...+ +1 +Mn+1 += [Mm, Mm+1, . . . , Mn, 1], +(7) +if n < m we set xn +m +def += 1 by convention. +When K = ∞ the continuous fraction in (7) is converging when n → ∞ +toward an irrational number because of Remark 2.1 that will be denoted by +x∞ +m. +Please remark that if YK(0) = a and YK(1) = b, at most one digit bk +of b is different of ak in any block of constancy A(m) of a, because of the +alternating sign rule. Let (ǫm)1≤m≤N be Bernoulli random variables such +that ǫm = 1 if and only if there is one change of digits between a and +b in the m-th block of constancy A(m). Let (Em)1≤k≤N be a sequence of +independent random variables uniformly distributed on {1, . . . , Mm} which +encode the digit that is changed in A(m). One further constraint due to +the alternating sign rule is that when ǫm = 1, ǫm+2k = 0 on the event that +ǫm+1 = 0, . . . , ǫm+(2k−1) = 0. In other words there cannot be change of digits +in two consecutive blocks of constancy of a that have an even difference +of indexes, since the ak are the same on those blocks. +The conditional +probability P(YK(1) = b|YK(0) = a) is then described by the following +Theorem that yields the distribution of the (ǫm)1≤m≤N. +Theorem 3.1. The distribution of (ǫm)1≤m≤N is given by : +• Initializing phase +P(ǫ1 = 1|a) = +M1 +M1 + 1 + xN +2 +(8) +where N is the number of blocks of constancy of a. +• Subsequent phase when previously there is no change +P(ǫm = 1|ǫm−1 = 0, . . . , ǫ1 = 0, a) = +Mm +Mm + 1 + xN +m+1 +(9) +where N is the number of blocks of constancy of a. +• Subsequent phase when previously there is at least one change +P(ǫm = 1|ǫm−1 = 0, . . . , ǫm−(2k−1) = 1, a) = MmxN +m, +where N is the number of blocks of constancy of a. +6 + +• Loop in GK. If ǫm = 0 for 1 ≤ m ≤ N which is equivalent to b = a. +P((YK(0), YK(1)) = (a, a)+|YK(0) = a) += P((YK(0), YK(1)) = (a, a)−|YK(0) = a) += 1 +2P(ǫm = 0, for 1 ≤ m ≤ N|a). +(10) +∀m ≥ 1 the distribution of the Em’s is uniform on {1, . . . , Mm} conditionally +to the event ǫm = 1. +Remark 3.2. Remember that the distribution P(.|a) is the uniform distri- +bution on the neighbors of a in EK, this fact is not obvious from the previous +theorem. Indeed, if a given a in EK has N blocks of constancy, it leads for +instance to +1 +degK(a) = P((ǫ1, . . . , ǫN) = (1, . . . , 1)|a) +�N +k=1 Mk +(11) += +1 +M1 + 1 + xN +2 +N +� +k=2 +xN +k . +Let us now suppose that for a given 2 ≤ k0 ≤ k0 + 1 < N, ǫk0 = 0, it implies +that ǫk0+1 = 0, and the other ǫk = 1. The equation +1 +degK(a) = P((ǫ1, . . . , ǫN) = (1, . . . , 1, 0, 0, 1, . . . , 1)|a) +�N +k=1 Mk +(12) +is still true when the 0, 0 are for this k0. In equation (11) we only have to +change the factors for k0 and k0 +1, so xN +k0 becomes xN +k0+1xN +k0 and the factor +xN +k0+1 becomes 1. Then we check +P((ǫ1, . . . , ǫN) = (1, . . . , 0, 0, . . . , 1)|a) +�N +k=1 Mk += +1 +degK(a). +Tedious computations can show that actually the probability to jump from a +to each of his neighbor is the same. +When K = ∞, the previous Theorem still holds true when we consider +that the number of constancy blocks of a is infinite and use the definition +of the continuous fraction as a limit . +Theorem 3.2. The distribution of (ǫm)1≤m is given by : +7 + +• Initializing phase +P(ǫ1 = 1|a) = +M1 +M1 + 1 + x∞ +2 +. +(13) +• Subsequent phase when previously there is no change +P(ǫm = 1|ǫm−1 = 0, . . . , ǫ1 = 0, a) = +Mm +Mm + 1 + x∞ +m+1 +. +(14) +• Subsequent phase when previously there is at least one change +P(ǫm = 1|ǫm−1 = 0, . . . , ǫm−(2k−1) = 1, a) = Mmx∞ +m. +∀m ≥ 1 the distribution of the Em’s is uniform on {1, . . . , Mm} conditionally +to the event ǫm = 1. +Remark 3.3. Please note that the probability of a loop in G∞ is vanishing. +Hence there is no loop case in the last Theorem. +4 +Proof of the result +Let us now introduce the conditional independence with respect of γk, +which is an important tool for our computations. Let us denote by σm +l +def += +σ(ak, bk, l ≤ k ≤ m), for 2 ≤ l ≤ m. By convention σm +1 +def += σ(ak, bk, 1 ≤ +k ≤ m, ǫ). +Lemma 4.1. For every 1 ≤ l ≤ m ≤ n σm +l +is independent of σn +m+1 condi- +tionally to γm+1. +The proof of this Lemma comes from the definitions of Lemma 2.1. +For every m ≤ n, let us define +un +m +def += P(γSm−1+1 = im ∩ A(m) ∩ . . . ∩ A(n)) +and +vn +m +def += P(γSm−1+1 = −ik ∩ A(m) ∩ . . . ∩ A(n)). +With the help of Lemma 4.1 we can compute un +m, vn +m, by induction starting +from un +n, vn +n, and we get the following result. +8 + +Lemma 4.2. For every m < n, +�un +m +vn +m +� += +1 +3Mm +�Mm +1 +1 +0 +� �un +m+1 +vn +m+1 +� +Proof. By conditioning and Lemma 4.1 we may write +un +m = P(γSm−1+1 = im, A(m)|γSm+1 = im) +× P(γSm+1 = −im+1, A(m+1) ∩ . . . ∩ A(n)) ++ P(γSm−1+1 = im, A(m)|γSm+1 = −im) +× P(γSm+1 = im+1, A(m+1) ∩ . . . ∩ A(n)). +This equation yields +un +m = P(γSm−1+1 = im ∩ A(m)|γSm+1 = im)vn +m+1 +(15) ++ P(γSm−1+1 = im ∩ A(m)|γSm+1 = −im)un +m+1. +Observe that +{γSm−1+1 = im ∩ A(m) ∩ γSm+1 = im} = +{αl = 0, βl = im for Sm−1 + 1 ≤ l ≤ Sm} ∩ {γSm−1+1 = im}. +(16) +The two events on the right hand side are independent and the probability +of the first one is +1 +3Mm . Therefore +P(γSm−1+1 = im ∩ A(m)|γSm+1 = im) = +P(γSm−1+1 = im ∩ A(m) ∩ γSm+1 = im) +P(γSm+1 = im) +. +Hence +P(γSm−1+1 = im ∩ A(m)|γSm+1 = im) = +1 +3Mm . +(17) +With a similar argument we get +P(γSm−1+1 = im ∩ A(m) ∩ γSm+1 = −im) = Mm +3Mm . +(18) +Moreover +vn +m = P(γSm−1+1 = −im ∩ A(m)|γSm+1 = −im) +× P(γSm+1 = im+1 ∩ A(m+1) ∩ . . . ∩ A(n)) ++ P(γSm−1+1 = −im ∩ A(m)|γSm+1 = im) +× P(γSm+1 = −im+1 ∩ A(m+1) ∩ . . . ∩ A(n)). +9 + +Note that P(γSm−1+1 = −im ∩ A(m) ∩ γSm+1 = im) = 0, since, on this event, +the value −im of γSm−1+1 does not fit the value of al’s on A(m). Hence all +the αl = 0 and γSm+1 = γSm−1+1. It follows that +vn +m = P(γSm−1+1 = −im ∩ A(m)|γSm+1 = −im)un +m+1. +(19) +The Lemma 4.2 is the consequence of equations (15), (19), (17), (18). +Let us recall (7) xn +m +def += +vn +m +unm . Lemma 4.2 yields for m + 1 ≤ n, +xn +m = +1 +Mm + xn +m+1 +. +(20) +Moreover +un +n = P(γSn−1+1 = in, aj = in, Sn−1 + 1 ≤ j ≤ Sn) +(21) += P(γSn−1+1 = in, βj = in, αj = 0, Sn−1 + 1 ≤ j ≤ Sn) ++ +Sn +� +j=Sn−1+1 +P(γSn−1+1 = in, ∀j ̸= j0βj = in, αj = 0, and αj0 = 1) += 1 +2 +1 +3Mn + 1 +2 +Mn +3Mn . +(22) +Moreover vn +n = 1 +2 +1 +3Mn and xn +n = +1 +Mn+1. Hence for k ≤ n +xn +m = +1 +Mm + +1 +Mm+1+ +1 +...+ +1 +Mn+1 += [Mm, Mm+1, . . . , Mn, 1]. +(23) +Remark 4.1. Please note that we can use the induction of Lemma 4.2 even +if K = ∞. Moreover when n → ∞ xn +m is converging to the irrational number +that we will denote by x∞ +m +5 +Initializing phase +In this part we will compute the conditional probability given a that there is +a digit bj ̸= aj in the first block of constancy of a namely A(1). Hence when K +is finite if a has N blocks of constancy we are aiming for P(ǫ1 = 1|∩N +m=1A(m)). +When K is infinite we want to compute the limit of the previous probability +10 + +when N → ∞. For N ≥ n ≥ 2, we start by computing +P(ǫ1 = 0, A(1), γS1+1 = ±1, ∩n +m=2A(m)) += P(ǫ1 = 0, A(1), ∩n +m=2A(m)| γS1+1 = ±1)P(γS1+1 = ±1) += P(ǫ1 = 0, A(1)|γS1+1 = ±1) +× P(∩n +m=2A(m), γS1+1 = ±1|γS1+1 = ±1)P(γS1+1 = ±1) += P(ǫ1 = 0, A(1), γS1+1 = ±1)P(∩n +m=2A(m), γS1+1 = ±1) +P(γS1+1 = ±1) +. +Since P(γS1+1 = ±1) = 1 +2 and +P(ǫ1 = 0, A(1), γS1+1 = ±1) = P(ǫ1 = 0, A(1), ǫ = ±1)); +P(ǫ1 = 0, A(1), γS1+1 = ±1) = +P(∀k = 1 to M1, αk = 0 S1βk = a1, ǫ = ±1), +P(ǫ1 = 0, A(1), γS1+1 = ±1) = +1 +3M1 +1 +2, +this yields +P(ǫ1 = 0, A(1), γS1+1 = ±1, ∩n +m=2A(m))) = +1 +3Mn P(∩n +m=2A(m), γS1+1 = ±1), +which can be written +P(ǫ1 = 0, ∩n +m=1A(m)) = +1 +3M1 (un +2 + vn +2 ). +Similarly +P(ǫ1 = 1, A(1), γS1+1 = i2, ∩n +m=2A(m)) += P(ǫ1 = 1, A(1), γS1+1 = i2)P(∩n +m=2A(m), γS1+1 = i2) +P(γS1+1 = i2) += M1 +3M1 un +2. +Since P(γS1+1 = −i2, ǫ1 = 1, A(1)) = 0, by summing the previous probabil- +ities, we obtain. +P(∩n +m=1A(m)) = +1 +3M1 ((M1 + 1)un +2 + vn +2 ) +11 + +and +P(ǫ1 = 1| ∩n +m=1 A(m)) = +M1 +M1 + 1 + xn +2 +. +(24) +Equation (24) yields equation (8) and becomes +P(ǫ1 = 1|a) = +M1 +M1 + 1 + x∞ +2 +. +(25) +when K is infinite by letting n → ∞. When N = 1, xN +2 is not defined but +we may compute P(ǫ1 = 1|A(1)) as follows. If N = 1 it means that the +vertex a has a single block of constancy. Hence A(1) = {a}, and the vertex a +has M1 neighbors in the graph which are different from a and there are two +edges that starts from a and ends at a. Hence, when N = 1, equation (24) +becomes +P(ǫ1 = 1|A(1)) = +M1 +M1 + 2, +(26) +which is coherent with the convention x1 +2 = 1. +6 +Subsequent phases +6.1 +P(ǫn = 1|ǫn−1 = 0, . . . , ǫn−(2k−1) = 1, ∩N +m=1A(m)) +Here we consider N > n ≥ 2. The numerator of the conditional probability +is +P(ǫn = 1, ǫn−1 = 0, . . . , ǫn−(2k−1) = 1, ∩N +m=1A(m)) += P(ǫn = 1, ǫn−1 = 0, . . . , ǫn−(2k−1) = 1, ∩N +m=1A(m), γSn+1 = in+1) += 2P(ǫn = 1, ǫn−1 = 0, . . . , ǫn−(2k−1) = 1, ∩n +m=1A(m)) +× P(γSn+1 = in+1, ∩N +m=n+1A(m)) += 4P(ǫn−1 = 0, . . . , ǫn−(2k−1) = 1, ∩n−1 +m=1A(m), γSn−1+1 = in) +× P(ǫn = 1, A(n))uN +n+1, +where the last two equalities come from conditional independence with re- +spect to γ taken at the convenient index. For the denominator of the condi- +tional probability of the title of the section, the same kind of manipulations +yield +P(ǫn−1 = 0, . . . , ǫn−(2k−1) = 1, ∩N +m=1A(m)) = +2P(ǫn−1 = 0, . . . , ǫn−(2k−1) = 1, ∩n−1 +m=1A(m))P(γSn−1+1 = in, ∩N +m=nA(m)) += 2P(ǫn−1 = 0, . . . , ǫn−(2k−1) = 1, ∩n−1 +m=1A(m))uN +n . +12 + +Then +P(ǫn = 1|ǫn−1 = 0, . . . , ǫn−(2k−1) = 1, ∩N +m=1A(m)) = 2P(ǫn = 1, A(n))uN +n+1 +uN +n +, +P(ǫn = 1|ǫn−1 = 0, . . . , ǫn−(2k−1) = 1, ∩N +m=1A(m)) = +Mn +3Mn +uN +n+1 +1 +3Mn (MnuN +n+1 + vN +n+1), +P(ǫn = 1|ǫn−1 = 0, . . . , ǫn−(2k−1) = 1, ∩N +m=1A(m)) = Mn +1 +Mn + xN +n+1 += MnxN +n . +(27) +When K is infinite, we can let N → ∞ to obtain +P(ǫn = 1|ǫn−1 = 0, . . . , ǫn−(2k−1) = 1, ∩∞ +m=1A(m)) = Mnx∞ +n . +(28) +When N = n, we have to change the computation of the numerator. +Actually in this case +P(ǫn = 1, ǫn−1 = 0, . . . , ǫn−(2k−1) = 1, ∩n +m=1A(m)) = +2P(ǫn−1 = 0, . . . , ǫn−(2k−1) = 1, ∩n−1 +m=1A(m))P(ǫn = 1, A(n)). +Since +P(ǫn−1 = 0, . . . , ǫn−(2k−1) = 1, ∩n +m=1A(m)) = += 2P(ǫn−1 = 0, . . . , ǫn−(2k−1) = 1 ∩n−1 +m=1 A(m))P(γSn−1+1 = in, ∩A(n)) += 2P(ǫn−1 = 0, . . . , ǫn−(2k−1) = 1, ∩n−1 +m=1A(m))un +n, +we get +P(ǫn = 1|ǫn−1 = 0, . . . , ǫn−(2k−1) = 1, ∩n +m=1A(m)) = 2P(ǫn = 1, A(n)) +unn += Mn +3Mn +1 +unn +. +Because of (22) +P(ǫn = 1|ǫn−1 = 0, . . . , ǫn−(2k−1) = 1, ∩n +m=1A(m)) = +Mn +Mn + 1. +(29) +Hence equation (27) is always verified. +13 + +6.2 +P(ǫn = 1|ǫn−1 = 0, . . . , ǫ1 = 0, ∩N +m=1A(m)) +Let us start with the denominator and assume until further notice that +N > n ≥ 2, +P(ǫn−1 = 0, . . . , ǫ1 = 0, ∩N +m=1A(m)) += ++1 +� +ν=−1 +P(γ1 = ν, ǫn−1 = 0, . . . , ǫ1 = 0, ∩N +m=1A(m)). +Then +P(γ1 = ν, ǫn−1 = 0, . . . , ǫ1 = 0, ∩N +m=1A(m)) = +2P(γ1 = ν, ǫ1 = 0, A(1))P(ǫn−1 = 0, . . . , ǫ2 = 0, γS1+1 = ν, ∩N +m=2A(m)) += P(ǫ1 = 0, A(1))P(ǫn−1 = 0, . . . , ǫ2 = 0, γS1+1 = ν, ∩N +m=2A(m)) += +n−1 +� +m=1 +P(ǫm = 0, A(m))P(γSn−1+1 = ν, ∩N +m=nA(m)). +Hence +P(ǫn−1 = 0, . . . , ǫ1 = 0, ∩N +m=1A(m)) = +n−1 +� +m=1 +P(ǫm = 0, A(m))(uN +n +vN +n ). (30) +The previous formula is also true for N = n. Moreover +P(ǫn = 1, ǫn−1 = 0, . . . , ǫ1 = 0, ∩N +m=1A(m)) += P(ǫ1 = 0, γ1 = in, ǫ2 = 0, γS1+1 = in, . . . , ǫn−1 = 0, . . . +. . . , γSn−1+1 = in ǫn = 1, γSn+1 = in+1 ∩N +m=1 A(m)) += +n−1 +� +m=1 +P(ǫm = 0, A(m))P(ǫn = 1, γSn+1 = in+1 ∩N +m=n A(m))) += 2 +n−1 +� +m=1 +P(ǫm = 0, A(m))P(ǫn = 1, A(n))uN +n+1, +for N > n. Hence +P(ǫn = 1|ǫn−1 = 0, . . . , ǫ1 = 0, ∩N +m=1A(m)) = P(ǫn = 1, A(n)) 2uN +n+1 +uN +n + vN +n += 1 +2 +Mn +3Mn +2 × 3MnuN +n+1 +(Mn + 1)uN +n+1 + vN +n+1 += +Mn +Mn + 1 + xN +n+1 +(31) +14 + +Where we have used +P(ǫn = 1, A(n)) = 1 +2 +Mn +3Mn +and Lemma 4.2 in the previous computations. When K = ∞ we let N → ∞ +in (31) and we get +P(ǫn = 1|ǫn−1 = 0, . . . , ǫ1 = 0, ∩∞ +m=1A(m)) = +Mn +Mn + 1 + x∞ +n+1 +. +(32) +When N = n, +P(ǫn = 1, ǫn−1 = 0, . . . , ǫ1 = 0, ∩N +m=1A(m)) += +n−1 +� +m=1 +P(ǫm = 0, A(m))P(ǫn = 1, A(n)). +P(ǫn = 1|ǫn−1 = 0, . . . , ǫ1 = 0, ∩n +m=1A(m)) = P(ǫn = 1, A(n)) +unn + vnn +(33) += 1 +2 +Mn +3Mn +1 +unn + vnn +(34) += +Mn +Mn + 2. +(35) +Equations (31) and (35) yield (9) in view of the convention xn +n+1 = 1. The +equation (10) is a consequence of the definition of ǫm. +To conclude the proof let us prove that ∀m ≥ 1 the distribution of the +Em’s is uniform on {1, . . . , Mm} conditionally to the event ǫm = 1. Let us +fix m0 such that 2 ≤ m0 ≤ N. We have to show that ∀j0 ∈ {1, . . . , Mm0} +P(Em0 = j0|a, ǫm0) +does not depend on j0. We have +P(Em0 = j0, ∩N +m=1A(m), ǫm0 = 1) = +P(Em0 = j0, ǫm0 = 1, γSm0−1+1 = im0, γSm0+1 = im0+1, ∩N +m=1A(m)) += 2P(∩m0−1 +m=1 A(m), γSm0−1+1 = im0) +× P(Em0 = j0, ǫm0 = 1, γSm0−1+1 = im0, γSm0+1 = im0+1, ∩N +m=m0+1A(m)) +15 + += 4P(∩m0−1 +m=1 A(m), γSm0−1+1 = im0) +× P(Em0 = j0, ǫm0 = 1, A(m0))P(γSm0−1+1 = im0+1, ∩N +m=m0+1A(m)) +thanks to the conditional independence when γ is given. Furthermore +P(Em0 = j0, A(m0), ǫm0 = 1) = P(αj0 = 1, ∀i ̸= j0, αi = 0, A(m0)) +which does not depend on j0 and consequently P(Em0 = j0, A(m0), ǫm0 = 1) +does not depend on j0. It is also the case for P(Em0 = j0, ∩N +m=1A(m), ǫm0 = +1|a, ǫm0 = 1) since P(∩N +m=1A(m), ǫm0 = 1) does not depend on j0. The proof +is easier when m0 = 1. +Acknowledgment +Th authors would like to thank James Norris for fruitful discussions con- +cerning a previous version of the article. +References +[1] Itai Benjamini, Ariel Yadin, and Amir Yehudayoff. +Random graph- +homomorphisms and logarithmic degree. +Electron. J. Probab., 12:no. +32, 926–950, 2007. +[2] Emmanuel Boissard, Serge Cohen, Thibault Espinasse, and James Nor- +ris. Diffusivity of a random walk on random walks. Random Structures +Algorithms, 47(2):267–283, 2015. +[3] Duminil-Copin, Karrila, Manolescu, and Oulamara. Delocalization of +the height function of the six-vertex model. +On arXiv:2012.13750v1 +[math.PR], 2021. +[4] T. Espinasse, N. Guillotin-Plantard, and P. Nadeau. A combinatorial +approach to a model of constrained random walkers. Combin. Probab. +Comput., 25(2):222–235, 2016. +[5] P. Lammers. +Diffusivity of a walk on fractures of a hypertorus. +arXiv:1706.05690v2 [math.PR], 2022. +[6] E. H. Lieb. Residual entropy of square ice. Physical Review, 162(1):162– +172, 1967. +[7] Fabien Montegut. Limite d’´echelle de marche al´eatoires contraintes. PhD +thesis, Ecole doctorale MITT Universit´e de Toulouse, 2020. +16 + diff --git a/ptA0T4oBgHgl3EQfKf9E/content/tmp_files/load_file.txt b/ptA0T4oBgHgl3EQfKf9E/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9e7a68a620b2a1e3fdc2c8d0c2f05770b6924246 --- /dev/null +++ b/ptA0T4oBgHgl3EQfKf9E/content/tmp_files/load_file.txt @@ -0,0 +1,471 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf,len=470 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='02104v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='PR] 5 Jan 2023 Transition of the simple random walk on the graph of the ice-model Serge Cohen ∗ Xavier Bressaud ∗ January 6, 2023 Abstract The 6-vertex model is a seminal model for many domains in Math- ematics and Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' The sets of configurations of the 6-vertex model can be described as the sets of paths in multigraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' In this article the transition probability of the simple random walk on the multigraphs is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' The unexpected point of the results is the use of continuous fractions to compute the transition probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Keywords: Random walk, Markov Chain AMS classification (2000): 05C81, 60F05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' 1 Introduction In this article we are interested in a simple random walk Y on particular (multi)graphs GK indexed by an integer K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' The set of vertices of GK is VK def = {−1, 1}K and the set EK of edges is defined so that the set of paths of length n is isomorphic with the set of configurations of the so-called 6- vertex model, on a rectangle K×n in the particular case of the ice-model [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Please note that the uniform distribution is usually considered in the 6- vertex model, with various admissible boundary conditions on the rectangle when n, K → ∞ of a rectangular lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' In this paper we endow the sets of configurations with the distribution of Y0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , Yn which is actually easier to study that the uniform distribution on the set of paths of length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' In the 6-vertex model the height function h which is a map from the rectangles K × n to the set of integers Z has physical meaning, and more precisely ∗Institut de Math´ematiques de Toulouse;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' UMR 5219, Universit´e de Toulouse;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' CNRS, UT3 F-31062 Toulouse Cedex 9, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' First-Name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='Name@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='univ-toulouse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='fr 1 the decay of the variance of the difference of height function between two distant points on the rectangle is of special importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' This question is also related to random graph homomorphisms [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' See [3] where the decay is shown to be of logarithmic order for periodic boundary conditions on the rectangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' For the Markov chain Y the height fonction is related to an additive functional Z of the Y, which collects a particular vector field defined on the edges of GK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' In a previous paper [2] it is shown that the variance of the height function decays like 2 (2+K)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Still for finite K the variance is computed in [4] for periodic boundary condition in the variable between 1 and K, and in [5] for other constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' The easiest way to describe the simple random walk on GK in the sta- tionary regime is to state that the distribution of the pair (YK(0), YK(1)) is the uniform distribution on the edges EK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' One aim of this article is to provide a formula for the transition probability of the simple random walk that starts from a given vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Surprisingly enough the formula in the Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='1 uses the continuous fraction associated to the length of the constancy blocks of the digits of the given vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' The authors see two use- ful consequences of this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' The first one is the fact that the transition probability of the simple random walk will considerably make easier and faster simulations of the walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' See [7] for a survey of previous simulation methods, which are both memories and computationally intensive due the difficulty to describe easily the neighbors of a given vertex in GK for large K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Another interesting consequence of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='1 is that it can be ex- tended to the case K = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' This in turns gives a sense to the decay of the variance of the height function in G∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' The definitions and the models are given in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' In section 3 the main theorems are written and they are proved in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' 2 The model Let � Z(1) t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , Z(K+1) t � t∈N ∈ ZK+1 denote the heights of K + 1 simple ran- dom walks on Z, conditioned on satisfying ∀t ∈ N, ∀i ∈ [1, K] , ���Z(i+1) t − Z(i) t ��� = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' (1) More precisely, the random walk is a Markov chain on the state space of K-step walks in Z SK = {(z(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , z(K+1)) ∈ ZK+1, ∀i ∈ [1, K] , |z(i+1) − z(i)| = 1} (2) 2 where the next step from z0 ∈ SK is selected uniformly among the z1s that belong to SK such that ∀i ∈ [1, K + 1], z(i) 1 − z(i) 0 ∈ {−1, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' In other words, we consider K + 1 simple random walks on the lattice Z coupled under a shape condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' One can associate to a path of length n (Z(i) t )0≤t≤n−1 a height function h on a rectangle ((t, i))0≤t≤n−1, 1≤i≤K+1 by h(t, i) = Z(i) t , which makes the link with the height function of the 6-vertex model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' In [2] an equivalent depiction is provided as a simple random walk on a (multi)-graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Let VK def = {−1, 1}K, E+ K, E− K ∈ VK×VK will be respectively set of postive, negative edges on VK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' A pair (a, b) ∈ VK × VK such that a ̸= b belongs to E+ K if non vanishing coordinates of the vector (b − a) ∈ {−2, 0, 2}K have alternate signs with the first sign negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' For every vertex a ∈ VK there is an edge from a to a E+ K denoted by (a, a)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' In a similar manner there is a pair (a, b) ∈ VK × VK such that a ̸= b belongs to E− K if non vanishing coordinates of the vector (b−a) ∈ {−2, 0, 2}K have alternate signs with the first sign positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' For every vertex a ∈ VK theres is an edge from a to a E− K denoted by (a, a)−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' For every K ∈ N the (multi)graph GK def = (VK, EK), where EK def = E+ K ∪ E− K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' The study of (Z(1) t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=', Z(K+1) t ) can be split in the study of the first coordinate Z(1) t and of the increments YK(t) def = � Z(2) t − Z(1) t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , Z(K+1) t − Z(K) t � , which is always an element of VK (because of (1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Let us remark that the uniform distribution on the set of edges EK is the same as the distribution of (YK(0), YK(1)) if the Markov process YK is stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' We have another useful characterization of this distribution given by the following Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Let ǫ be a random variable such that P(ǫ = −1) = P(ǫ = 1) = 1 2), let (αk)k≥1 be an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' sequence of Bernoulli random variables with parameter 1 3 and let (βk)k≥1 be an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' sequence of random variables such that P(βk = −1) = P(βk = 1) = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' The previous random variables are mutually independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Let us define ∀k ≥ 2 : γk def = ǫ(−1) �k−1 i=1 αi, 3 with the convention that γ1 = ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Let us also define k ≥ 1 : Ak = (1 − αk)βk + αkγk (3) Bk = (1 − αk)βk − αkγk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' (4) The distribution of the pair ((Ak)1≤k≤K, (Bk)1≤k≤K) is the uniform dis- tribution on EK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' If we denote by (A, B)K the edge defined by : (A, B)K def = � ((Ak)1≤k≤K, (Bk)1≤k≤K) if (Ak)1≤k≤K ̸= (Bk)1≤k≤K ((Ak)1≤k≤K, (Bk)1≤k≤K)ǫ if (Ak)1≤k≤K = (Bk)1≤k≤K then (A, B)K L= (YK(0), YK(1)) if YK(0) ̸= YK(1) and (A, A)ǫ K L= (YK(0), YK(0))Z(1) 1 −Z(1) 0 if YK(0) = YK(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Moreover (Ak)1≤k≤K L= YK(0) and (Bk)1≤k≤K L= YK(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' The proof is by induction and can be found in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Let us first prove that (A, B)K ∈ EK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' We remark that Bk −Ak = −2αkγk, then it is vanishing if αk = 0, and the alternating rule sign is fulfilled because every time αk = 1, γk has a different sign from γk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Let us denote by DK the cardinal of EK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' A simple computation yields D1 = 6 and by induction DK = 2 × 3K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' It is also obvious to check P((A, B)1 = e) = 1 6 for every edge in E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Let us assume that P((A, B)K = e) = 1 DK is true for every edge in EK Let us consider u′, v′ ∈ VK and denote by u′ ± 1 the vertex in VK+1 obtained by concatenating ±1 on the right of u′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Then for P((A, B)K+1 = (u′1, v′1)) = P((A, B)K = (u′, v′) ∩ αK+1 = 0 ∩ βK+1 = 1) = 1 DK 2 3 1 2 = 1 DK+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' The same holds for (u′ −1, v′ −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' If the concatenated digit to u′ is different from the one concatenated to v′ and u′ ̸= v′ there is only one possible choice which leads to a non vanishing probability depending on the last digit that differs between u′ and v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Then for this choice P((A, B)K+1 = (u′ ± 1, v′ ∓ 1)) = P((A, B)K = (u′, v′) ∩ αK+1 = 1) = 1 DK 1 3 = 1 DK+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' 4 The proof is complete when we consider the case u′ = v′ and in this case we get also P((A, B)K+1 = (u′ ± 1, v′ ∓ 1)) = 1 DK+1 thanks to the distribution of ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' One important consequence of the previous result is the fact that the graph GK is defined for K = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Let us be more precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' When K = ∞, V∞ def = {−1, 1}N, and E+ ∞, E− ∞ are defined with the same alternating rules as for finite K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' The Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='1 is still true when K infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' 3 Results The aim of this article is to compute the transition probability of the sta- tionary Markov chain associated with the simple random walks on the (multi)graphs GK for K ∈ N ∪ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Hence we will compute the conditional probability that (Bk)k≤K takes a particular value in VK once the sequence (Ak)k≤K is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Please note that when K = ∞, the law of large number implies for both sequences (Ak)k∈N, (Bk)k∈N that there are not constant for k big enough almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Let us assume that the sequence (ak)k∈N ∈ {−1, 1}N starts with a1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Let us fix the consecutive times where a is constant and denote by im = (−1)m+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' (If a1 = −1, then im = (−1)m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' By convention we set S0 = 0, and for m ≥ 1, we assume that the m-th block of constancy of a starts with Sm−1 + 1 and stops with Sm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Let us denote for m ≥ 1, the event A(m) = {a ∈ {−1, 1}N such that aSm−1+1 = im, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , aSm = im} (5) of sequences which are equal to a on the m-th block of constancy of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' In the following we are conditioning the distribution of B with respect of events of the form {A = a}, where a is a deterministic sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Once a is given, so is the sequence S and the conditioning with respect of A(m) actually means with respect of the event {ASm−1+1 = im, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ASn = im}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' We will use the abuse of notation P(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='|a), P(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='|A(m)) in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Hence {a} = {(ak)k≤K} = ∩N m=1A(m) where N is the number of blocks of constancy of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' By definition of Mm the length of the m-th block of constancy of a is equal to Mm = Sm − Sm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' (6) 5 Let for 1 ≤ m ≤ n ≤ N xn m def = 1 Mm + 1 Mm+1+ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='+ 1 Mn+1 = [Mm, Mm+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , Mn, 1], (7) if n < m we set xn m def = 1 by convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' When K = ∞ the continuous fraction in (7) is converging when n → ∞ toward an irrational number because of Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='1 that will be denoted by x∞ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Please remark that if YK(0) = a and YK(1) = b, at most one digit bk of b is different of ak in any block of constancy A(m) of a, because of the alternating sign rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Let (ǫm)1≤m≤N be Bernoulli random variables such that ǫm = 1 if and only if there is one change of digits between a and b in the m-th block of constancy A(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Let (Em)1≤k≤N be a sequence of independent random variables uniformly distributed on {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , Mm} which encode the digit that is changed in A(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' One further constraint due to the alternating sign rule is that when ǫm = 1, ǫm+2k = 0 on the event that ǫm+1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫm+(2k−1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' In other words there cannot be change of digits in two consecutive blocks of constancy of a that have an even difference of indexes, since the ak are the same on those blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' The conditional probability P(YK(1) = b|YK(0) = a) is then described by the following Theorem that yields the distribution of the (ǫm)1≤m≤N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' The distribution of (ǫm)1≤m≤N is given by : Initializing phase P(ǫ1 = 1|a) = M1 M1 + 1 + xN 2 (8) where N is the number of blocks of constancy of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Subsequent phase when previously there is no change P(ǫm = 1|ǫm−1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫ1 = 0, a) = Mm Mm + 1 + xN m+1 (9) where N is the number of blocks of constancy of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Subsequent phase when previously there is at least one change P(ǫm = 1|ǫm−1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫm−(2k−1) = 1, a) = MmxN m, where N is the number of blocks of constancy of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' 6 Loop in GK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' If ǫm = 0 for 1 ≤ m ≤ N which is equivalent to b = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' P((YK(0), YK(1)) = (a, a)+|YK(0) = a) = P((YK(0), YK(1)) = (a, a)−|YK(0) = a) = 1 2P(ǫm = 0, for 1 ≤ m ≤ N|a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' (10) ∀m ≥ 1 the distribution of the Em’s is uniform on {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , Mm} conditionally to the event ǫm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Remember that the distribution P(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='|a) is the uniform distri- bution on the neighbors of a in EK, this fact is not obvious from the previous theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Indeed, if a given a in EK has N blocks of constancy, it leads for instance to 1 degK(a) = P((ǫ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫN) = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , 1)|a) �N k=1 Mk (11) = 1 M1 + 1 + xN 2 N � k=2 xN k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Let us now suppose that for a given 2 ≤ k0 ≤ k0 + 1 < N, ǫk0 = 0, it implies that ǫk0+1 = 0, and the other ǫk = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' The equation 1 degK(a) = P((ǫ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫN) = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , 1, 0, 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , 1)|a) �N k=1 Mk (12) is still true when the 0, 0 are for this k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' In equation (11) we only have to change the factors for k0 and k0 +1, so xN k0 becomes xN k0+1xN k0 and the factor xN k0+1 becomes 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Then we check P((ǫ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫN) = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , 0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , 1)|a) �N k=1 Mk = 1 degK(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Tedious computations can show that actually the probability to jump from a to each of his neighbor is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' When K = ∞, the previous Theorem still holds true when we consider that the number of constancy blocks of a is infinite and use the definition of the continuous fraction as a limit .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' The distribution of (ǫm)1≤m is given by : 7 Initializing phase P(ǫ1 = 1|a) = M1 M1 + 1 + x∞ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' (13) Subsequent phase when previously there is no change P(ǫm = 1|ǫm−1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫ1 = 0, a) = Mm Mm + 1 + x∞ m+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' (14) Subsequent phase when previously there is at least one change P(ǫm = 1|ǫm−1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫm−(2k−1) = 1, a) = Mmx∞ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' ∀m ≥ 1 the distribution of the Em’s is uniform on {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , Mm} conditionally to the event ǫm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Please note that the probability of a loop in G∞ is vanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Hence there is no loop case in the last Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' 4 Proof of the result Let us now introduce the conditional independence with respect of γk, which is an important tool for our computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Let us denote by σm l def = σ(ak, bk, l ≤ k ≤ m), for 2 ≤ l ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' By convention σm 1 def = σ(ak, bk, 1 ≤ k ≤ m, ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' For every 1 ≤ l ≤ m ≤ n σm l is independent of σn m+1 condi- tionally to γm+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' The proof of this Lemma comes from the definitions of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' For every m ≤ n, let us define un m def = P(γSm−1+1 = im ∩ A(m) ∩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' ∩ A(n)) and vn m def = P(γSm−1+1 = −ik ∩ A(m) ∩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' ∩ A(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' With the help of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='1 we can compute un m, vn m, by induction starting from un n, vn n, and we get the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' 8 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' For every m < n, �un m vn m � = 1 3Mm �Mm 1 1 0 � �un m+1 vn m+1 � Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' By conditioning and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='1 we may write un m = P(γSm−1+1 = im, A(m)|γSm+1 = im) × P(γSm+1 = −im+1, A(m+1) ∩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' ∩ A(n)) + P(γSm−1+1 = im, A(m)|γSm+1 = −im) × P(γSm+1 = im+1, A(m+1) ∩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' ∩ A(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' This equation yields un m = P(γSm−1+1 = im ∩ A(m)|γSm+1 = im)vn m+1 (15) + P(γSm−1+1 = im ∩ A(m)|γSm+1 = −im)un m+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Observe that {γSm−1+1 = im ∩ A(m) ∩ γSm+1 = im} = {αl = 0, βl = im for Sm−1 + 1 ≤ l ≤ Sm} ∩ {γSm−1+1 = im}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' (16) The two events on the right hand side are independent and the probability of the first one is 1 3Mm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Therefore P(γSm−1+1 = im ∩ A(m)|γSm+1 = im) = P(γSm−1+1 = im ∩ A(m) ∩ γSm+1 = im) P(γSm+1 = im) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Hence P(γSm−1+1 = im ∩ A(m)|γSm+1 = im) = 1 3Mm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' (17) With a similar argument we get P(γSm−1+1 = im ∩ A(m) ∩ γSm+1 = −im) = Mm 3Mm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' (18) Moreover vn m = P(γSm−1+1 = −im ∩ A(m)|γSm+1 = −im) × P(γSm+1 = im+1 ∩ A(m+1) ∩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' ∩ A(n)) + P(γSm−1+1 = −im ∩ A(m)|γSm+1 = im) × P(γSm+1 = −im+1 ∩ A(m+1) ∩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' ∩ A(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' 9 Note that P(γSm−1+1 = −im ∩ A(m) ∩ γSm+1 = im) = 0, since, on this event, the value −im of γSm−1+1 does not fit the value of al’s on A(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Hence all the αl = 0 and γSm+1 = γSm−1+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' It follows that vn m = P(γSm−1+1 = −im ∩ A(m)|γSm+1 = −im)un m+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' (19) The Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='2 is the consequence of equations (15), (19), (17), (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Let us recall (7) xn m def = vn m unm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='2 yields for m + 1 ≤ n, xn m = 1 Mm + xn m+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' (20) Moreover un n = P(γSn−1+1 = in, aj = in, Sn−1 + 1 ≤ j ≤ Sn) (21) = P(γSn−1+1 = in, βj = in, αj = 0, Sn−1 + 1 ≤ j ≤ Sn) + Sn � j=Sn−1+1 P(γSn−1+1 = in, ∀j ̸= j0βj = in, αj = 0, and αj0 = 1) = 1 2 1 3Mn + 1 2 Mn 3Mn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' (22) Moreover vn n = 1 2 1 3Mn and xn n = 1 Mn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Hence for k ≤ n xn m = 1 Mm + 1 Mm+1+ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='+ 1 Mn+1 = [Mm, Mm+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , Mn, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' (23) Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Please note that we can use the induction of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='2 even if K = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Moreover when n → ∞ xn m is converging to the irrational number that we will denote by x∞ m 5 Initializing phase In this part we will compute the conditional probability given a that there is a digit bj ̸= aj in the first block of constancy of a namely A(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Hence when K is finite if a has N blocks of constancy we are aiming for P(ǫ1 = 1|∩N m=1A(m)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' When K is infinite we want to compute the limit of the previous probability 10 when N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' For N ≥ n ≥ 2, we start by computing P(ǫ1 = 0, A(1), γS1+1 = ±1, ∩n m=2A(m)) = P(ǫ1 = 0, A(1), ∩n m=2A(m)| γS1+1 = ±1)P(γS1+1 = ±1) = P(ǫ1 = 0, A(1)|γS1+1 = ±1) × P(∩n m=2A(m), γS1+1 = ±1|γS1+1 = ±1)P(γS1+1 = ±1) = P(ǫ1 = 0, A(1), γS1+1 = ±1)P(∩n m=2A(m), γS1+1 = ±1) P(γS1+1 = ±1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Since P(γS1+1 = ±1) = 1 2 and P(ǫ1 = 0, A(1), γS1+1 = ±1) = P(ǫ1 = 0, A(1), ǫ = ±1));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' P(ǫ1 = 0, A(1), γS1+1 = ±1) = P(∀k = 1 to M1, αk = 0 S1βk = a1, ǫ = ±1), P(ǫ1 = 0, A(1), γS1+1 = ±1) = 1 3M1 1 2, this yields P(ǫ1 = 0, A(1), γS1+1 = ±1, ∩n m=2A(m))) = 1 3Mn P(∩n m=2A(m), γS1+1 = ±1), which can be written P(ǫ1 = 0, ∩n m=1A(m)) = 1 3M1 (un 2 + vn 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Similarly P(ǫ1 = 1, A(1), γS1+1 = i2, ∩n m=2A(m)) = P(ǫ1 = 1, A(1), γS1+1 = i2)P(∩n m=2A(m), γS1+1 = i2) P(γS1+1 = i2) = M1 3M1 un 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Since P(γS1+1 = −i2, ǫ1 = 1, A(1)) = 0, by summing the previous probabil- ities, we obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' P(∩n m=1A(m)) = 1 3M1 ((M1 + 1)un 2 + vn 2 ) 11 and P(ǫ1 = 1| ∩n m=1 A(m)) = M1 M1 + 1 + xn 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' (24) Equation (24) yields equation (8) and becomes P(ǫ1 = 1|a) = M1 M1 + 1 + x∞ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' (25) when K is infinite by letting n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' When N = 1, xN 2 is not defined but we may compute P(ǫ1 = 1|A(1)) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' If N = 1 it means that the vertex a has a single block of constancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Hence A(1) = {a}, and the vertex a has M1 neighbors in the graph which are different from a and there are two edges that starts from a and ends at a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Hence, when N = 1, equation (24) becomes P(ǫ1 = 1|A(1)) = M1 M1 + 2, (26) which is coherent with the convention x1 2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' 6 Subsequent phases 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='1 P(ǫn = 1|ǫn−1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫn−(2k−1) = 1, ∩N m=1A(m)) Here we consider N > n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' The numerator of the conditional probability is P(ǫn = 1, ǫn−1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫn−(2k−1) = 1, ∩N m=1A(m)) = P(ǫn = 1, ǫn−1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫn−(2k−1) = 1, ∩N m=1A(m), γSn+1 = in+1) = 2P(ǫn = 1, ǫn−1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫn−(2k−1) = 1, ∩n m=1A(m)) × P(γSn+1 = in+1, ∩N m=n+1A(m)) = 4P(ǫn−1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫn−(2k−1) = 1, ∩n−1 m=1A(m), γSn−1+1 = in) × P(ǫn = 1, A(n))uN n+1, where the last two equalities come from conditional independence with re- spect to γ taken at the convenient index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' For the denominator of the condi- tional probability of the title of the section, the same kind of manipulations yield P(ǫn−1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫn−(2k−1) = 1, ∩N m=1A(m)) = 2P(ǫn−1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫn−(2k−1) = 1, ∩n−1 m=1A(m))P(γSn−1+1 = in, ∩N m=nA(m)) = 2P(ǫn−1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫn−(2k−1) = 1, ∩n−1 m=1A(m))uN n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' 12 Then P(ǫn = 1|ǫn−1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫn−(2k−1) = 1, ∩N m=1A(m)) = 2P(ǫn = 1, A(n))uN n+1 uN n , P(ǫn = 1|ǫn−1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫn−(2k−1) = 1, ∩N m=1A(m)) = Mn 3Mn uN n+1 1 3Mn (MnuN n+1 + vN n+1), P(ǫn = 1|ǫn−1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫn−(2k−1) = 1, ∩N m=1A(m)) = Mn 1 Mn + xN n+1 = MnxN n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' (27) When K is infinite, we can let N → ∞ to obtain P(ǫn = 1|ǫn−1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫn−(2k−1) = 1, ∩∞ m=1A(m)) = Mnx∞ n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' (28) When N = n, we have to change the computation of the numerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Actually in this case P(ǫn = 1, ǫn−1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫn−(2k−1) = 1, ∩n m=1A(m)) = 2P(ǫn−1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫn−(2k−1) = 1, ∩n−1 m=1A(m))P(ǫn = 1, A(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Since P(ǫn−1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫn−(2k−1) = 1, ∩n m=1A(m)) = = 2P(ǫn−1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫn−(2k−1) = 1 ∩n−1 m=1 A(m))P(γSn−1+1 = in, ∩A(n)) = 2P(ǫn−1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫn−(2k−1) = 1, ∩n−1 m=1A(m))un n, we get P(ǫn = 1|ǫn−1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫn−(2k−1) = 1, ∩n m=1A(m)) = 2P(ǫn = 1, A(n)) unn = Mn 3Mn 1 unn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Because of (22) P(ǫn = 1|ǫn−1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫn−(2k−1) = 1, ∩n m=1A(m)) = Mn Mn + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' (29) Hence equation (27) is always verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' 13 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='2 P(ǫn = 1|ǫn−1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫ1 = 0, ∩N m=1A(m)) Let us start with the denominator and assume until further notice that N > n ≥ 2, P(ǫn−1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫ1 = 0, ∩N m=1A(m)) = +1 � ν=−1 P(γ1 = ν, ǫn−1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫ1 = 0, ∩N m=1A(m)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Then P(γ1 = ν, ǫn−1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫ1 = 0, ∩N m=1A(m)) = 2P(γ1 = ν, ǫ1 = 0, A(1))P(ǫn−1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫ2 = 0, γS1+1 = ν, ∩N m=2A(m)) = P(ǫ1 = 0, A(1))P(ǫn−1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫ2 = 0, γS1+1 = ν, ∩N m=2A(m)) = n−1 � m=1 P(ǫm = 0, A(m))P(γSn−1+1 = ν, ∩N m=nA(m)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Hence P(ǫn−1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫ1 = 0, ∩N m=1A(m)) = n−1 � m=1 P(ǫm = 0, A(m))(uN n +vN n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' (30) The previous formula is also true for N = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Moreover P(ǫn = 1, ǫn−1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫ1 = 0, ∩N m=1A(m)) = P(ǫ1 = 0, γ1 = in, ǫ2 = 0, γS1+1 = in, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫn−1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , γSn−1+1 = in ǫn = 1, γSn+1 = in+1 ∩N m=1 A(m)) = n−1 � m=1 P(ǫm = 0, A(m))P(ǫn = 1, γSn+1 = in+1 ∩N m=n A(m))) = 2 n−1 � m=1 P(ǫm = 0, A(m))P(ǫn = 1, A(n))uN n+1, for N > n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Hence P(ǫn = 1|ǫn−1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫ1 = 0, ∩N m=1A(m)) = P(ǫn = 1, A(n)) 2uN n+1 uN n + vN n = 1 2 Mn 3Mn 2 × 3MnuN n+1 (Mn + 1)uN n+1 + vN n+1 = Mn Mn + 1 + xN n+1 (31) 14 Where we have used P(ǫn = 1, A(n)) = 1 2 Mn 3Mn and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='2 in the previous computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' When K = ∞ we let N → ∞ in (31) and we get P(ǫn = 1|ǫn−1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫ1 = 0, ∩∞ m=1A(m)) = Mn Mn + 1 + x∞ n+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' (32) When N = n, P(ǫn = 1, ǫn−1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫ1 = 0, ∩N m=1A(m)) = n−1 � m=1 P(ǫm = 0, A(m))P(ǫn = 1, A(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' P(ǫn = 1|ǫn−1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , ǫ1 = 0, ∩n m=1A(m)) = P(ǫn = 1, A(n)) unn + vnn (33) = 1 2 Mn 3Mn 1 unn + vnn (34) = Mn Mn + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' (35) Equations (31) and (35) yield (9) in view of the convention xn n+1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' The equation (10) is a consequence of the definition of ǫm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' To conclude the proof let us prove that ∀m ≥ 1 the distribution of the Em’s is uniform on {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , Mm} conditionally to the event ǫm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Let us fix m0 such that 2 ≤ m0 ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' We have to show that ∀j0 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' , Mm0} P(Em0 = j0|a, ǫm0) does not depend on j0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' We have P(Em0 = j0, ∩N m=1A(m), ǫm0 = 1) = P(Em0 = j0, ǫm0 = 1, γSm0−1+1 = im0, γSm0+1 = im0+1, ∩N m=1A(m)) = 2P(∩m0−1 m=1 A(m), γSm0−1+1 = im0) × P(Em0 = j0, ǫm0 = 1, γSm0−1+1 = im0, γSm0+1 = im0+1, ∩N m=m0+1A(m)) 15 = 4P(∩m0−1 m=1 A(m), γSm0−1+1 = im0) × P(Em0 = j0, ǫm0 = 1, A(m0))P(γSm0−1+1 = im0+1, ∩N m=m0+1A(m)) thanks to the conditional independence when γ is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Furthermore P(Em0 = j0, A(m0), ǫm0 = 1) = P(αj0 = 1, ∀i ̸= j0, αi = 0, A(m0)) which does not depend on j0 and consequently P(Em0 = j0, A(m0), ǫm0 = 1) does not depend on j0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' It is also the case for P(Em0 = j0, ∩N m=1A(m), ǫm0 = 1|a, ǫm0 = 1) since P(∩N m=1A(m), ǫm0 = 1) does not depend on j0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' The proof is easier when m0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Acknowledgment Th authors would like to thank James Norris for fruitful discussions con- cerning a previous version of the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' References [1] Itai Benjamini, Ariel Yadin, and Amir Yehudayoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Random graph- homomorphisms and logarithmic degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=', 12:no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' 32, 926–950, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' [2] Emmanuel Boissard, Serge Cohen, Thibault Espinasse, and James Nor- ris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Diffusivity of a random walk on random walks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Random Structures Algorithms, 47(2):267–283, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' [3] Duminil-Copin, Karrila, Manolescu, and Oulamara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Delocalization of the height function of the six-vertex model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' On arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='13750v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='PR], 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' [4] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Espinasse, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Guillotin-Plantard, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Nadeau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' A combinatorial approach to a model of constrained random walkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=', 25(2):222–235, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' [5] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Lammers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Diffusivity of a walk on fractures of a hypertorus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' arXiv:1706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='05690v2 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content='PR], 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' [6] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Lieb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Residual entropy of square ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Physical Review, 162(1):162– 172, 1967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' [7] Fabien Montegut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' Limite d’´echelle de marche al´eatoires contraintes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' PhD thesis, Ecole doctorale MITT Universit´e de Toulouse, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} +page_content=' 16' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptA0T4oBgHgl3EQfKf9E/content/2301.02104v1.pdf'} diff --git a/ptE0T4oBgHgl3EQf9QJL/content/tmp_files/load_file.txt b/ptE0T4oBgHgl3EQf9QJL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9fd22d1e480a871ecdb16365f60670df4d3791ba --- /dev/null +++ b/ptE0T4oBgHgl3EQf9QJL/content/tmp_files/load_file.txt @@ -0,0 +1,1809 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf,len=1808 +page_content='1 Modulation-Doping a Correlated Electron Insulator Debasish Mondal1, Smruti Rekha Mahapatra1, Abigail M Derrico2, Rajeev Kumar Rai3, Jay R Paudel2, Christoph Schlueter4, Andrei Gloskovskii4, Rajdeep Banerjee1, Frank M F DeGroot5, Dipankar D Sarma1, Awadhesh Narayan1, Pavan Nukala3, Alexander X Gray2* and Naga Phani B Aetukuri1* Affiliations: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Solid State and Structural Chemistry Unit, Indian Institute of Science, Bengaluru, Karnataka, 560012, India 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Department of Physics, Temple University, 1925 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 12th St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=', Philadelphia, PA 19122, USA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Centre for Nano Science and Engineering, Indian Institute of Science, Bangalore, Karnataka, 560012, India 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Deutsches Elektronen-Synchrotron, DESY, 22607 Hamburg, Germany 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Utrecht University, Inorganic Chemistry and Catalysis Group Universiteitsweg 99, 3584 CA Utrecht, The Netherlands Corresponding authors emails: phani@iisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='in (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' axgray@temple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='edu (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='G) ABSTRACT Correlated electron materials (CEMs) host a rich variety of condensed matter phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Vanadium dioxide (VO2) is a prototypical CEM with a temperature-dependent metal-to-insulator (MIT) transition with a concomitant crystal symmetry change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' External control of MIT in VO2 – especially without inducing structural changes - has been a long-standing challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' In this work, we design and synthesize modulation-doped VO2-based thin film heterostructures that closely emulate a textbook example of filling control in a correlated electron insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Using a combination of charge transport, hard x-ray photoelectron spectroscopy, and structural characterization, we show that the insulating state can be doped to achieve carrier densities greater than 5x1021 cm-3 without inducing any measurable structural changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' We find that the MIT temperature (TMIT) continuously decreases with increasing carrier concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Remarkably, the insulating state is robust even at doping concentrations as high as ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='2 e- /vanadium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Finally, our work reveals modulation-doping as a viable method for electronic control of phase transitions in correlated electron oxides with the potential for use in future devices based on electric-field controlled phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 2 INTRODUCTION Strong electron-electron correlations within narrow d- or f-orbitals underpin a variety of condensed matter phenomena, such as metal-to-insulator transitions (MITs), high-temperature superconductivity, magnetism, and multiferroicity, often observed in correlated electron materials (CEMs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='1,2 VO2 is a prototypical example of a CEM with a temperature-dependent metal-to- insulator phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The electronic phase transition in bulk VO2, which occurs at a MIT temperature (TMIT) of ~340 K, is accompanied by a structural phase transition from a metallic rutile phase to an insulating monoclinic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='3,4 The origin of the MIT in VO2 – whether gap-opening is driven by the symmetry-lowering structural transition or by electron-electron correlations - has been widely studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5–8 In particular, there is widespread interest in the nature of the insulating state and its external control via doping9,10, strain11, oxygen vacancy creation12, hydrogenation13, light-and-pulse-induced modulation14,15, and via electric-fields in a field-effect transistor geometry16,17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' For example, n-type doping of VO2 with dopants such as W6+, Mo5+, and Nb5+ was shown to decrease TMIT and stabilize the metallic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='9,10,18 By contrast, p-type doping of VO2 with dopants such as Cr3+, Ga3+, and Al3+ was shown to increase TMIT, thereby stabilizing the insulating phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='19–21 Similarly, both oxygen vacancy creation and hydrogenation were shown to n-dope VO2 and decrease TMIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='12,13,22 Finally, in VO2 thin films, macroscopic tensile strain along the rutile a- axis was also shown to decrease TMIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='11,12 In all these previous approaches, modulation of TMIT was always associated with macroscopic changes to the lattice parameters (due to strain) and/or dopant-induced local structural distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='9,10,12,13,19,22 In such experiments, where both the lattice strain and carrier concentration change, it is challenging and, sometimes, impossible to disentangle the role of carrier concentration changes from the role of lattice strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' For instance, in the case of W-doped VO2, an increase in W-doping concentration increases both the carrier density and lattice strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='23 Other techniques utilizing external stimuli, such as electric-field induced metallization of VO2 in a field-effect transistor geometry, could, in theory, enable the modulation of its conductivity without inducing macroscopic structural changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' However, previous attempts at electric-field-driven metallization of VO2 have not been successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='12,16,17,24,25 The absence of any 3 field-effect, even when gated through high-K dielectrics, was attributed to the presence of strong correlations in the insulating VO2 phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='16 Further, ionic-liquid gating of VO2, which could enable accessibility to large interfacial electric-fields, led to oxygen vacancy creation and/or hydrogenation of VO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='12,16 Modulation- or remote-doping of oxide semiconductors is an alternative method for achieving high dopant carrier densities without inducing local structural distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='26–29 In modulation-doping, a chemical potential mismatch between a high band gap heavily-doped layer (dopant-layer) and a lower band gap undoped layer (channel) leads to charge transfer from the heavily doped dopant-layer to the undoped channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' In general, the dopant layer and the channel are spatially separated by a barrier (or a spacer) layer that kinetically limits the interdiffusion of the dopants from the dopant layer to the channel layer while allowing charge transfer via quantum mechanical tunneling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='30,31 Modulation-doping was successfully applied to semiconductors and band-insulating oxides such as ZnO and SrTiO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='27,29,32–35 However, modulation-doping of correlated electron insulators has had limited success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' For example, Stemmer and colleagues reported modulation-doping of NdNiO3, but this did not lead to any significant changes in TMIT of the nickelate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='27 Whether modulation-doping could be a generic approach to induce phase transitions in oxides is unclear and several key questions remain unanswered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' For example, how do bands evolve in correlated oxides as a function of doping?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Can a rigid band model be applied to understand doping in oxides?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' How sensitive are the ground state properties in correlated oxides to carrier doping?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' In this work, we address some of these open questions using the MIT in VO2 as a model system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' We propose a modulation-doped heterostructure to n-dope VO2 without inducing any structural distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Commonly in modulation-doping, an epitaxial structure is grown with a spacer and the dopant layers epitaxially matched to the semiconducting channel layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Note that both the spacer and dopant layers must be insulating with a bandgap that is higher than that of the channel layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' However, the only stable rutile oxide that is both insulating with a compatible band mismatch that allows modulation doping of VO2 is TiO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The other rutile oxides such as CrO2, RuO2, and IrO2 are metallic and therefore not compatible as dopant layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='36,37 As an additional challenge, oxygen lattice continuity in epitaxial structures might also lead to oxygen vacancy diffusion across the layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='22,38,39 We note that oxygen vacancy formation, 4 which was shown to affect the MIT in VO2, is commonly observed in transition metal oxides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='12,40,41 Thus, in order to prevent oxygen vacancy diffusion across the wide band-gap spacer layer as well as to circumvent the lack of lattice-matching insulating rutile oxides, we have gone away from epitaxially-matched modulation-doped heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Instead, we propose an amorphous LaAlO3 (LAO) layer (with a reported electronic band gap of ~5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='6 eV)42 as the spacer layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Since LAO has a low oxygen vacancy-diffusivity43, we use an amorphous oxygen-deficient TiO2-x as the dopant layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Stoichiometric TiO2 has a bandgap of ~3 eV44 and TiO2-x is n-type conducting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Using TiO2-x instead of a conventionally doped TiO2 such as, Nb-doped TiO2, significantly simplified heterostructure deposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Furthermore, this approach avoids the interdiffusion of metallic dopants such as Nb in Nb-doped TiO2 and the associated unintentional doping of VO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The modulation-doped structure for all samples used in this work is comprised of a VO2 channel layer, a 2 nm thick LAO spacer layer, and a 3 nm thick TiO2-x dopant layer, as shown schematically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' All heterostructures were capped with a 1 nm thick LAO layer to prevent dopant passivation from atmospheric impurities as well as oxidation of the TiO2-x dopant layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Fermi level alignment across the structure is expected to lead to an electron accumulation region at the LAO/VO2 interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Expected band-alignments for this type-I heterojunction before and after heterostructure formation are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' To experimentally realize the proposed modulation-doped structure, all samples were grown using pulsed laser deposition (PLD) on single-crystalline TiO2 (001) substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' VO2 was deposited at 425 ºC, while all the other amorphous layers were deposited at room temperature (see methods section for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' It is important to note that room-temperature deposition of the spacer, dopant and capping layers also minimizes any interfacial interdiffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' A cross-sectional scanning transmission electron microscopy (STEM) image (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 1c) and the associated energy dispersive spectroscopy (EDS) elemental maps (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 1d) show abrupt high-quality interfaces between the TiO2 substrate and the VO2 film, and between the film and LAO spacer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' This data is consistent with in-situ RHEED patterns of the deposited VO2 films and suggests that the films are both atomically smooth and single-crystalline (SI Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The atomic force microscopy (AFM) images of the complete heterostructures further confirm the high quality of the growth by showing atomically smooth film surfaces (SI Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 5 To study the correlation between modulation-doping-induced carrier density changes and the changes in the MIT characteristics, we deposited several modulation-doped heterostructures with varying thicknesses of the VO2 layer ranging from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm, while keeping the thickness of the TiO2-x layer unchanged at 3 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Thomas-Fermi screening lengths in VO2 are expected to be on the order of 1 nm (SI section S3) and, therefore, the highest n-type carrier densities are expected for the lowest VO2 film thickness used in this study (~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' (a) Schematic diagram of the heterostructures used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The thickness of VO2 is varied while the thicknesses of all the other layers are as mentioned in the schematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' (b) Schematic energy band diagram for a VO2/LAO/TiO2-x heterostructure before and after the junction formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Electron accumulation is expected based on the known band offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The color intensities are chosen to be proportional to expected electron densities for better visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' EC, EV, and EF indicate the conduction band edge, valence band edge, and Fermi level, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' (c) High-resolution cross-sectional high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) image showing abrupt interfaces between TiO2 substrate and VO2 film and VO2 film and the amorphous LAO spacer layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' (d) Elemental mapping using energy dispersive x-ray spectroscopy (EDS) showing the various layers in the heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Note that the scales of (c) and (d) are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' AO LAO XOll VO2 VO2 1nmLAO cappinglayer 3 nm TiO2-x dopantlayer 2 nm LAO spacerlayer 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm vO2 (001) channe TiO2 (001) TiO2 vO2 LAO nm nm 110nm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='10nm 10nm 6 Heterostructures with VO2 films thinner than ~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm were not attempted due to the expected titanium interdiffusion at the VO2 film and TiO2 substrate interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='45,46 We note that interfacial titanium interdiffusion will be present in thicker films as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' However, at VO2 thicknesses greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm, there is still an observable MIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' A summary of the \uf071-2\uf071 X-ray diffraction (XRD) measurements, performed at room temperature, for a 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 film and the VO2/LAO/TiO2-x/LAO heterostructures on TiO2(001) substrates are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Clearly, the angular positions of the VO2 (002) Bragg reflection peaks are identical for both the 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 film (purple spectrum) and the 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 heterostructure (blue spectrum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Furthermore, it is evident that the angular position of the Bragg reflection is essentially independent of the thickness of the VO2 film in the heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Additionally, there were no significant changes in the \uf071-2\uf071 X-ray diffractograms between VO2 films and heterostructures with the same VO2 thickness (SI Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Reciprocal space maps also confirm that all samples are coherently strained in the plane of the TiO2(001) substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The out- of-plane rutile c-axis lattice parameter is identical for the thin film and the heterostructures for all thicknesses of VO2 (SI Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Based on the \uf071-2\uf071 XRD measurements, reciprocal space maps, and cross-sectional STEM imaging we conclude that the lattice parameter changes, if any, are within the instrumental resolution (better than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='1 pm) for all the VO2 heterostructures used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' We also note that the reflections for the LAO spacer and capping layers and the TiO2-x dopant layer are absent, suggesting that these layers are not crystalline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Next, we discuss the variations in the MIT characteristics for the same set of samples as used in the XRD studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 2b, TMIT systematically decreases with decreasing VO2 thickness in the heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Note that the decrease in TMIT for thin films of VO2 is thickness independent (SI Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S6), suggesting that the decrease in TMIT for VO2 heterostructures is intrinsic to heterostructure formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Furthermore, the sheet resistance of VO2 heterostructures in the insulating state also decreases (SI Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Except for the heterostructure with the thinnest VO2 layer (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5nm), all films continue to show a positive temperature coefficient of resistance, suggesting metallicity above TMIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' (a) High-resolution θ-2θ XRD spectra for the 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 thin film and for VO2 heterostructures with varying thicknesses, t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' For nomenclature simplicity, we distinguish VO2 thin films and heterostructures with a VO2 thickness of ‘t’ as tVO2 and tVO2-het, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' For example, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO2-het corresponds to a heterostructure with 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm thick VO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' (b) Resistance versus temperature plots for the same set of samples as shown in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Resistance values presented here are normalized to the resistance at 330 K and were measured in Van der Pauw geometry (also see SI Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Clearly, TMIT decreases with decreasing VO2 thickness in the heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' (c) A comparison of the changes in TMIT versus the changes in the rutile C- axis lattice parameter (ΔTMIT vs ΔCR) for this work and other previously published work using W9- and Mo10-doping (red and blue respectively), oxygen vacancy doping22 (gray) and strain11 (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The relative changes are compared to the undoped and unstrained states in the case of bulk doping and for strained VO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' For the modulation-doped heterostructures, the reference state is taken as a 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 thin film on TiO2 (001) substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' (a) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,-het 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,- het TTiO,(002)R 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,-het VO,(002)R 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,- het 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,-het Intensity 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,- het 60 62 64 66 68 70 20(°) (b) 10 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,-het 105 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,- het 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,- het K 10* 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,- het R(T) / R(at 330 h 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5vO,- het 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5vO,- het 10 10 10° 200 220 240 260 280300320340 Temperature (K) 40 O-VO, Heterostructures (this work) O-MoxV1-x02 0 0:000 40 MIT 80 △T 120 OWxV1x02 O- TiO2-x / VO2 160 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='.Strained VO, 0 1 2 △C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' (pm) 8 We summarize our observations and compare the changes in TMIT with other previously published studies in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 2c (also see SI Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S8 for details regarding TMIT calculation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Significantly, there is a nearly 60 K change in TMIT for the thinnest heterostructures without any measurable changes to the rutile C-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' In contrast, any comparable change in TMIT in the literature is associated with \uf044CR greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' This demonstrates control over the MIT in VO2 without any measurable structural changes in the VO2 heterostructures proposed and synthesized in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' We note that a decrease in TMIT was observed for elemental doping of VO2 with n-type dopants such as W and Mo, while an increase in TMIT was observed for hole-doping with elements such as Cr and Al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' There is no W or Mo in any of the heterostructures in this work, and both La and/or Al doping can be ruled out because they would (if anything) lead to hole-doping resulting in an increase in the TMIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' This is contrary to the decrease in TMIT observed in these VO2 heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' To measure the extent and type of doping, we performed temperature-dependent Hall measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Hall measurements show an enhancement in the carrier densities in the insulating state with the Hall coefficient indicative of electron-doping (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' On the other hand, carrier densities in the insulating phase increased from ~6×1017 cm-3 (for 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 thin film) to ~2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='8×1019 cm-3 (for 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 heterostructure) to a highest of ~5×1021 cm-3 for the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Contrastingly, the metallic state carrier densities remain identical (~6×1022 cm-3) across all the VO2 heterostructures and are consistent with previous reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='12,24 However, carrier mobility decreases in both the insulating and metallic states as the thickness of the VO2 layer decreases (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' In the metallic state, this is potentially due to contributions from interfacial scattering, which increases with decreasing film thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' In the insulating state, the decrease in carrier mobility could result in part from increased electron- electron scattering and interfacial scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' A summary of the changes in carrier concentration is plotted against TMIT in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' There is a clear correlation between the TMIT and the carrier density with the highest carrier density of ~5x1021 cm-3 stabilizing the metallic state of VO2 to a TMIT of ~237 K (SI Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Importantly, the continuous increase in carrier density with decreasing VO2 thickness without any lattice parameter changes is suggestive of modulation-doping in VO2 heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Plots of temperature-dependent (a) carrier densities and (b) carrier mobilities for tVO2 and tVO2-het samples as mentioned in the legends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Carrier density in the insulating state increases with decreasing VO2 thickness while carrier mobility decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' (c) A phase diagram from the results in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The dotted lines connected across the data points are guide to the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' (a) 10 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO2 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5vO,- het 79 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='.7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='- het .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='.4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5vO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='- het .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='.3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5vO,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' het .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,- het 1017 200 220 240 260 280 300 320 Temperature (K) (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='.9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='.9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' het .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='.7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5vO, het .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='.4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5vO,- hei 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5vO-het .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,-het 0 200 220 240 260 280 300 320 Temperature (K) (c) 10 8 6 Metallic Phase 4 2 Insulating Phase 0 230 240 250 260 270 280 290 300 MIT(K) 10 To further establish that most of the carriers are induced by modulation doping, we prepared two additional control samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The first is a 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 thin film with a 2 nm LAO cap layer (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO2-LAO) and the second is a 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 heterostructure with a 3 nm stoichiometric TiO2 layer (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO2-LAO-TiO2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' We compared the MIT characteristics of these two samples with the MIT characteristics of the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 thin film (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO2) and a 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 heterostructure with a TiO2-x dopant layer (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO2-het).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' A summary of sheet resistance versus temperature data is shown in SI Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 heterostructure with a TiO2-x dopant layer has the lowest sheet resistance and the lowest TMIT with a TMIT change of ~20 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' By contrast, the decrease in TMIT was restricted to ~6 K after the deposition of the 2 nm LAO layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Remarkably, there is no further decrease in TMIT in the heterostructure with stoichiometric TiO2 layer, suggesting that the TiO2-x dopant layer is required for the observed large change in TMIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Consistent with this, the carrier density in the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 heterostructure with the TiO2-x dopant layer is ~7x1019 cm-3 compared to a carrier density of 2x1019 cm-3 for the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 capped with 2 nm LAO (SI Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' It is possible that amorphous (disordered) LAO could host ionized donors and enable modulation-doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='29 However, we found that the amorphous LAO deposited for these experiments is insulating, suggesting that any ionized donors should be below the measurement threshold of electrical resistivity measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' We estimate that such ionized donors in LAO, if any, should have a carrier density of ~1019 cm-3 (assuming a carrier mobility of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='01 cm2/V-s) or lower, putting an upper bound on the number of carriers contributed by the spacer layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' To probe the electronic band bending that enables electron accumulation in the VO2 channel layer, we performed bulk-sensitive hard X-ray photoelectron spectroscopy (HAXPES)47 measurements at the P22 beamline in the PETRA III synchrotron at DESY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' We note that standard ultra-violet photoemission (UPS) or soft X-ray photoemission measurements cannot facilitate a probing depth sufficient to reach the VO2/LAO interface that is buried beneath multiple layers of the heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' To capture interfacial band bending in VO2, HAXPES measurements were performed in both the insulating phase (at 200 K) and the metallic phase (at 310 K) for VO2 heterostructures with VO2 thicknesses of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm, and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm, and for a VO2 thin film with a thickness of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm as a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 11 For the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 film measured at 200 K (insulating state), the binding energies of the V 2p3/2 and V 2p1/2 core-level peaks were observed to be ~515.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='8 eV and ~523.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='1 eV, respectively (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' These measured binding energies (see SI Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S11 for binding energy calibration procedure) are consistent with previous reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='48–50 Importantly, a systematic shift of the main component of the V 2p3/2 peak to higher binding energies is observed for the VO2 heterostructures, with the highest increase in binding energy (~250 meV) observed for the heterostructure with the thinnest VO2 layer (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm), as seen in the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' This is also in agreement with the highest carrier densities and the lowest TMIT being observed for the heterostructures with the thinnest VO2 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' For measurements performed on VO2 in the metallic state, no such binding energy shift was observed (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 4b and SI Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' This is consistent with the complete screening of interfacial electric fields at the metallic VO2/LAO interface (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 4c and d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The presence of binding energy shifts observed in the insulating state of VO2 and their absence in the metallic state of VO2 further support carrier doping by chemical potential shifts (modulation-doping) in the insulating state for VO2 heterostructures as proposed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Photoemission data also showed two remarkable features in the V 2p spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The first, labelled P1 in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 4a and 4b, is a lower binding energy shoulder around ~514.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 eV in the insulating and metallic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The presence of this spectral feature at lower binding energies was proposed to signify non-local screening from coherent 3d1 states near EF in the metallic phase of VO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='50,51 Interestingly, the intensity of P1 in the insulating state increases with increasing carrier density and decreasing VO2 thickness in the heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The emergence of this peak in the insulating state spectra for VO2 heterostructures suggests that the additional charge transferred to the VO2 channel layer enables non-local screening that was previously observed only in the metallic phase of VO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' To further quantify the evolution of the P1 peak across the metallic and insulating phases, we compared the metallic and insulating state spectra of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 heterostructures in SI Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The intensity of the P1 peak in the insulating state of the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 heterostructure is observable but small in comparison to the P1 peak in the metallic state (SI Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S13a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The difference spectrum shows a large difference at ~514.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 eV (SI Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S13c) further confirming that the non-locally screened shoulder is negligibly small in the insulating state when compared to the metallic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Remarkably, the non-locally screened shoulder is quite predominant in the 12 insulating state spectra of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 heterostructure (SI Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S13b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The intensity difference spectra between the metallic and insulating states shows a very small difference between the two phases (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S13d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' These trends are consistent with the differences in the carrier densities between the metallic and insulating phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The carrier density ratio between the metallic and insulating states is about 10 for the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 heterostructure compared to about 1000 for the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The second remarkable feature in the photoemission data is a high binding energy spectral feature at ~517.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' This feature is associated with the V 2p3/2 peak and labelled P2 in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 4a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' A corresponding feature is also observed for the V 2p1/2 peak but is more smeared out and appears as broadening on the higher-binding-energy side at ~525 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' In general, higher binding energy spectral features are associated with higher oxidation states in photoemission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The presence Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' A comparison of V 2p core-level spectra of modulation doped VO2 heterostructures for (a) the insulating (200 K) and (b) the metallic states (at 310 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' A clear shift in the V 2p levels is seen in the insulating state spectra but not in the metallic state spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Schematics in (c) and (d) show the expected band-bending in the modulation-doped heterostructures for the insulating and metallic states respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Band-bending is expected in the insulating state and not in the metallic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Two additional spectral features not seen in VO2 thin films are labelled P1 and P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' (a) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='0 (b) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' izedintesity 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5vO,- het 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,-het 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5vO,- het 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5vO,- het 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='0 P2 Tintensi P2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5vO,- het Normali 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,-het 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,- het 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='918 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,- het 517 516 515 517 516 515 Binding energy (eV) Binding energy (eV) lized P1 P1 Measured at Measured at Normal 200 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 310K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 V2P/2 V2p1/2 V2p, V2p 3/2 3/2 0 0 525 520 515 510 525 520 515 510 Binding energy (eV) Binding energy (eV) (c) LAO LAO (d) LAO LAO 2o!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' VO2 (1) TiO 2-X VO2 (I) x2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' VO2 (M) 2o!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' VO2 (M) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO, film 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO, film 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO, film 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO, film 13 of V5+ is a possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' However, an increase in the oxidation state from V4+ to V5+ cannot explain the observed electron-doping in VO2 heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Generally, electron doping should decrease the V4+ oxidation state in VO2 and therefore, an increase in the oxidation state of vanadium cannot explain the increase in electron density in the VO2 heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Therefore, V5+, even if present, has no bearing on the MIT observed in heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' P2 was also observed in VO2 samples capped with 2 nm LAO (SI Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Therefore, we have also inspected the La 3d5/2 and Al 1s spectra to look for any chemical shifts associated with a redox or chemical reaction at the VO2/LAO interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' As shown in SI Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S15, there are no observable changes to the spectra across heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Finally, interfacial oxygen vacancy creation remains a possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' However, any oxygen vacancy creation should lead to V3+ and an associated low-binding-energy feature in both the metallic and insulating state spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' However, the spectra do not show any signatures of oxygen vacancies in VO2 heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Therefore, we rule out any oxygen-vacancy induced carrier doping in these heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Furthermore, the intensity of P2 is carrier density dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' For all V 2p spectra in the metallic state, where the carrier density is independent of the VO2 thickness, the intensity of this additional peak relative to the main V 2p3/2 peak is also independent of the VO2 thickness, with the intensity ratio of P2 to V 2p3/2 being close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Contrastingly, the intensity of P2 increases with the decreasing film thickness in the insulating state of VO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The intensity ratio of P2 to V 2p3/2 approaches the intensity ratio observed for the metallic state spectra at the highest carrier density of ~5x1021 cm-3 in the insulating state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' These carrier-density-dependent changes suggest that this new spectral feature is intrinsic to the heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' However, this additional spectral feature might benefit from further spectroscopic investigation with complementary techniques such as X-ray absorption spectroscopy to confirm its origins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The combination of electron transport and HAXPES data show that VO2 heterostructures facilitated effective modulation doping and carrier densities as high as 5x1021 cm-3 could be achieved using this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The highest carrier densities correspond to electron doping of ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='2 e-/vanadium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' This is an extremely high dopant density at which conventional rigid band models predict metallization in doped correlated insulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='52 Bulk-sensitive valence-band HAXPES spectra recorded for the same set of heterostructures as discussed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 4 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 5 and SI Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The insulating-state spectra for all VO2 14 heterostructures (blue) exhibit nearly zero spectral intensity at the Fermi level while an appreciable non-zero spectral intensity is observed for the higher-temperature metallic-state spectra (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' These spectra further confirm that VO2 continues to undergo a MIT even in the presence of electron densities as high as ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='2 e-/vanadium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The presence of MIT at such high doping levels, without any observable changes in the lattice parameters (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 2a and SI Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S4), points to a possible renormalization of the electronic structure with doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Conclusions: In summary, we demonstrated a purely electronic control of the MIT in modulation-doped VO2 heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Our work shows that the insulating state in VO2 is surprisingly robust even in the presence of electron doping as high as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='2 e-/vanadium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Notably, all the films meet the Mott criterion (𝑎𝐵 ∙ 𝑛𝐶 1 3 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='25, where 𝑎𝐵 is the effective Bohr radius and 𝑛𝐶 is the carrier density).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Therefore, metallicity is expected at all temperatures based on the carrier densities achieved in these experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' We note that a similar robust insulating state had also been found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' A comparison of the V 3d valence band (VB) spectra of modulation-doped VO2 heterostructures for the insulating (200 K, blue) and metallic states (at 310 K, orange) for different VO2 film thicknesses of (a) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 film and (b) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm, (c) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm, (d) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm, and (e) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' There is a clear spectral weight shift across the MIT for all the samples with the insulating state being robust even for the heterostructure with a VO2 thickness of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm, corresponding to carrier doping of ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='2 e-/Vanadium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' (a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='.7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,at200K (b) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='-hetat200K (c) @4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,-hetat200K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,at310K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,-hetat310K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='.4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='-hetat310K 60 Intensity o cco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='o Intensity Intensity : GO CO V3d V3d V3d 2 1 0 2 1 0 2 1 0 Binding Energy (eV) Binding Energy (eV) Binding Energy (eV) (d) o---3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,-hetat200K (e) ---- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='-hetat200K --3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO-hetat310K @.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='. 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,-hetat310K EOD Intensity Intensity co V3d V3d 2 1 0 2 0 BindingEnergy(eV) BindingEnergy(eV) 15 modulation-doped nickelate thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='27 Perhaps, the development of theoretical models that go beyond the conventional carrier concentration independent rigid-band models will be required to understand electronic phase transitions in correlated electron oxides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' An alternate explanation could be that the insulating state is favored at lower thicknesses due to interfacial disorder-induced Anderson-like localization of carriers, which will be more pronounced for the thinnest films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Further experiments will be needed to assess whether the insulating state is stabilized by the interfacial disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' A remarkable feature of this work is the possibility of bulk metallization in modulation- doped VO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' In general, in band semiconductors such as SrTiO3, conductivity modulation is achieved over a thickness of 1-2 nm in the vicinity of the channel/spacer interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='29,53 In this work, a sharp MIT is observed for heterostructures with VO2 thicknesses as high as 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm, which are much higher than the Thomas Fermi screening length of ~1 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' This is suggestive of the entirety of the film being metallized at the lowered TMIT after modulation doping, even though the charge transfer densities are the highest at the interface (within 1-2 nm of the interface).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' While further studies are required to establish this beyond doubt, interfacial-doping induced bulk metallization of correlated electron insulators has implications for low-power electronics with high ON/OFF ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='28,54 Finally, our work shows that modulation doping is a powerful technique for achieving high carrier densities, close to those possible with elemental doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Since our approach does not need any epitaxially matched spacer and dopant layers, it expands the library of materials that can be explored for the study of modulation-doping-induced electronic phase transitions of other related CEMs including complex oxides2 and pyrochlores55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' This methodology, therefore, paves the way for exploring ‘pure’ electronic effects in correlated oxides and related systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Such studies could also enable a fundamental understanding of band matching and relevant energy scales in complex oxides and, perhaps enable the discovery of new interfacial phases and devices that rely on phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Methods Prior to deposition, single-crystalline rutile TiO2 (001) substrates (Shinkosa, Japan) were treated using the procedure discussed previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='56 All thin film samples were deposited using PLD (NEOCERA) with a 248 nm KrF laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' All VO2 thin films (both pristine films and 16 heterostructures) were deposited on treated TiO2 substrates from a sintered V2O5 target with a laser fluence of ~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 J/cm2, 8 mTorr of oxygen pressure, and a growth rate of ~4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='7×10-2 Å/pulse at a substrate temperature of 425 °C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='56 For all heterostructure samples, 2 nm thick LAO spacer layers were deposited at 10 mTorr of O2 pressure at a growth rate of ~5×10-2 Å/pulse using a single- crystalline LAO target (Shinkosa Japan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 3 nm thick TiO2-x dopant layers were then deposited using a TiO2-x single-crystalline target (Shinkosa Japan) at a growth rate of ~4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='2×10-2 Å/pulse in 10-5 Torr of background vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Finally, a 1 nm thick LAO layer was deposited under the same conditions used for the LAO spacer layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Depositions of the spacer, dopant, and capping layers were all done at room temperature at a laser fluence of ~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='2 J/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' For all depositions, the substrate-to-target distance was maintained at 55 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' High-resolution Cu-Kα X-ray diffraction spectra for both the pristine and heterostructure VO2 films were recorded in standard θ–2θ geometry using a Rigaku Smart Lab X-ray diffractometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' LEPTOS 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='8 software by Bruker was used to determine film thicknesses and used to calculate the differential strain between pristine and heterostructure films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Cross-sectional scanning transmission electron microscopy (STEM) imaging and electron dispersive x-ray spectroscopy (EDS) mapping were performed using TITAN Themis microscope (60-300 kV) equipped with a probe corrector and super-X four quadrant EDS detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The high angle annual dark field (HAADF)-STEM images were acquired at an operating potential of 300 kV with a convergence angle of 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 mrad, 160 mm camera length, and a dwell time of 12 μs per pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The images were further processed with Gatan digital micrograph software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The EDS maps were acquired using Velox software under similar microscopic conditions with a dwell time of 2 μs per pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Sheet resistance vs temperature measurements were performed in Van der Pauw geometry using Keithley 2450 SMU and Eurotherm 2408 PID temperature controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Continuous temperature scanning was carried out at a rate of 4 K/minute for both heating and cooling cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' To extract carrier density and mobility, Hall measurements for all films and heterostructures were performed using Van der Pauw geometry in a PPMS-Dynacool equipment from Quantum Design and Keithley SMU 2450 from Tektronix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' For these measurements, the magnetic field was swept from 0 T to 2 T to -2 T to 0 T at a scan rate of 100 Oe/s for different temperatures ranging from 200 K to 320 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 17 High-resolution hard X-ray photoelectron spectroscopy (HAXPES) measurements57 were carried out with an incident photon energy of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='2 keV at the sample temperatures of 200 K (insulating phase) and 310 K (metallic phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Binding energy calibration was carried out using a high-resolution Fermi-edge measurement on a standard Au sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Core-level and valence-band spectra were measured using a wide acceptance angle SPECS Phoibos 225HV hemispherical electrostatic analyzer in a near-normal emission geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The total energy resolution was estimated to be approximately 320 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Preliminary HAXPES measurements and sample screening were carried out using a lab-based HAXPES instrument at Temple University equipped with a wide acceptance angle ScientaOmicron EW4000 analyzer at a total experimental energy resolution of 450 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Acknowledgments: AXG, AMD, and JRP acknowledge support from the DOE, Office of Science, Office of Basic Energy Sciences, Materials Sciences, and Engineering Division under Award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' DE-SC0019297.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The electrostatic photoelectron analyzer for the lab-based HAXPES measurements at Temple University was acquired through an Army Research Office DURIP grant (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' W911NF-18- 1-0251).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' We acknowledge DESY (Hamburg, Germany), a member of the Helmholtz Association HGF, for the provision of experimental facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Beamtime at DESY was allocated for proposal I-20210142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Funding for the HAXPES instrument at beamline P22 by the Federal Ministry of Education and Research (BMBF) under framework program ErUM is gratefully acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' PN and RKR acknowledge the Advanced Facility for Microscopy and Microanalysis (AFMM) for providing the electron microscope and FIB facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' acknowledges support from the startup grant at Indian Institute of Science (SG/MHRD-19-0001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The authors acknowledge CeNSE, IISc for access to HR-XRD, wire bonding, and clean-room facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' acknowledges the new faculty startup grant provided by the Indian Institute of Science under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 12-0205-0618- 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' is thankful to Professor Anil Kumar for access to the PLD system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' want to thank Jibin J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Samuel and Mithun Ghosh for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' We thank Professor Satish Patil for providing access to facilities supported by the Swarnajayanti fellowship under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' DST/SJF/CSA-01/2013-14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' AFM measurements were performed on a Cypher-ES AFM funded by the DST-FIST program and Hall measurements were performed on a PPMS-Dynacool system funded under the UGC-CAS program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 18 References: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Dagotto, E.' metadata={'source': 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single-crystalline VO2 (101) thin films with sharp metal- insulator transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Journal of Applied Physics 126, 215302 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Schlueter, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The new dedicated HAXPES beamline P22 at PETRAIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' AIP Conference Proceedings 2054, 040010 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 1 Supplementary Information Modulation-Doping a Correlated Electron Insulator Debasish Mondal1, Smruti Rekha Mahapatra1, Abigail M Derrico2, Rajeev Kumar Rai3, Jay R Paudel2, Christoph Schlueter4, Andrei Gloskovskii4, Rajdeep Banerjee1, Frank M F DeGroot5, Dipankar D Sarma1, Awadhesh Narayan1, Pavan Nukala3, Alexander X Gray2* and Naga Phani B Aetukuri1* Affiliations: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Solid State and Structural Chemistry Unit, Indian Institute of Science, Bengaluru, Karnataka, 560012, India 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Department of Physics, Temple University, 1925 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 12th St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=', Philadelphia, PA 19122, USA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Centre for Nano Science and Engineering, Indian Institute of Science, Bangalore, Karnataka, 560012, India 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Deutsches Elektronen-Synchrotron, DESY, 22607 Hamburg, Germany 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Utrecht University, Inorganic Chemistry and Catalysis Group Universiteitsweg 99, 3584 CA Utrecht, The Netherlands Corresponding authors emails: phani@iisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='in (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' axgray@temple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='edu (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='G) 2 Table of Contents: S1: Reflection High Energy Electron Diffraction (RHEED) of VO2 thin films .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 3 S2: AFM images of VO2 thin film and heterostructure .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='. 3 S3: Thomas-Fermi screening length calculation for VO2 heterostructure .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='. 4 S4: A comparison of X-ray diffractograms for VO2 thin films and heterostructures .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 5 S5: Reciprocal space maps (RSM) of VO2 thin film and heterostructures .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 6 S6: Temperature-dependent sheet resistance of VO2 thin films .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 7 S7: Temperature-dependent sheet resistance of VO2 thin film and heterostructures .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 8 S8: Calculation of TMIT from sheet resistance vs temperature curves .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='. 9 S9: Temperature-dependent sheet resistance of VO2 heterostructures and controls .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='. 10 S10: Carrier density and carrier mobility for 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 thin films and heterostructures .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='. 11 S11: Binding energy calibration of VO2 spectra across the MIT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 12 S12: Summary of Binding energy changes for modulation-doped VO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 13 S13: Evolution of P1 peak in metallic and insulating VO2 heterostructures .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='. 14 S14: HAXPES spectra of V 2p3/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='. 15 S15: HAXPES spectra of La 3d and Al 1s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 16 S16: Valence band spectra in insulating state for VO2 heterostructures and thin films .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 17 3 S1: Reflection High Energy Electron Diffraction (RHEED) of VO2 thin films Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' RHEED pattern of VO2 thin films on TiO2 (001) substrate along the <110> direction for two different thicknesses of (a) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm and (b) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' These patterns were captured at 425 °C just after the VO2 deposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' RHEED patterns are indicative of single-crystalline VO2 films with smooth film surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S2: AFM images of VO2 thin film and heterostructure Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' AFM images of (a) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 film and (b) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 heterostructure on TiO2 (001) substrate show smooth 2D surfaces with a root mean-square roughness in the range of 80-100 pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The images were taken in the tapping mode (AC air topography) using an Asylum Cypher ES AFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' (a) (b) <110> <110>(a) (b) pm pm 750 750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='00 400nm 750 400nm 750 4 S3: Thomas-Fermi screening length calculation for VO2 heterostructures Thomas-Fermi screening length (L) can be calculated as follows 𝐿 = √𝐾𝜀0𝑇𝑘𝐵 𝑒2𝑛𝑒 where critical carrier density, 𝑛𝑒 ≈ ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='25 𝑎𝐵 ) 3 and 𝐾, 𝜀0, 𝑘𝐵, 𝑇, 𝑒, 𝑎𝐵 are the dielectric constant (≈ 36)1,2 of VO2, free space permittivity (≈ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='854x10-12 F/m ), Boltzmann constant (≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='380x10-23 m2 kg s-2 K-1), temperature of VO2, electronic charge (≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='602x10-19 C) and effective Bohr radius respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=" Effective Bohr radius (𝑎𝐵) can be calculated as follows 𝑎𝐵 = ℎ2𝐾𝜀0 𝜋𝑚∗𝑒2 where ℎ is the Planck's constant (≈ 6." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='626x10-34 m2kg/s) effective mass of electron in VO2, 𝑚∗ ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5𝑚𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='2,3 By plugging these above values and keeping T ≈ 300 K, calculated Thomas-Fermi screening length (L) is ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='73 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 5 S4: A comparison of X-ray diffractograms for VO2 thin films and heterostructures Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Comparison of high-resolution \uf071-2\uf071 X-ray diffractograms of VO2 (001) thin films (black) and heterostructures (red) at different VO2 thicknesses of (a) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm, (b) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm, (c) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm and (d) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' We showed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 2a of the main manuscript that there is no measurable change in the VO2 lattice parameter, CR, between VO2 thin films and heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' To further confirm this, we performed and compared high-resolution X-ray diffractograms for thin films and heterostructures with identical VO2 thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Clearly, there is excellent overlap of the two diffractograms, including thickness fringes, for all thicknesses measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' This is clear indication that depositing heterostructures on ultra-thin VO2 films did not affect the lattice parameters of VO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' a 0 0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO, (b 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO TiO,(002) TiO, (002) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='-het 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=" het 10-1 10' Normalized intensity Normalized intensity 3 VO,(002)R VO,(002)R 10 10 10 10 5 10-6 10~6 10° 10 60 62 64 66 68 70 60 62 64 66 68 70 20(°) 20(°) c 4." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' TiO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' (002) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='-het TiO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' (002) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=" het 10-1 10 Normalized intensity 10° 2 Normalized intensity 10'2 10° 3 VO,(002)R 10 10° 2 VO,(002)R 10 10 5 106 106 10 10-7 60 62 64 66 68 70 60 62 64 66 68 70 20(°) 20() 6 S5: Reciprocal space maps (RSM) of VO2 thin film and heterostructures Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Asymmetrical RSM images around (112) plane for (a) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 thin film, and VO2 heterostructures with VO2 thicknesses of (b) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm (c) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm and (d) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The vertical and horizontal axis of all the plots is scaled relative to the miller indices of the TiO2 (001) substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The RSM data presented here is further evidence that there are no measurable changes to the unit cell volume across all heterostructures and that all films are coherently strained to the TiO2 (001) substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' (b) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='12 (c) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='12 (d) a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='10 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='10 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='08 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='08 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='08 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='08 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='06 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='06 100 100 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='04 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='00 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='05 HHO HHO HHO HHO 7 S6: Temperature-dependent sheet resistance of VO2 thin films Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Temperature-dependent sheet resistance of VO2 thin films with varying VO2 thicknesses as mentioned in the legends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' For nomenclature simplicity, VO2 thin films are written as tVO2 where ‘t’ is the thickness of VO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' For example, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO2 corresponds to a 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm thick VO2 thin film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The increase in metallic state resistance as the thickness of VO2 film is decreased is suggestive of interfacial scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' This also shows all the films, except 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO2, have nearly the same transition temperature (~295 K) which again confirms the reduction in transition temperature in VO2 heterostructures is entirely after the formation of the heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The slight increase in transition temperature for the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO2 film is attributed to be due to titanium interdiffusion at the VO2/TiO2(substrate) interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='4,5 We were not able to measure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 thin film as it was found to be not stable when exposed to ambient without a capping layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 10 106 104 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 102 200 240 280 320 Temperature (K) 8 S7: Temperature-dependent sheet resistance of VO2 thin film and heterostructures Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Temperature-dependent sheet resistance of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 thin film and VO2 heterostructures with varying VO2 thicknesses, t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' For nomenclature simplicity, we distinguish VO2 thin films and heterostructures with a VO2 thickness of ‘t’ as tVO2 and tVO2-het, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' For example, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO2-het corresponds to a heterostructure with 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm thick VO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The sheet resistance data in this figure corresponds to the normalized resistance presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 2b of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The increase in metallic state resistance as the thickness of VO2 film is decreased is suggestive of interfacial scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Importantly, it can be seen that the metallic state resistance is identical for the 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 heterostructure and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 thin film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' This is also an indirect indication of the sharp interfaces across the heterostructure layers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Rough interfaces could lead to increased interfacial scattering and a higher metallic state resistance for the 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 heterostructure in comparison with the 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 thin film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 10 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='- het 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='- het (Q/sq-) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='- het 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='- het Sheet resistance ( 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='- het 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='- het 10° 200 220 240 260 280 300 320 340 Temperature (K) 9 S8: Calculation of TMIT from sheet resistance vs temperature curves Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' d(ln(R))/dT plots for both the heating and cooling cycles for VO2 heterostructures with VO2 thicknesses of (a) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm, (b) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm, (c) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm, (d) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm, (e) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm, (f) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm, and, (g) for a 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 thin film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Here, ln(R) is the natural log of the temperature-dependent sheet resistance of the films as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S7 and d/dT is the derivative operator with respect to temperature, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Peak positions of the d(ln(R))/dT plots correspond to the transition temperature for the respective thermal cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' All the transition temperatures discussed in the main text are the average transition temperature of the transition temperature for heating and cooling cycle given by TMIT = ((TMIT)heating +(TMIT)cooling)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The estimated TMIT are: 233 K for 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 237 K for 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 250 K for 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 260 K for 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 275 K for 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 282 K for 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 in VO2 heterostrtuctures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' By comparison, the TMIT is 295 K for 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 thin film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' (a) 0 Heating (b) (c) o Cooling .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='- Heating ---o-- Cooling Cooling 0-00000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='2 d(Ln(R)/dT d(Ln(R)dT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='04 d(Ln(R)/ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,-het 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5vO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='-het 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,-het 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='8 220 240 260 280 300 220 240 260 280 300 220 240 260 280 300 T (K) T (K) T (K) (d) (f) o e Heating Heating 8 Heating 0 Cooling Cooling Cooling 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 d(Ln(R)/dT d(Ln(R)dT d(Ln(R)/dT 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='9 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,-het 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,-het 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,-het 220 240 260 280 300 220 240 260 280 300 220 240 260 280 300 T (K) T (K) T (K) (g) 0 : Heating Q: Cooling : 2 d(Ln(R)/dT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='4 6 220 240 260 280 300 T (K) 10 S9: Temperature-dependent sheet resistance of VO2 heterostructures and controls Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' A comparison of temperature-dependent sheet resistance characteristics of modulation- doped 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 heterostructure (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO2-het) and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 thin film (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO2) with various VO2 heterostructure ‘controls’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The control heterostructures include 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 thin film capped with 2 nm LAO (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO2-LAO) and a 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 heterostructure with a near stoichiometric TiO2 deposited at 10 mTorr of oxygen pressure with a 1 nm LAO capping layer over TiO2 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO2- LAO-TiO2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The number prefixed with each of the layers represent the respective film thickness in nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' For the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO2-LAO heterostructure, there is a reduction in resistivity in the insulating phase and a small but observable reduction in transition temperature (TMIT) ~6 K compared to the TMIT of VO2 thin film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' However, for the modulation-doped heterostructure, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO2-het), there is a further reduction in resistivity with a decrease in TMIT of ~20 K compared to the TMIT of VO2 thin film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' For the heterostructure where TiO2 is near stoichiometric (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO2-LAO-TiO2), the reduction in TMIT is also ~6 K, similar to the reduction in TMIT for 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO2-LAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Clearly, a majority of the reduction in TMIT comes from the charge transfer from the TiO2-x dopant layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 10° (Q/sq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=') 106 Sheet resistance ( 10° 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='-het 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='-LAO-TiO 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,-LAO 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO 102 2 220 240 260 280 300 320 Temperature (K) 11 S10: Carrier density and carrier mobility for 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 thin films and heterostructures Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Plots of temperature-dependent (a) carrier densities and (b) carrier mobilities for 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 heterostructure (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO2-het), 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 thin film (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO2) and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 capped with 2 nm LAO (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO2-LAO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' For the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO2-LAO, there is a noticeable increase in carrier density (and a decreased in carrier mobility) compared to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 thin film, but this is less than the carrier density observed in the comparable modulation-doped heterostructure (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO2-het).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='-het 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,-het 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,-LAO 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,-LAO 1022 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO 10 10 1018 200 220 240 260 280 300 320 200 220 240 260 280 300 320 Temperature (K) Temperature (K) 12 S11: Binding energy calibration of VO2 spectra across the MIT Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Binding energy calibration for the insulating state spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' We noticed charging effects in some of the insulating state HAXPES spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' In order to correct for charging effects, we used O 2p binding energy across insulating and metallic states of the samples as an internal reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' We note that metallic state spectra are unaffected by charging affects and several previous studies showed that binding energy of O 2p spectra does not change across the MIT6–9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Further, no changes to the O 2p contributions from the LAO and TiO2 layers are expected across the MIT in VO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The valence band spectra for (a) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 film and VO2 heterostructures with VO2 thicknesses of (b) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm, (c) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm, (d) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm, and (e) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm are shown for both the metallic phase (at 310 K) and the insulating phase (at 200 K) after binding energy correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' All the V 2p spectra shown in the main text are based on this calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='.7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5vO, at200K b 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,-het at 200K @.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='.4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5vO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='-het at200K 0 @.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='.7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5vO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='at310K --7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5vO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='-het at 310K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='0 2p 28 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='.5vO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='-het at 310K 2p 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='0 I Intensity Normalized Intensity Normalized Intensity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='8 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='2 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 3d 0 0 0 12 10 8 6 4 2 0 2 12 10 8 6 4 2 0 12 10 8 6 4 2 0 2 Binding Energy (eV) Binding Energy (eV) Binding Energy (eV) d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='- 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,-het at 200K e : 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5vO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='-het at 200K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='. 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='. hetat310k 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='0 2p @.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='-het at 310K 2p 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='0 Intensity Intensity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='6 Normalized Normalized I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='2 V V 3d 3d 0 0 12 10 8 6 2 0 12 10 8 6 4 0 2 Binding Energy (eV) Binding Energy (eV) 13 S12: Summary of Binding energy changes for modulation-doped VO2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The change in binding energy of the V 2p3/2 peak for VO2 heterostructures with respect to its binding energy in VO2 thin films (\uf044BE) for various thicknesses of VO2 in heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' There is no binding energy change in the metallic state (orange, at 310 K), while there is a binding energy increase for the same set of samples in the insulating state (blue, at 200 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' This is suggestive of band-bending leading to modulation-doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' (a) At 310 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='3 At 200 K (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='2 3/2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='1 0 2 4 6 8 VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' thickness (nm) 14 S13: Evolution of P1 peak in metallic and insulating VO2 heterostructures Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' V 2p core level spectra for (a) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm and (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 heterostructures in the metallic (at 310 K) and insulating (at 200K) states normalized to the V 2p area under the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Corresponding intensity difference plots in (c) showing that the P1 peak (at ~514.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 eV) is prominent in the metallic phase for the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 heterostructure (negative intensity difference) while for the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm heterostructure in (d) the intensity difference decreases suggesting an increase in the P1 intensity for the insulating state spectrum for the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' (a) (b) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='. 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,-het at 200K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='-het at 200K 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,-het at 310K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='-het at 310K Normalized Intensity Normalized Intensity 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='0 P2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 V2p 12p V21 312 0 0 525 520 515 510 525 520 515 510 Binding Energy (eV) Binding Energy (eV) (c) (d) (at 200 K) - I(at 310 K) t 200 K) - I(at 310 K) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='-het 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO-het 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 525 520 515 510 525 520 515 510 Binding Energy (eV) Binding Energy (eV) 15 S14: HAXPES spectra of V 2p3/2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' A comparison of V 2p core-level spectra of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm modulation doped VO2 heterostructure, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 with 2 nm LAO capping layer and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 thin film in (a) the insulating (200 K) and (b) the metallic states (310 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The additional peak P2 is observed after the deposition of LAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' However, the change in the peak P1 in the insulating state spectra is not significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' (a) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5V0 6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='0 P2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='- LAC het 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='- het Normalized intensity Normalized intensity Measured at 200 K Measured at P2 P1 Pr 310K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 V2p V2P3/2 V2p, 12 3/2 0 0 525 520 515 510 525 520 515 510 Binding energy (eV) Binding energy (eV) 16 S15: HAXPES spectra of La 3d and Al 1s Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' A comparison of core level HAXPES spectra of modulation doped VO2 heterostructures of (a) La 3d5/2 and (b) Al 1s at 200 K, and (c) La 3d5/2 (d) Al 1s at 310 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' In the insulating state (at 200 K), the spectra were collected for VO2 heterostructures corresponding to all VO2 thicknesses used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' In the metallic state (at 310 K), the spectra were collected for 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 nm VO2 heterostructures and for 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO2/2LAO heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' No significant changes to the La and Al core level were observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='- het b Al 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,- het 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,- het 15 La.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5/2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,- het d intensity 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,- het I intensity 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,- het 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='8 —7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,- het 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,- het 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,- LAO 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='- LAO Normalized 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='6 Measured at Normalized 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='6 Measured at 200 K 200 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='2 0 0 845 840 835 830 1558 1560 1562 1564 Binding energy (eV) Binding energy (eV) (c) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,- het 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='- het 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='0 Al 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,- het La .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5/2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,- het intensity 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO LAO Normalized intensity 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='- LAO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='8 Measured at Measured at 310 K 310 K Normalized 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='2 0 0 845 840 835 830 1558 1560 1562 1564 Binding energy (eV) Binding energy (eV) 17 S16: Valence band spectra in insulating state for VO2 heterostructures and thin films Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' S16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' (a) Valence band spectra (O 2p and V 3d) and, (b) V 3d spectra (zoom in of figure S16 a) measured at 200 K are shown here after the binding energy correction as discussed in S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' The V 3d peak position at ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='9 eV is in good agreement with existing literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='6–9 The vertical dotted line at 0 eV corresponds to the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Supplementary References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Yang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=', Ko, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=', Balakrishnan, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=', Gopalakrishnan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' & Ramanathan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Dielectric and carrier transport properties of vanadium dioxide thin films across the phase transition utilizing gated capacitor devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' B 82, 205101 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Paik, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Transport properties of ultra-thin VO2 films on (001) TiO2 grown by reactive molecular-beam epitaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 107, 163101 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Rosevear, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' & Paul, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Hall Effect in VO2 near the Semiconductor-to-Metal Transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} 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single-crystalline VO2 (101) thin films with sharp metal- insulator transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' Journal of Applied Physics 126, 215302 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' (a) b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='08 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='0 2p 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='-het 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='-het V 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,-het 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,-het 3d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='06 intensity 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO het Normalized intensity 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5VO,-het 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content=' het Measured at Measured at Normalized 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 200 K 200 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='02 V 3d 0 0 12 8 4 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQf9QJL/content/2301.02798v1.pdf'} +page_content='5 1.' metadata={'source': 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b/sNE5T4oBgHgl3EQfKQ5P/content/tmp_files/2301.05464v1.pdf.txt @@ -0,0 +1,852 @@ +arXiv:2301.05464v1 [gr-qc] 13 Jan 2023 +PDM KG-oscillators in cosmic string rainbow gravity spacetime in a non-uniform +magnetic field +Omar Mustafa∗ +Department of Physics, Eastern Mediterranean University, G. Magusa, north Cyprus, Mersin 10 - Turkey. +Abstract: We consider position-dependent mass (PDM) Klein-Gordon (KG) particles in cosmic +string rainbow gravity spacetime in a non-uniform magnetic field. The corresponding KG-equation +is reduced into the one-dimensional form of the two-dimensional radial Schr¨odinger-oscillator like +equation (hence the notion KG-oscillator). We first report on the effects of rainbow gravity on the +energy levels of KG-oscillators with constant mass. Next, we include the PDM settings so that +KG-oscillators like interaction are introduced. The effects of PDM on the spectra of KG-oscillators +in cosmic string rainbow gravity spacetime are also reported. In both cases four pairs of rainbow +functions are considered: (a) g0 (y) = 1, g1 (y) = +� +1 − ǫy2, (b) g0 (y) = 1, g1 (y) = √1 − ǫy, (c) +g0 (y) = g1 (y) = (1 − ǫy)−1, and (d) g0 (y) = (eǫy − 1) /ǫy, g1 (y) = 1. +PACS numbers: 05.45.-a, 03.50.Kk, 03.65.-w +Keywords: Klein-Gordon (KG) particles, position-dependent mass, cosmic string spacetime, +rainbow gravity, non-uniform magnetic field. +I. +INTRODUCTION +In quantum gravity (QG), the semi-classical model of rainbow gravity (RG) (an extension of the deformed or +doubly special relativity into a general relativity (GR) framework) has attracted research attention over the years +[1–5]. Under the RG-model, the energy of the probe particles is assumed to affect the spacetime background, at the +ultra-high energy regime, so that the spacetime metric become an energy-dependent one [5–12]. Hereby, the Planck +energy Ep = +� +ℏc5/G plays the role of a threshold separating the classical description from the quantum mechanical +one and introduces itself as another invariant energy scale alongside the speed of light. Consequently, rainbow gravity +justifies the modified relativistic energy-momentum dispersion relation +E2g0 (y)2 − p2c2g1 (y)2 = m2c4; 0 ≤ (y = E/Ep) ≤ 1, +(1) +where g0 (y), g1 (y) are the rainbow functions, E is the energy of the probe particle and mc2 is its rest mass energy. +Such a modification in the energy-momentum relation is significant in the ultraviolet limit and is constrained to +reproduce the standard GR dispersion relation in the infrared limit so that +lim +y→0 gk (y) = 1; k = 0, 1. +(2) +The effects of such modifications could be observed, for example, in the tests of thresholds for ultra high-energy cosmic +rays [6, 13–15], TeV photons [16], gamma-ray bursts [6], nuclear physics experiments [17]. +∗Electronic address: omar.mustafa@emu.edu.tr + +2 +In the rainbow gravity setings, recent studies on the quantum mechanical gravity effects are carried out. Amongst, +the thermodynamical properties of black holes [18–22], the dynamical stability conditions of neutron stars [23], thermo- +dynamic stability of modified black holes [24], charged black holes in massive RG [25], on geometrical thermodynamics +and heat engine of black holes in RG [26], on RG and f(R) theories [27], the initial singularity problem for closed rain- +bow cosmology [28], the black hole entropy [29], the removal of the singularity of the early universe [30], the Casimir +effect in the rainbow Einstein’s universe [8], massive scalar field in RG Schwarzschild metric [31], five-dimensional +Yang–Mills black holes in massive RG [32]. +Moreover, recent studies are carried out the on effects of the RG on the dynamics of Klein-Gordon (KG) particles +(i.e., spin-0 mesons), Dirac particles (spin-1/2 fermionic particles), and Duffen-Kemmer-Peatiau (DKP) particles +(spin-1 particles like bosons and photons) in the background of different spacetime models. For example, in a cosmic +string spacetime background in rainbow gravity, Bezzerra et al. [8] have studied Landau levels via Schr¨odinger and KG +equations, Bakke and Mota [33] have studies the Dirac oscillator, they have also studied the Aharonov-Bohm effect +[34]. Hosseinpour et al. [5] have studied the DKP-particles,, Sogut et al. [11] have studied the quantum dynamics of +photon, and Kangal et al. [12] have studied KG-particles in a topologically trivial G¨odel-type spacetime in rainbow +gravity. In the current proposal, however, we extend such studies to include position-dependent mass (PDM) settings. +We shall focus on PDM KG-particles in cosmic string rainbow gravity spacetime in a non-uniform magnetic field. +However, one should be reminded that PDM is a metaphoric notion that emerges as a manifestation of coordinate +transformation/deformation that renders the mass to become effectively position-dependent [35–42]. Very recently, +PDM concept has been introduced to study PDM KG-oscillators in cosmic string spacetime within Kaluza-Klein +theory [43], in (2+1)-dimensional G¨urses spacetime backgrounds [44], and in Minkowski spacetime with space-like +dislocation [45]. Basically, for the PDM von Roos Schr¨odinger Hamiltonian [35], it has been shown (c.f., e.g., [36–38]) +that the PDM momentum operator takes the form +ˆp (r) = −i +� +∇−∇f (r) +4f (r) +� +⇒ ˆpj (r) = −i +� +∂j − ∂jf (r) +4f (r) +� +; j = 1, 2, 3, +(3) +where f (r) is a positive valued dimensionless scalar multiplier. For more details on this issue the reader may refer to +[36, 38, 43–45]. Which, in turn, would allow one to cast the PDM von Roos kinetic energy operator (using ℏ = 2m = 1 +units in the von Roos Hamiltonian) as +ˆT (r) = +� ˆp (r) +f (r) +�2 +ψ (r) = −f (r)−1/4 � +∇ f (r)−1/2� +· +� +∇ f (r)−1/4 ψ (r) +� +. +(4) +Such operator is known in the literature as Mustafa-Mazharimousavi’s PDM kinetic energy operator [37]. In short, +the momentum operator for constant mass setting ˆpj (r)−i ∂j is to be replaced by the PDM operator (3) in our study +of PDM KG-particles in cosmic string rainbow gravity spacetime. +Apriori, the cosmic string spacetime is described, in the natural units c = ℏ = G = 1, by the line element +ds2 = −dt2 + dr2 + α2 r2dϕ2 + dz2, +(5) +with a constant α related to the deficit angle of the conical spacetime and defined as α = 1 − 4Gµ, where G is the +Newton’s constant and µ is the linear mass density of the cosmic string so that α < 1. This would suggest that the +cosmic string metric (5) in the rainbow gravity takes the energy-dependent form +ds2 = − +1 +g0 (y)2 dt2 + +1 +g1 (y)2 +� +dr2 + α2 r2dϕ2 + dz2� +, +(6) + +3 +where the signature of the line element (6) is (−, +, +, +). The corresponding metric tensor gµν is given by +gµν = diag +� +− +1 +g0 (y)2 , +1 +g1 (y)2 , α2 r2 +g1 (y)2 , +1 +g1 (y)2 +� +; µ, ν = t, r, ϕ, z, +(7) +with +det (gµν) = − +α2 r2 +g0 (y)2 g1 (y)6 =⇒ gµν = diag +� +−g0 (y)2 , g1 (y)2 , g1 (y)2 +α2 r2 , g1 (y)2 +� +. +(8) +In the current methodical proposal, we study the effects of such cosmic string rainbow gravity on PDM KG-particles +in a non-uniform magnetic field. In so doing, we shall be interested in three pairs of rainbow functions: (i) g0 (y) = 1, +g1 (y) = +� +1 − ǫy2, and g0 (y) = 1, g1 (y) = √1 − ǫy, which belong to the set of rainbow functions g0 (y) = 1, +g1 (y) = √1 − ǫyn (where ǫ is a dimensionless constant of order unity) used to describe the geometry of spacetime +in loop quantum gravity [46, 47], (ii) g0 (y) = g1 (y) = (1 − ǫy)−1, a suitable set used to resolve the horizon problem +[13, 48], and (iii) g0 (y) = (eǫy − 1) /ǫy and g1 (y) = 1, which are obtained from the spectra of gamma-ray bursts at +cosmological distances [6]. +Our paper is organized as follows. In section 2, we discuss the PDM KG-particles in the cosmic string rainbow gravity +spacetime (6) in a non-uniform magnetic field. We bring the corresponding KG-equation into the one-dimensional +form of the two-dimensional radial Sch¨odinger. Using the above mentioned sets of the rainbow functions, we report +and discuss, in section 3, the effects of rainbow gravity on the energy levels of KG-particles with constant mass. In +section 4, we discuss the effects of rainbow gravity as well as PDM on the energy levels of a PDM KG-oscillator using +the three sets of rainbow functions above. We conclude in section 5. +II. +PDM KG-PARTICLES IN COSMIC STRING RAINBOW GRAVITY SPACETIME IN A +NON-UNIFORM MAGNETIC FIELD +In the cosmic string rainbow gravity spacetime background (6), a KG-particle of charge e in a 4-vector potential +Aµ is described (in c = ℏ = 1 units) by the KG-equation +1 +√−g Dµ +�√−ggµνDνΨ +� += m2Ψ, +(9) +where Dµ is the gauge-covariant derivative given by Dµ = ∂µ − ieAµ, and m is the rest mass energy of the KG- +particle. +At this point, we may also include position-dependent mass (PDM) settings (a metaphoric description +of deformed coordinates and inherited from the von Roos Hamiltonian [35] ) using the PDM-momentum operator +ˆpj (r) = −i +� +∂j − ∂j f(r) +4 f(r) +� +[36–38, 43–45]. In this case, Dµ −→ ˜Dµ = Dµ + Fµ = ∂µ + Fµ − ieAµ, where Fµ = +(0, Fr, 0, 0), Fr = +f ′(r) +4 f(r) and f (r) = f (r) is only radially dependent. +One should notice that a KG-oscillator is +obtained using f (r) = exp +� +2βr2� +, where f (r) is a positive dimensionless scalar multiplier. Under such new structure +our KG-equation (9) now describes PDM KG-particles in the cosmic string rainbow gravity spacetime and reads +1 +√−g +˜Dµ +�√−ggµν ˜Dν +� +Ψ = m2Ψ =⇒ +1 +√−g (Dµ + Fµ) √−ggµν (Dν − Fν) Ψ = m2Ψ. +(10) +Which, in a straightforward manner, yields +� +−g0 (y)2 ∂2 +t + g1 (y)2 +� +∂2 +r + 1 +r ∂r − M (r) + +1 +α2 r2 (∂ϕ − ieAϕ)2 + ∂2 +z +�� +Ψ (t, r, ϕ, z) = m2Ψ (t, r, ϕ, z) , +(11) + +4 +where +M (r) = F′ +r + Fr +r + F2 +r = − 3 +16 +�f ′ (r) +f (r) +�2 ++ f ′ (r) +4rf (r) + f ′′ (r) +4f (r) +(12) +We now use the substitution +Ψ (t, r, ϕ, z) = exp (i [ℓϕ + kzz − Et]) ψ (r) , +(13) +in Eq. (11) to obtain +� +˜E2 + g1 (y)2 +� +∂2 +r + 1 +r ∂r − M (r) − (ℓ − eAϕ)2 +α2 r2 +�� +ψ (r) = 0, +(14) +where +˜E2 = g0 (y)2 E2 − g1 (y)2 k2 +z − m2 +(15) +In what follows we shall consider Aϕ = 1 +2B◦r2, which in turn yields a non-uniform magnetic field B = ∇ × A = +3 +2B◦r ˆz. Consequently, Eq.(14) becomes +� +λ + ∂2 +r + 1 +r ∂r − M (r) − +˜ℓ2 +r2 − 1 +4 +˜B2r2 +� +ψ (r) = 0, +(16) +where +λ = +g0 (y)2 E2 + g1 (y)2 � +˜B˜ℓ − k2 +z +� +− m2 +g1 (y)2 +, ˜ℓ = ℓ +α, ˜B = eB◦ +α . +(17) +Moreover, with ψ (r) = R (r) /√r we obtain + + +∂2 +r − +� +˜ℓ2 − 1/4 +� +r2 +− M (r) − 1 +4 +˜B2r2 + λ + + + R (r) = 0. +(18) +Under such spacetime and magnetic field structures, we shall consider two types of KG-particles: constant mass and +PDM ones. +A. +Constant mass KG-particles in cosmic string rainbow gravity spacetime +It is convenient to discuss the KG-particles with a standard constant mass, i.e., f (r) = 1 ⇐⇒ M (r) = 0, so that +Eq.(18) reduces into the two-dimensional Schr¨odinger-oscillator form + + +∂2 +r − +� +˜ℓ2 − 1/4 +� +r2 +− 1 +4 +˜B2r2 + λ + + + R (r) = 0. +(19) +Which obviously admits exact solution in the form of hypergeometric function so that +R (r) = C r|˜ℓ|+1/2 exp +� +−| ˜B| +4 r2 +� +1F1 +� +1 +2 + |˜ℓ| +2 − +λ +2| ˜B| +, 1 + |˜ℓ|, | ˜B| +4 r2 +� +. +(20) + +5 +However, to secure finiteness and square integrability we need to terminate the hypergeometric function into a poly- +nomial of degree nr ≥ 0 so that the condition +1 +2 + |˜ℓ| +2 − +λ +2| ˜B| += −nr +(21) +is satisfied. This would in turn imply that +λnr,ℓ = | ˜B| +� +2nr + |˜ℓ| + 1 +� +, R (r) = C r|˜ℓ|+1/2 exp +� +−| ˜B| +4 r2 +� +1F1 +� +−nr, 1 + |˜ℓ|, | ˜B| +4 r2 +� +. +(22) +Consequently, Eq.(17) would read +g0 (y)2 E2 − m2 = g1 (y)2 Knr,ℓ; Knr,ℓ = +� +| ˜B| +� +2nr + |˜ℓ| + 1 +� +− ˜B˜ℓ + k2 +z +� +. +(23) +Before we proceed, it is convenient to observe that there are degeneracies associated with this relativistic energy +relation. Obviously, all states with ˜B˜ℓ = +| ˜B||˜ℓ| (i.e., for ˜B = ±| ˜B| = ±|e|B◦/α and ˜ℓ = ±|˜ℓ|) collapse into the +S-state (i.e., ℓ = 0; ˜ℓ = ℓ/α.) for a given radial quantum number nr so that the energy dispersion relation (23) reads +g0 (y)2 E2 − m2 = g1 (y)2 � +| ˜B| (2nr + 1) + k2 +z +� +. +(24) +Whereas, for ˜B˜ℓ = −| ˜B||˜ℓ| (i.e., for ˜B = ±| ˜B| = B◦ +α (±|e|) and ˜ℓ = ∓|˜ℓ|) the energy dispersion relation (23) yields +g0 (y)2 E2 − m2 = g1 (y)2 � +| ˜B| +� +2nr + 2|˜ℓ| + 1 +� ++ k2 +z +� +. +(25) +Which suggests that for ℓ ̸= 0 there are degeneracies for every ℓ = ±1, ±2, · · ·. These may very well be called charge +associated degeneracies. It should be made clear that such observations have nothings to do with the rainbow gravity +effects as can be concluded from (23). However, in what follows, we shall consider positively charged KG-particles so +that +g0 (y)2 E2 − m2 = g1 (y)2 � +| ˜B| (2nr + 1) + k2 +z +� +(26) +for ℓ = +|ℓ|, and +g0 (y)2 E2 − m2 = g1 (y)2 � +| ˜B| +� +2nr + 2|˜ℓ| + 1 +� ++ k2 +z +� +(27) +for ℓ = −|ℓ|. Consequently, Eq. (26) would allow all positive/negative energies with ℓ = +|ℓ| to cluster and merge +with the corresponding positive/negative S-states, whereas Eq. (27) would allow the positive/negative energies with +ℓ = −|ℓ| to appear in the corresponding energy spectra. +We may at this point consider different rainbow functions and discuss the effect of each on the energy levels of +Eq.(23). +Case (i)-(a): g0 (y) = 1 and g1 (y) = +� +1 − ǫy2: +For such rainbow functions structures Eq.(23) results +E2 − m2 = +� +1 − ǫE2 +E2p +� +Knr,ℓ =⇒ E = ± +� +Knr,ℓ + m2 +1 + δ Knr,ℓ +; δ = +ǫ +E2p +. +(28) +In Figures 1(a), and (b), we plot the corresponding energies against δ = ǫ/E2 +p. We observe that, for a given radial +quantum number nr, all positive/negative energy levels seem to converge into E = 0 as δ → ∞ (note that δ = 0 is + +6 +FIG. 1: +The energy levels of (28), using α = 1/2, m = kz = 1, so that (a) shows E against δ = ǫ/E2 +p for |eB◦| = 1, nr = 0, +ℓ = 0, −1, −2, −3, −4, (b) shows E against δ = ǫ/E2 +p for |eB◦| = 1, nr = 2, ℓ = 0, −1, −2, −3, −4, and (c) shows E against +|eB◦| for δ = 0.1, nr = 0, ℓ = 0, −1, −2, −3, −4. +FIG. 2: +The energy levels of (29), using α = 1/2, m = kz = 1, so that (a) shows E against β = ǫ/2Ep for |eB◦| = 1, nr = 0, +ℓ = 0, −1, −2, −3, −4, and (b) shows E against |eB◦| for β = 0.1, nr = 0, ℓ = 0, −1, −2, −3, −4. +the cosmic string spacetime limit). This is, in fact, the natural asymptotic convergence of E in (28) as δ → ∞. Yet, +it has been noted that the energies converge into E = 0 more rapidly for larger nr values, as shown in Fig. 1(b). +In Figure 1(c), moreover, we plot the energies against |eB◦|. We observe that the energy levels splitting occurs as +|eB◦| increases from the zero value and the separations between energy levels increase with increasing magnetic field +strength for all ℓ = −|ℓ| states (as a consequence of Eq. (27)). One should notice that all positive/negative energy +states with ℓ = +|ℓ| are already embedded in the corresponding positive/negative S-states. +Case (i)-(b): g0 (y) = 1 and g1 (y) = √1 − ǫy: +This rainbow functions model in Eq.(23) implies +E2 − m2 = +� +1 − ǫ E +Ep +� +Knr,ℓ =⇒ E = −βKnr,ℓ ± +� +β2K2 +nr,ℓ + Knr,ℓ + m2; β = +ǫ +2Ep +. +(29) +In Figures 2(a) and (b), we plot the energy levels against β = ǫ/2Ep and |eB◦|, respectively. It is obvious that the +symmetry of the energy levels about E = 0 is broken as an effect of such rainbow functions structure . Yet, in Fig.2(a) +we notice that the convergence of the positive energies into E = 0 for β → ∞ is an obvious consequence of lim +β→∞ E ≈ 0 + +7 +FIG. 3: +The energy levels of (30), using α = 1/2, m = kz = 1, so that (a) shows E against γ = ǫm/Ep < 1 for |eB◦| = 1, +nr = 0, ℓ = 0, −1, −2, −3, −4, (b) shows E against |eB◦| for γ = 0.1, nr = 0, ℓ = 0, −1, −2, −3, −4, and (c) shows E against +|eB◦| for γ = 0.5, nr = 3, ℓ = 0, −1, −2, −3, −4. +of (29), whereas for the negative energies of (29) we get lim +β→∞ E ≈ −2βKnr,ℓ . Therefore, in the lower half, we observe +that the splitting in the negative energy levels increases as β increases from zero. Moreover, in the lower half of +Fig.2(b) we see that the energy levels separation increases as the magnetic field strength increases from zero, for a +given β = 0.1 value. This effect is obvious from the energy levels in Eq.(29), as the first negative term increases the +negativity of the energy levels and breaks the symmetry of the energy around E = 0 line. +Case (ii): g0 (y) = g1 (y) = (1 − ǫy)−1: +Such rainbow functions assumption in Eq.(23) yields +E2 − Knr,ℓ = +� +1 − ǫ E +Ep +�2 +m2 =⇒ E = −mγ ± +� +Knr,ℓ (1 − γ2) + m2 +1 − γ2 +; γ = ǫm +Ep +< 1. +(30) +In Figures 3(a) we plot the energy levels against γ = ǫm/Ep < 1 to observe the rainbow gravity effect. We clearly see +that the symmetry in the energy levels is broken as an effect of the first term +� +−mγ/ +� +1 − γ2�� +in Eq.(30). In Figures +3(b) and (c) the energy levels are plotted against |eB◦| so that the magnetic field effect on the energy levels is shown, +where the broken energy levels symmetry is more obvious in Fig.3(c) since nr = 0 is used for 3(b) whereas we used +nr = 3 for 3(c). +Case (iii): g0 (y) = (eǫy − 1) /ǫy and g1 (y) = 1: +This rainbow functions structure in Eq.(23) implies +E2 +�eǫE/Ep − 1 +ǫE/Ep +�2 +− m2 = Knr,ℓ =⇒ E = 1 +2β ln +� +1 ± +� +4β2 (Knr,ℓ + m2) +� +; β = +ǫ +2Ep +(31) +In Figure 4(a) we plot the energy levels against β = ǫ/2Ep and observe eminent clustering in the positive energies as +β grows up from just above zero (i.e., β ≥ 0.001), whereas the negative energies are rapidly pushed further into the +negative energy region. This is very much related to the nature of the logarithmic structure of the energies in (31). +This is also reflected on Fig. 4(b), where we show the effect of the magnetic field on the energy levels. + +8 +FIG. 4: +The energy levels of (31), using α = 1/2, m = kz = 1, so that (a) shows E against β = ǫ/2Ep for |eB◦| = 1, nr = 0, +ℓ = 0, −1, −2, −3, −4, and (b) shows E against |eB◦| for β = 0.1, nr = 0, ℓ = 0, −1, −2, −3, −4. +B. +PDM KG-particles in cosmic string rainbow gravity spacetime +We now consider a positive-valued dimensionless scalar multiplier f (r) = exp +� +2ηr2� +in Eq.(12) so that M (r) = +η2r2 + 2η. This would suggest that Eq.(18) now reads + + +∂2 +r − +� +˜ℓ2 − 1/4 +� +r2 +− 1 +4Ω2r2 + ˜λ + + + R (r) = 0; +(32) +where +˜λ = +g0 (y)2 E2 + g1 (y)2 � +˜B˜ℓ − k2 +z − 2η +� +− m2 +g1 (y)2 +, |Ω| = | ˜B| +� +1 + 4η2 +˜B2 +(33) +In this case, its exact solution is in the form of +R (r) = C r|˜ℓ|+1/2 exp +� +−|Ω| +4 r2 +� +1F1 +� +1 +2 + |˜ℓ| +2 − +˜λ +2|Ω|, 1 + |˜ℓ|, |Ω| +4 r2 +� +, +(34) +which takes the form of a polynomial of degree nr ≥ 0 for +1 +2 + |˜ℓ| +2 − +˜λ +2|Ω| = −nr. +(35) +Consequently, +˜λnr,ℓ = |Ω| +� +2nr + |˜ℓ| + 1 +� +, R (r) = C r|˜ℓ|+1/2 exp +� +−|Ω| +4 r2 +� +1F1 +� +−nr, 1 + |˜ℓ|, |Ω| +4 r2 +� +. +(36) +and +g0 (y)2 E2 − m2 = g1 (y)2 ˜Knr,ℓ; ˜Knr,ℓ = +�� +˜B2 + 4η2 +� +2nr + |˜ℓ| + 1 +� +− ˜B˜ℓ + k2 +z + 2η +� +. +(37) +Notably, we observe that the degeneracies discussed in the preceding section are removed as an effect of PDM settings. +However, one should notice that as ˜B → 0 our ˜Knr,ℓ → 2η +� +2nr + |˜ℓ| + 1 +� ++ k2 +z + 2η and consequently states with a +specific ℓ = ±|ℓ| will emerge from the same ˜B = 0 value and split as ˜B grows up from zero. We may now discuss the +effects of different rainbow functions on the energy levels under the current PDM-settings of Eq.(37). + +9 +FIG. 5: +The energy levels of (38), using α = 1/2, m = kz = 1, so that (a) shows E against δ = ǫ/E2 +p for |eB◦| = η = 1, nr = 0, +ℓ = 0, ±1, ±2, (b) shows E against η for |eB◦| = 1, δ = 0.1, nr = 0, ℓ = 0, ±1, ±2, ±3, and (c) shows E against |eB◦| for η = 1, +δ = 0.1, nr = 0, ℓ = 0, ±1, ±2, ±3. +Case (i)-(a): g0 (y) = 1 and g1 (y) = +� +1 − ǫy2: +For such rainbow functions structures Eq.(23) results +E2 − m2 = +� +1 − ǫE2 +E2p +� +˜Knr,ℓ =⇒ E = ± +� +˜Knr,ℓ + m2 +1 + δ ˜Knr,ℓ +; δ = +ǫ +E2p +. +(38) +In Figures 5(a), we plot the energies against δ = ǫ/E2 +p. We observe that, for a given radial quantum number nr, +all positive/negative energy levels seem to converge into E = 0 as δ → ∞. This is, in fact, the natural convergence +of E in (38) as δ → ∞ (similar tendency as that observed for Fig.1(a), but here all states with ℓ = ±|ℓ| appear in +the spectrum as a consequence of PDM parameter η ̸= 0). However, both halves of the energy levels asymptotically +converge to E = 0 value as δ → ∞ for a fixed value of the PDM parameter η. In Fig. 5(b), we plot the energies +against the PDM parameter η and notice that the degeneracies associated with ℓ = +|ℓ| are removed as η increases +from zero (i.e., it is obvious the all states with .ℓ = +|ℓ| emerge form the same η = 0 point and split as η grows +up). In Figure 5(c), moreover, we plot the energies against |eB◦| and observe that the energy levels with a specific +ℓ = ±|ℓ| split as |eB◦| increases from the zero value and the separations between energy levels increase with increasing +magnetic field strength. +Case (i)-(b): g0 (y) = 1 and g1 (y) = √1 − ǫy: +This rainbow functions model in Eq.(23) implies +E2 − m2 = +� +1 − ǫ E +Ep +� +˜Knr,ℓ =⇒ E = −β ˜Knr,ℓ ± +� +β2 ˜K2 +nr,ℓ + ˜Knr,ℓ + m2; β = +ǫ +2Ep +. +(39) +In Figures 6(a) and (b), we plot the energy levels against β = ǫ/2Ep and |eB◦|, respectively. It is obvious that the +symmetry of the energy levels about E = 0 is broken as an effect of such rainbow functions structure. Yet, in Fig.6(a) +we notice that, for a given radial quantum number nr, only positive energy levels converge into E = 0 as δ → ∞, +whereas in the lower half we observe that the splitting in the energy levels increases as β increases from zero. In the +lower half of Fig. 6(b), we see that the energy levels separation increases as the magnetic field strength increases +from zero. This effect is obvious from the form of the energy levels in Eq.(39), as the first negative term increases the +negativity of the energy levels and breaks the symmetry of the energy part of the second term. In Fig.6(c) we show +the effect of the PDM settings on the energy levels and follow the same scenario as that for Fig.5(c). + +10 +FIG. 6: +The energy levels of (39), using α = 1/2, m = kz = 1, so that (a) shows E against β = ǫ/2Ep for |eB◦| = η = 1, +nr = 0, ℓ = 0, ±1, ±2, (b) shows E against |eB◦| for η = 1, β = 0.1, nr = 0, ℓ = 0, ±1, ±2, and (c) shows E against η for +|eB◦| = 1, β = 0.1, nr = 0, ℓ = 0, ±1, ±2. +FIG. 7: +The energy levels of (40), using α = 1/2, m = kz = 1, so that (a) shows E against γ = ǫm/Ep < 1 for |eB◦| = η = 1, +nr = 0, ℓ = 0, ±1, ±2, (b) shows E against |eB◦| for η = 1, γ = 0.1, nr = 0, ℓ = 0, ±1, ±2, and (c) shows E against η for +|eB◦| = 1, γ = 0.1, nr = 0, ℓ = 0, ±1, ±2. +Case (ii): g0 (y) = g1 (y) = (1 − ǫy)−1: +Such rainbow functions assumption in Eq.(23) yields +E2 − ˜Knr,ℓ = +� +1 − ǫ E +Ep +�2 +m2 =⇒ E = +−mγ ± +� +˜Knr,ℓ (1 − γ2) + m2 +1 − γ2 +; γ = ǫm +Ep +< 1. +(40) +In Figure 7(a) we plot the energy levels against γ = ǫm/Ep to observe the rainbow gravity effect. We clearly see that +the symmetry in the energy levels is broken as an effect of the first term +� +−mγ/ +� +1 − γ2�� +in Eq.(40). In Fig. 7(b), +the energy levels are plotted against |eB◦| so that the magnetic field effect on the energy levels is shown, and in Fig. +7(c) we show the effect of the PDM settings on the energy levels. +Case (iii): g0 (y) = (eǫy − 1) /ǫy and g1 (y) = 1: +This rainbow functions structure in Eq.(23) implies +E2 +�eǫE/Ep − 1 +ǫE/Ep +�2 +− m2 = ˜Knr,ℓ =⇒ E = 1 +2β ln +� +1 ± +� +4β2 +� +˜Knr,ℓ + m2 +�� +; β = +ǫ +2Ep +(41) + +11 +FIG. 8: +The energy levels of (41), using α = 1/2, m = kz = 1, so that (a) shows E against β = ǫ/2Ep for |eB◦| = η = 1, +nr = 0, ℓ = 0, ±1, ±2, (b) shows E against |eB◦| for η = 1, β = 0.1, nr = 0, ℓ = 0, ±1, ±2, and (c) shows E against η for +|eB◦| = 1, β = 0.1, nr = 0, ℓ = 0, ±1, ±2. +In Figure 8(a) we plot the energy levels against β = ǫ/2Ep and observe eminent clustering in the positive energies +as β grows up from just above zero (i.e., β ≥ 0.001), whereas the negative energies are rapidly pushed further into +the negative energy region. In Figures 8(b) we show the effect of the magnetic field and in 8(c) we show the effect of +the PDM settings on the energy levels. +III. +CONCLUDING REMARKS +In this paper, we started with the PDM KG-particles in the cosmic string rainbow gravity spacetime (6) in a non- +uniform magnetic field B = ∇×A = 3 +2B◦r ˆz introduced by Aϕ = 1 +2B◦r2. We reduced the corresponding KG-equation +into the one-dimensional form of the two-dimensional radial Sch¨odinger-oscillator like equation. We have first studied +the effects of rainbow gravity on the energy levels of KG-particles with constant mass. Next, we have included the +PDM settings through the dimensionless positive-valued scalar multiplier f (r) = exp +� +2ηr2� +, in Eq.(12), so that an +oscillator-like interaction is introduced as M (r) = η2r2 + 2η (documented in Eq.(32)). This would, in turn, suggest +that a KG-particle in a cosmic string spacetime (5) in a non-uniform magnetic field B = ∇ × A = 3 +2B◦r ˆz and/or +in M (r) = η2r2 + 2η would introduce the so called KG-oscillator. Consequently, the current methodical proposal +discusses the rainbow gravity effect on KG-oscillators manifested by the non-uniform magnetic field and/or the PDM +setting M (r) mentioned above. In so doing, we have considered four pairs of rainbow functions: (i)-(a) g0 (y) = 1, +g1 (y) = +� +1 − ǫy2, (i)-(b) g0 (y) = 1, g1 (y) = √1 − ǫy, (ii) g0 (y) = g1 (y) = (1 − ǫy)−1, and (iii) g0 (y) = (eǫy − 1) /ǫy, +g1 (y) = 1. +In the light of our experience above, we recollect that the most general KG-oscillators’ energies in cosmic string +rainbow gravity spacetime are given by Eq. +(37) (which yields the constant mass energies of (23) for the PDM +parameter η = 0). and may summarize the effects of rainbow gravity on the spectra of KG-oscillators in cosmic string +spacetime as follows. Whilst, for the pair of rainbow functions +� +g0 (y) = 1, g1 (y) = +� +1 − ǫy2 +� +the KG-oscillators +energies about E = 0 are observed to be symmetric (documented in Figures 1 and 5), the symmetry of KG-oscillators +energies for the rest of the rainbow functions pairs (i.e., +� +g0 (y) = 1, g1 (y) = √1 − ǫy +� +, +� +g0 (y) = g1 (y) = (1 − ǫy)−1� +, + +12 +and [g0 (y) = (eǫy − 1) /ǫy, g1 (y) = 1]) are observed broken (documented in Figures 2, 3, 4, 6, 7, and 8). Moreover, +rainbow gravity appeared to have no effect, whatsoever, on either introducing new type of degeneracies or removing +natural degeneracies associated with the spectra of the KG-oscillators in cosmic string spacetime (i.e., at lim +y→0 gk (y) = +1; k = 0, 1). Finally, it could be interesting to report that for the rainbow functions +� +g0 (y) = g1 (y) = (1 − ǫy)−1� +, +one may observe that when γ = ǫm/Ep = 1 is considered (although we considered γ < 1 in (30) and (40), it remains +a feasible limit) there are energy states to fly away and disappear from the spectrum (a phenomenon that have been +observed in the non-Hermitian PT - symmetric Schr¨odinger Coulomb problem [49] ). To the best of our knowledge, +the current study has never been published elsewhere (within the lines discussed above, of course). +[1] J. Magueijo, L Smolin, Phys. Rev. Lett. 88 (2002) 190403. +[2] P. Galan, G. A. Mena Marugan, Phys. Rev. D 70 (2004) 124003. +[3] G. Amelino-Camelia, Int. J. Mod. Phys. D 11 (2002) 35. +[4] G. Amelino-Camelia, Int. J. Mod. Phys. D 11 (2002) 1643. +[5] .H. Hosseinpour, H. Hassanabadi, J. Kˇr´ıˇz, S. Hassanabadi. B. C. L¨utf¨uoˆglu, Int. J. Geom. Methods Mod. Phys. 18 (2021) +2150224. +[6] G. Amelino-Camelia, J. R. Ellis, N. 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Gen. 35 (2002) 8929. + diff --git a/sNE5T4oBgHgl3EQfKQ5P/content/tmp_files/load_file.txt b/sNE5T4oBgHgl3EQfKQ5P/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ac9136acb85ee80b7dbf7e42a74889d00adcf033 --- /dev/null +++ b/sNE5T4oBgHgl3EQfKQ5P/content/tmp_files/load_file.txt @@ -0,0 +1,621 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf,len=620 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='05464v1 [gr-qc] 13 Jan 2023 PDM KG-oscillators in cosmic string rainbow gravity spacetime in a non-uniform magnetic field Omar Mustafa∗ Department of Physics, Eastern Mediterranean University, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Magusa, north Cyprus, Mersin 10 - Turkey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Abstract: We consider position-dependent mass (PDM) Klein-Gordon (KG) particles in cosmic string rainbow gravity spacetime in a non-uniform magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' The corresponding KG-equation is reduced into the one-dimensional form of the two-dimensional radial Schr¨odinger-oscillator like equation (hence the notion KG-oscillator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' We first report on the effects of rainbow gravity on the energy levels of KG-oscillators with constant mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Next, we include the PDM settings so that KG-oscillators like interaction are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' The effects of PDM on the spectra of KG-oscillators in cosmic string rainbow gravity spacetime are also reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' In both cases four pairs of rainbow functions are considered: (a) g0 (y) = 1, g1 (y) = � 1 − ǫy2, (b) g0 (y) = 1, g1 (y) = √1 − ǫy, (c) g0 (y) = g1 (y) = (1 − ǫy)−1, and (d) g0 (y) = (eǫy − 1) /ǫy, g1 (y) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' PACS numbers: 05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='-a, 03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='Kk, 03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='-w Keywords: Klein-Gordon (KG) particles, position-dependent mass, cosmic string spacetime, rainbow gravity, non-uniform magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' INTRODUCTION In quantum gravity (QG), the semi-classical model of rainbow gravity (RG) (an extension of the deformed or doubly special relativity into a general relativity (GR) framework) has attracted research attention over the years [1–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Under the RG-model, the energy of the probe particles is assumed to affect the spacetime background, at the ultra-high energy regime, so that the spacetime metric become an energy-dependent one [5–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Hereby, the Planck energy Ep = � ℏc5/G plays the role of a threshold separating the classical description from the quantum mechanical one and introduces itself as another invariant energy scale alongside the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Consequently, rainbow gravity justifies the modified relativistic energy-momentum dispersion relation E2g0 (y)2 − p2c2g1 (y)2 = m2c4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' 0 ≤ (y = E/Ep) ≤ 1, (1) where g0 (y), g1 (y) are the rainbow functions, E is the energy of the probe particle and mc2 is its rest mass energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Such a modification in the energy-momentum relation is significant in the ultraviolet limit and is constrained to reproduce the standard GR dispersion relation in the infrared limit so that lim y→0 gk (y) = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' k = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (2) The effects of such modifications could be observed, for example, in the tests of thresholds for ultra high-energy cosmic rays [6, 13–15], TeV photons [16], gamma-ray bursts [6], nuclear physics experiments [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' ∗Electronic address: omar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='mustafa@emu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='tr 2 In the rainbow gravity setings, recent studies on the quantum mechanical gravity effects are carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Amongst,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' the thermodynamical properties of black holes [18–22],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' the dynamical stability conditions of neutron stars [23],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' thermo- dynamic stability of modified black holes [24],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' charged black holes in massive RG [25],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' on geometrical thermodynamics and heat engine of black holes in RG [26],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' on RG and f(R) theories [27],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' the initial singularity problem for closed rain- bow cosmology [28],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' the black hole entropy [29],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' the removal of the singularity of the early universe [30],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' the Casimir effect in the rainbow Einstein’s universe [8],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' massive scalar field in RG Schwarzschild metric [31],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' five-dimensional Yang–Mills black holes in massive RG [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Moreover, recent studies are carried out the on effects of the RG on the dynamics of Klein-Gordon (KG) particles (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=', spin-0 mesons), Dirac particles (spin-1/2 fermionic particles), and Duffen-Kemmer-Peatiau (DKP) particles (spin-1 particles like bosons and photons) in the background of different spacetime models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' For example, in a cosmic string spacetime background in rainbow gravity, Bezzerra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' [8] have studied Landau levels via Schr¨odinger and KG equations, Bakke and Mota [33] have studies the Dirac oscillator, they have also studied the Aharonov-Bohm effect [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Hosseinpour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' [5] have studied the DKP-particles,, Sogut et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' [11] have studied the quantum dynamics of photon, and Kangal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' [12] have studied KG-particles in a topologically trivial G¨odel-type spacetime in rainbow gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' In the current proposal, however, we extend such studies to include position-dependent mass (PDM) settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' We shall focus on PDM KG-particles in cosmic string rainbow gravity spacetime in a non-uniform magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' However, one should be reminded that PDM is a metaphoric notion that emerges as a manifestation of coordinate transformation/deformation that renders the mass to become effectively position-dependent [35–42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Very recently, PDM concept has been introduced to study PDM KG-oscillators in cosmic string spacetime within Kaluza-Klein theory [43], in (2+1)-dimensional G¨urses spacetime backgrounds [44], and in Minkowski spacetime with space-like dislocation [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Basically, for the PDM von Roos Schr¨odinger Hamiltonian [35], it has been shown (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=', e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=', [36–38]) that the PDM momentum operator takes the form ˆp (r) = −i � ∇−∇f (r) 4f (r) � ⇒ ˆpj (r) = −i � ∂j − ∂jf (r) 4f (r) � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' j = 1, 2, 3, (3) where f (r) is a positive valued dimensionless scalar multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' For more details on this issue the reader may refer to [36, 38, 43–45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Which, in turn, would allow one to cast the PDM von Roos kinetic energy operator (using ℏ = 2m = 1 units in the von Roos Hamiltonian) as ˆT (r) = � ˆp (r) f (r) �2 ψ (r) = −f (r)−1/4 � ∇ f (r)−1/2� � ∇ f (r)−1/4 ψ (r) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (4) Such operator is known in the literature as Mustafa-Mazharimousavi’s PDM kinetic energy operator [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' In short, the momentum operator for constant mass setting ˆpj (r)−i ∂j is to be replaced by the PDM operator (3) in our study of PDM KG-particles in cosmic string rainbow gravity spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Apriori, the cosmic string spacetime is described, in the natural units c = ℏ = G = 1, by the line element ds2 = −dt2 + dr2 + α2 r2dϕ2 + dz2, (5) with a constant α related to the deficit angle of the conical spacetime and defined as α = 1 − 4Gµ, where G is the Newton’s constant and µ is the linear mass density of the cosmic string so that α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' This would suggest that the cosmic string metric (5) in the rainbow gravity takes the energy-dependent form ds2 = − 1 g0 (y)2 dt2 + 1 g1 (y)2 � dr2 + α2 r2dϕ2 + dz2� , (6) 3 where the signature of the line element (6) is (−, +, +, +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' The corresponding metric tensor gµν is given by gµν = diag � − 1 g0 (y)2 , 1 g1 (y)2 , α2 r2 g1 (y)2 , 1 g1 (y)2 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' µ, ν = t, r, ϕ, z, (7) with det (gµν) = − α2 r2 g0 (y)2 g1 (y)6 =⇒ gµν = diag � −g0 (y)2 , g1 (y)2 , g1 (y)2 α2 r2 , g1 (y)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (8) In the current methodical proposal, we study the effects of such cosmic string rainbow gravity on PDM KG-particles in a non-uniform magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' In so doing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' we shall be interested in three pairs of rainbow functions: (i) g0 (y) = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' g1 (y) = � 1 − ǫy2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' and g0 (y) = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' g1 (y) = √1 − ǫy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' which belong to the set of rainbow functions g0 (y) = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' g1 (y) = √1 − ǫyn (where ǫ is a dimensionless constant of order unity) used to describe the geometry of spacetime in loop quantum gravity [46,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' 47],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (ii) g0 (y) = g1 (y) = (1 − ǫy)−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' a suitable set used to resolve the horizon problem [13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' 48],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' and (iii) g0 (y) = (eǫy − 1) /ǫy and g1 (y) = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' which are obtained from the spectra of gamma-ray bursts at cosmological distances [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Our paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' In section 2, we discuss the PDM KG-particles in the cosmic string rainbow gravity spacetime (6) in a non-uniform magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' We bring the corresponding KG-equation into the one-dimensional form of the two-dimensional radial Sch¨odinger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Using the above mentioned sets of the rainbow functions, we report and discuss, in section 3, the effects of rainbow gravity on the energy levels of KG-particles with constant mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' In section 4, we discuss the effects of rainbow gravity as well as PDM on the energy levels of a PDM KG-oscillator using the three sets of rainbow functions above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' We conclude in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' PDM KG-PARTICLES IN COSMIC STRING RAINBOW GRAVITY SPACETIME IN A NON-UNIFORM MAGNETIC FIELD In the cosmic string rainbow gravity spacetime background (6), a KG-particle of charge e in a 4-vector potential Aµ is described (in c = ℏ = 1 units) by the KG-equation 1 √−g Dµ �√−ggµνDνΨ � = m2Ψ, (9) where Dµ is the gauge-covariant derivative given by Dµ = ∂µ − ieAµ, and m is the rest mass energy of the KG- particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' At this point, we may also include position-dependent mass (PDM) settings (a metaphoric description of deformed coordinates and inherited from the von Roos Hamiltonian [35] ) using the PDM-momentum operator ˆpj (r) = −i � ∂j − ∂j f(r) 4 f(r) � [36–38, 43–45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' In this case, Dµ −→ ˜Dµ = Dµ + Fµ = ∂µ + Fµ − ieAµ, where Fµ = (0, Fr, 0, 0), Fr = f ′(r) 4 f(r) and f (r) = f (r) is only radially dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' One should notice that a KG-oscillator is obtained using f (r) = exp � 2βr2� , where f (r) is a positive dimensionless scalar multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Under such new structure our KG-equation (9) now describes PDM KG-particles in the cosmic string rainbow gravity spacetime and reads 1 √−g ˜Dµ �√−ggµν ˜Dν � Ψ = m2Ψ =⇒ 1 √−g (Dµ + Fµ) √−ggµν (Dν − Fν) Ψ = m2Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (10) Which, in a straightforward manner, yields � −g0 (y)2 ∂2 t + g1 (y)2 � ∂2 r + 1 r ∂r − M (r) + 1 α2 r2 (∂ϕ − ieAϕ)2 + ∂2 z �� Ψ (t, r, ϕ, z) = m2Ψ (t, r, ϕ, z) , (11) 4 where M (r) = F′ r + Fr r + F2 r = − 3 16 �f ′ (r) f (r) �2 + f ′ (r) 4rf (r) + f ′′ (r) 4f (r) (12) We now use the substitution Ψ (t, r, ϕ, z) = exp (i [ℓϕ + kzz − Et]) ψ (r) , (13) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (11) to obtain � ˜E2 + g1 (y)2 � ∂2 r + 1 r ∂r − M (r) − (ℓ − eAϕ)2 α2 r2 �� ψ (r) = 0, (14) where ˜E2 = g0 (y)2 E2 − g1 (y)2 k2 z − m2 (15) In what follows we shall consider Aϕ = 1 2B◦r2, which in turn yields a non-uniform magnetic field B = ∇ × A = 3 2B◦r ˆz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Consequently, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (14) becomes � λ + ∂2 r + 1 r ∂r − M (r) − ˜ℓ2 r2 − 1 4 ˜B2r2 � ψ (r) = 0, (16) where λ = g0 (y)2 E2 + g1 (y)2 � ˜B˜ℓ − k2 z � − m2 g1 (y)2 , ˜ℓ = ℓ α, ˜B = eB◦ α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (17) Moreover, with ψ (r) = R (r) /√r we obtain \uf8f1 \uf8f2 \uf8f3∂2 r − � ˜ℓ2 − 1/4 � r2 − M (r) − 1 4 ˜B2r2 + λ \uf8fc \uf8fd \uf8fe R (r) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (18) Under such spacetime and magnetic field structures, we shall consider two types of KG-particles: constant mass and PDM ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Constant mass KG-particles in cosmic string rainbow gravity spacetime It is convenient to discuss the KG-particles with a standard constant mass, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=', f (r) = 1 ⇐⇒ M (r) = 0, so that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (18) reduces into the two-dimensional Schr¨odinger-oscillator form \uf8f1 \uf8f2 \uf8f3∂2 r − � ˜ℓ2 − 1/4 � r2 − 1 4 ˜B2r2 + λ \uf8fc \uf8fd \uf8fe R (r) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (19) Which obviously admits exact solution in the form of hypergeometric function so that R (r) = C r|˜ℓ|+1/2 exp � −| ˜B| 4 r2 � 1F1 � 1 2 + |˜ℓ| 2 − λ 2| ˜B| , 1 + |˜ℓ|, | ˜B| 4 r2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (20) 5 However, to secure finiteness and square integrability we need to terminate the hypergeometric function into a poly- nomial of degree nr ≥ 0 so that the condition 1 2 + |˜ℓ| 2 − λ 2| ˜B| = −nr (21) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' This would in turn imply that λnr,ℓ = | ˜B| � 2nr + |˜ℓ| + 1 � , R (r) = C r|˜ℓ|+1/2 exp � −| ˜B| 4 r2 � 1F1 � −nr, 1 + |˜ℓ|, | ˜B| 4 r2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (22) Consequently, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (17) would read g0 (y)2 E2 − m2 = g1 (y)2 Knr,ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Knr,ℓ = � | ˜B| � 2nr + |˜ℓ| + 1 � − ˜B˜ℓ + k2 z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (23) Before we proceed, it is convenient to observe that there are degeneracies associated with this relativistic energy relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Obviously, all states with ˜B˜ℓ = +| ˜B||˜ℓ| (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=', for ˜B = ±| ˜B| = ±|e|B◦/α and ˜ℓ = ±|˜ℓ|) collapse into the S-state (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=', ℓ = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' ˜ℓ = ℓ/α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=') for a given radial quantum number nr so that the energy dispersion relation (23) reads g0 (y)2 E2 − m2 = g1 (y)2 � | ˜B| (2nr + 1) + k2 z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (24) Whereas, for ˜B˜ℓ = −| ˜B||˜ℓ| (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=', for ˜B = ±| ˜B| = B◦ α (±|e|) and ˜ℓ = ∓|˜ℓ|) the energy dispersion relation (23) yields g0 (y)2 E2 − m2 = g1 (y)2 � | ˜B| � 2nr + 2|˜ℓ| + 1 � + k2 z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (25) Which suggests that for ℓ ̸= 0 there are degeneracies for every ℓ = ±1, ±2, · · ·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' These may very well be called charge associated degeneracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' It should be made clear that such observations have nothings to do with the rainbow gravity effects as can be concluded from (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' However, in what follows, we shall consider positively charged KG-particles so that g0 (y)2 E2 − m2 = g1 (y)2 � | ˜B| (2nr + 1) + k2 z � (26) for ℓ = +|ℓ|, and g0 (y)2 E2 − m2 = g1 (y)2 � | ˜B| � 2nr + 2|˜ℓ| + 1 � + k2 z � (27) for ℓ = −|ℓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Consequently, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (26) would allow all positive/negative energies with ℓ = +|ℓ| to cluster and merge with the corresponding positive/negative S-states, whereas Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (27) would allow the positive/negative energies with ℓ = −|ℓ| to appear in the corresponding energy spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' We may at this point consider different rainbow functions and discuss the effect of each on the energy levels of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Case (i)-(a): g0 (y) = 1 and g1 (y) = � 1 − ǫy2: For such rainbow functions structures Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (23) results E2 − m2 = � 1 − ǫE2 E2p � Knr,ℓ =⇒ E = ± � Knr,ℓ + m2 1 + δ Knr,ℓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' δ = ǫ E2p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (28) In Figures 1(a), and (b), we plot the corresponding energies against δ = ǫ/E2 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' We observe that, for a given radial quantum number nr, all positive/negative energy levels seem to converge into E = 0 as δ → ∞ (note that δ = 0 is 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' 1: The energy levels of (28), using α = 1/2, m = kz = 1, so that (a) shows E against δ = ǫ/E2 p for |eB◦| = 1, nr = 0, ℓ = 0, −1, −2, −3, −4, (b) shows E against δ = ǫ/E2 p for |eB◦| = 1, nr = 2, ℓ = 0, −1, −2, −3, −4, and (c) shows E against |eB◦| for δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='1, nr = 0, ℓ = 0, −1, −2, −3, −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' 2: The energy levels of (29), using α = 1/2, m = kz = 1, so that (a) shows E against β = ǫ/2Ep for |eB◦| = 1, nr = 0, ℓ = 0, −1, −2, −3, −4, and (b) shows E against |eB◦| for β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='1, nr = 0, ℓ = 0, −1, −2, −3, −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' the cosmic string spacetime limit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' This is, in fact, the natural asymptotic convergence of E in (28) as δ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Yet, it has been noted that the energies converge into E = 0 more rapidly for larger nr values, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' In Figure 1(c), moreover, we plot the energies against |eB◦|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' We observe that the energy levels splitting occurs as |eB◦| increases from the zero value and the separations between energy levels increase with increasing magnetic field strength for all ℓ = −|ℓ| states (as a consequence of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (27)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' One should notice that all positive/negative energy states with ℓ = +|ℓ| are already embedded in the corresponding positive/negative S-states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Case (i)-(b): g0 (y) = 1 and g1 (y) = √1 − ǫy: This rainbow functions model in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (23) implies E2 − m2 = � 1 − ǫ E Ep � Knr,ℓ =⇒ E = −βKnr,ℓ ± � β2K2 nr,ℓ + Knr,ℓ + m2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' β = ǫ 2Ep .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (29) In Figures 2(a) and (b), we plot the energy levels against β = ǫ/2Ep and |eB◦|, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' It is obvious that the symmetry of the energy levels about E = 0 is broken as an effect of such rainbow functions structure .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Yet, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='2(a) we notice that the convergence of the positive energies into E = 0 for β → ∞ is an obvious consequence of lim β→∞ E ≈ 0 7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' 3: The energy levels of (30), using α = 1/2, m = kz = 1, so that (a) shows E against γ = ǫm/Ep < 1 for |eB◦| = 1, nr = 0, ℓ = 0, −1, −2, −3, −4, (b) shows E against |eB◦| for γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='1, nr = 0, ℓ = 0, −1, −2, −3, −4, and (c) shows E against |eB◦| for γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='5, nr = 3, ℓ = 0, −1, −2, −3, −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' of (29), whereas for the negative energies of (29) we get lim β→∞ E ≈ −2βKnr,ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Therefore, in the lower half, we observe that the splitting in the negative energy levels increases as β increases from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Moreover, in the lower half of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='2(b) we see that the energy levels separation increases as the magnetic field strength increases from zero, for a given β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='1 value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' This effect is obvious from the energy levels in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (29), as the first negative term increases the negativity of the energy levels and breaks the symmetry of the energy around E = 0 line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Case (ii): g0 (y) = g1 (y) = (1 − ǫy)−1: Such rainbow functions assumption in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (23) yields E2 − Knr,ℓ = � 1 − ǫ E Ep �2 m2 =⇒ E = −mγ ± � Knr,ℓ (1 − γ2) + m2 1 − γ2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' γ = ǫm Ep < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (30) In Figures 3(a) we plot the energy levels against γ = ǫm/Ep < 1 to observe the rainbow gravity effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' We clearly see that the symmetry in the energy levels is broken as an effect of the first term � −mγ/ � 1 − γ2�� in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='(30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' In Figures 3(b) and (c) the energy levels are plotted against |eB◦| so that the magnetic field effect on the energy levels is shown, where the broken energy levels symmetry is more obvious in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='3(c) since nr = 0 is used for 3(b) whereas we used nr = 3 for 3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Case (iii): g0 (y) = (eǫy − 1) /ǫy and g1 (y) = 1: This rainbow functions structure in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (23) implies E2 �eǫE/Ep − 1 ǫE/Ep �2 − m2 = Knr,ℓ =⇒ E = 1 2β ln � 1 ± � 4β2 (Knr,ℓ + m2) � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' β = ǫ 2Ep (31) In Figure 4(a) we plot the energy levels against β = ǫ/2Ep and observe eminent clustering in the positive energies as β grows up from just above zero (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=', β ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='001), whereas the negative energies are rapidly pushed further into the negative energy region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' This is very much related to the nature of the logarithmic structure of the energies in (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' This is also reflected on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' 4(b), where we show the effect of the magnetic field on the energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' 8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' 4: The energy levels of (31), using α = 1/2, m = kz = 1, so that (a) shows E against β = ǫ/2Ep for |eB◦| = 1, nr = 0, ℓ = 0, −1, −2, −3, −4, and (b) shows E against |eB◦| for β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='1, nr = 0, ℓ = 0, −1, −2, −3, −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' PDM KG-particles in cosmic string rainbow gravity spacetime We now consider a positive-valued dimensionless scalar multiplier f (r) = exp � 2ηr2� in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (12) so that M (r) = η2r2 + 2η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' This would suggest that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (18) now reads \uf8f1 \uf8f2 \uf8f3∂2 r − � ˜ℓ2 − 1/4 � r2 − 1 4Ω2r2 + ˜λ \uf8fc \uf8fd \uf8fe R (r) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (32) where ˜λ = g0 (y)2 E2 + g1 (y)2 � ˜B˜ℓ − k2 z − 2η � − m2 g1 (y)2 , |Ω| = | ˜B| � 1 + 4η2 ˜B2 (33) In this case, its exact solution is in the form of R (r) = C r|˜ℓ|+1/2 exp � −|Ω| 4 r2 � 1F1 � 1 2 + |˜ℓ| 2 − ˜λ 2|Ω|, 1 + |˜ℓ|, |Ω| 4 r2 � , (34) which takes the form of a polynomial of degree nr ≥ 0 for 1 2 + |˜ℓ| 2 − ˜λ 2|Ω| = −nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (35) Consequently, ˜λnr,ℓ = |Ω| � 2nr + |˜ℓ| + 1 � , R (r) = C r|˜ℓ|+1/2 exp � −|Ω| 4 r2 � 1F1 � −nr, 1 + |˜ℓ|, |Ω| 4 r2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (36) and g0 (y)2 E2 − m2 = g1 (y)2 ˜Knr,ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' ˜Knr,ℓ = �� ˜B2 + 4η2 � 2nr + |˜ℓ| + 1 � − ˜B˜ℓ + k2 z + 2η � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (37) Notably, we observe that the degeneracies discussed in the preceding section are removed as an effect of PDM settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' However, one should notice that as ˜B → 0 our ˜Knr,ℓ → 2η � 2nr + |˜ℓ| + 1 � + k2 z + 2η and consequently states with a specific ℓ = ±|ℓ| will emerge from the same ˜B = 0 value and split as ˜B grows up from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' We may now discuss the effects of different rainbow functions on the energy levels under the current PDM-settings of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' 9 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' 5: The energy levels of (38), using α = 1/2, m = kz = 1, so that (a) shows E against δ = ǫ/E2 p for |eB◦| = η = 1, nr = 0, ℓ = 0, ±1, ±2, (b) shows E against η for |eB◦| = 1, δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='1, nr = 0, ℓ = 0, ±1, ±2, ±3, and (c) shows E against |eB◦| for η = 1, δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='1, nr = 0, ℓ = 0, ±1, ±2, ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Case (i)-(a): g0 (y) = 1 and g1 (y) = � 1 − ǫy2: For such rainbow functions structures Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (23) results E2 − m2 = � 1 − ǫE2 E2p � ˜Knr,ℓ =⇒ E = ± � ˜Knr,ℓ + m2 1 + δ ˜Knr,ℓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' δ = ǫ E2p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (38) In Figures 5(a), we plot the energies against δ = ǫ/E2 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' We observe that, for a given radial quantum number nr, all positive/negative energy levels seem to converge into E = 0 as δ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' This is, in fact, the natural convergence of E in (38) as δ → ∞ (similar tendency as that observed for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='1(a), but here all states with ℓ = ±|ℓ| appear in the spectrum as a consequence of PDM parameter η ̸= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' However, both halves of the energy levels asymptotically converge to E = 0 value as δ → ∞ for a fixed value of the PDM parameter η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' 5(b), we plot the energies against the PDM parameter η and notice that the degeneracies associated with ℓ = +|ℓ| are removed as η increases from zero (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=', it is obvious the all states with .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='ℓ = +|ℓ| emerge form the same η = 0 point and split as η grows up).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' In Figure 5(c), moreover, we plot the energies against |eB◦| and observe that the energy levels with a specific ℓ = ±|ℓ| split as |eB◦| increases from the zero value and the separations between energy levels increase with increasing magnetic field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Case (i)-(b): g0 (y) = 1 and g1 (y) = √1 − ǫy: This rainbow functions model in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (23) implies E2 − m2 = � 1 − ǫ E Ep � ˜Knr,ℓ =⇒ E = −β ˜Knr,ℓ ± � β2 ˜K2 nr,ℓ + ˜Knr,ℓ + m2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' β = ǫ 2Ep .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (39) In Figures 6(a) and (b), we plot the energy levels against β = ǫ/2Ep and |eB◦|, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' It is obvious that the symmetry of the energy levels about E = 0 is broken as an effect of such rainbow functions structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Yet, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='6(a) we notice that, for a given radial quantum number nr, only positive energy levels converge into E = 0 as δ → ∞, whereas in the lower half we observe that the splitting in the energy levels increases as β increases from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' In the lower half of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' 6(b), we see that the energy levels separation increases as the magnetic field strength increases from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' This effect is obvious from the form of the energy levels in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (39), as the first negative term increases the negativity of the energy levels and breaks the symmetry of the energy part of the second term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='6(c) we show the effect of the PDM settings on the energy levels and follow the same scenario as that for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='5(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' 10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' 6: The energy levels of (39), using α = 1/2, m = kz = 1, so that (a) shows E against β = ǫ/2Ep for |eB◦| = η = 1, nr = 0, ℓ = 0, ±1, ±2, (b) shows E against |eB◦| for η = 1, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='1, nr = 0, ℓ = 0, ±1, ±2, and (c) shows E against η for |eB◦| = 1, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='1, nr = 0, ℓ = 0, ±1, ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' 7: The energy levels of (40), using α = 1/2, m = kz = 1, so that (a) shows E against γ = ǫm/Ep < 1 for |eB◦| = η = 1, nr = 0, ℓ = 0, ±1, ±2, (b) shows E against |eB◦| for η = 1, γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='1, nr = 0, ℓ = 0, ±1, ±2, and (c) shows E against η for |eB◦| = 1, γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='1, nr = 0, ℓ = 0, ±1, ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Case (ii): g0 (y) = g1 (y) = (1 − ǫy)−1: Such rainbow functions assumption in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (23) yields E2 − ˜Knr,ℓ = � 1 − ǫ E Ep �2 m2 =⇒ E = −mγ ± � ˜Knr,ℓ (1 − γ2) + m2 1 − γ2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' γ = ǫm Ep < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (40) In Figure 7(a) we plot the energy levels against γ = ǫm/Ep to observe the rainbow gravity effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' We clearly see that the symmetry in the energy levels is broken as an effect of the first term � −mγ/ � 1 − γ2�� in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='(40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' 7(b), the energy levels are plotted against |eB◦| so that the magnetic field effect on the energy levels is shown, and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' 7(c) we show the effect of the PDM settings on the energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Case (iii): g0 (y) = (eǫy − 1) /ǫy and g1 (y) = 1: This rainbow functions structure in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (23) implies E2 �eǫE/Ep − 1 ǫE/Ep �2 − m2 = ˜Knr,ℓ =⇒ E = 1 2β ln � 1 ± � 4β2 � ˜Knr,ℓ + m2 �� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' β = ǫ 2Ep (41) 11 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' 8: The energy levels of (41), using α = 1/2, m = kz = 1, so that (a) shows E against β = ǫ/2Ep for |eB◦| = η = 1, nr = 0, ℓ = 0, ±1, ±2, (b) shows E against |eB◦| for η = 1, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='1, nr = 0, ℓ = 0, ±1, ±2, and (c) shows E against η for |eB◦| = 1, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='1, nr = 0, ℓ = 0, ±1, ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' In Figure 8(a) we plot the energy levels against β = ǫ/2Ep and observe eminent clustering in the positive energies as β grows up from just above zero (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=', β ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='001), whereas the negative energies are rapidly pushed further into the negative energy region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' In Figures 8(b) we show the effect of the magnetic field and in 8(c) we show the effect of the PDM settings on the energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' CONCLUDING REMARKS In this paper, we started with the PDM KG-particles in the cosmic string rainbow gravity spacetime (6) in a non- uniform magnetic field B = ∇×A = 3 2B◦r ˆz introduced by Aϕ = 1 2B◦r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' We reduced the corresponding KG-equation into the one-dimensional form of the two-dimensional radial Sch¨odinger-oscillator like equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' We have first studied the effects of rainbow gravity on the energy levels of KG-particles with constant mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Next, we have included the PDM settings through the dimensionless positive-valued scalar multiplier f (r) = exp � 2ηr2� , in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (12), so that an oscillator-like interaction is introduced as M (r) = η2r2 + 2η (documented in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='(32)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' This would, in turn, suggest that a KG-particle in a cosmic string spacetime (5) in a non-uniform magnetic field B = ∇ × A = 3 2B◦r ˆz and/or in M (r) = η2r2 + 2η would introduce the so called KG-oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Consequently, the current methodical proposal discusses the rainbow gravity effect on KG-oscillators manifested by the non-uniform magnetic field and/or the PDM setting M (r) mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' In so doing, we have considered four pairs of rainbow functions: (i)-(a) g0 (y) = 1, g1 (y) = � 1 − ǫy2, (i)-(b) g0 (y) = 1, g1 (y) = √1 − ǫy, (ii) g0 (y) = g1 (y) = (1 − ǫy)−1, and (iii) g0 (y) = (eǫy − 1) /ǫy, g1 (y) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' In the light of our experience above, we recollect that the most general KG-oscillators’ energies in cosmic string rainbow gravity spacetime are given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' (37) (which yields the constant mass energies of (23) for the PDM parameter η = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' and may summarize the effects of rainbow gravity on the spectra of KG-oscillators in cosmic string spacetime as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Whilst, for the pair of rainbow functions � g0 (y) = 1, g1 (y) = � 1 − ǫy2 � the KG-oscillators energies about E = 0 are observed to be symmetric (documented in Figures 1 and 5), the symmetry of KG-oscillators energies for the rest of the rainbow functions pairs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=', � g0 (y) = 1, g1 (y) = √1 − ǫy � , � g0 (y) = g1 (y) = (1 − ǫy)−1� , 12 and [g0 (y) = (eǫy − 1) /ǫy, g1 (y) = 1]) are observed broken (documented in Figures 2, 3, 4, 6, 7, and 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Moreover, rainbow gravity appeared to have no effect, whatsoever, on either introducing new type of degeneracies or removing natural degeneracies associated with the spectra of the KG-oscillators in cosmic string spacetime (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=', at lim y→0 gk (y) = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' k = 0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Finally, it could be interesting to report that for the rainbow functions � g0 (y) = g1 (y) = (1 − ǫy)−1� , one may observe that when γ = ǫm/Ep = 1 is considered (although we considered γ < 1 in (30) and (40), it remains a feasible limit) there are energy states to fly away and disappear from the spectrum (a phenomenon that have been observed in the non-Hermitian PT - symmetric Schr¨odinger Coulomb problem [49] ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' To the best of our knowledge, the current study has never been published elsewhere (within the lines discussed above, of course).' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Mena Marugan, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' D 70 (2004) 124003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' [3] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Amelino-Camelia, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Momennia, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' C 76 (2016) 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' [22] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Hamil, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' L¨utf¨uoˆglu, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Methods Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' 19 (2022) 2250047.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Momennia, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Eslam Panah, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Panahiyan, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Dark Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' 16 (2017) 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' 13 [31] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Bezerra, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Christiansen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Cunha, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Muniz, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' D 96 (2017) 024018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' [32] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Aounallah, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Pourhassan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Hendi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Faizal, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' C 82 (2022) 351.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' [33] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Bakke, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Mota, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Plus 133 (2018) 409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' [34] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Bakke, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Mota, Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' 52 (2020) 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' [35] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' von Roos, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' B 27 (1983) 7547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' [36] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Mustafa, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' A 384 (2020) 126265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' [37] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Mustafa, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Mazharimousavi, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Phys 46 (2007) 1786.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' [38] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Mustafa, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Algadhi, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Plus 134 (2019) 228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' [39] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' dos Santos, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Gomez, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' da Costa, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Mustafa, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Plus 136 (2021) 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' [40] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Khlevniuk, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Tymchyshyn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' 59 (2018) 082901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' [41] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Mustafa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' 48 (2015) 225206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' [42] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' dos Santos, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Gomez, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' da Costa, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Mustafa, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Plus 136 (2021) 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' [43] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Mustafa, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' 440 (2022) 168857.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' [44] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Mustafa, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' C 82 (2022) 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' [45] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Mustafa, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' 446 (2022) 169124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' [46] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Amelino-Camelia, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Ellis, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Mavromatos, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Nanopoulos, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' A 12 (1997) 607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' [47] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Amelino-Camelia, Living Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Relativ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' 16 (2013) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' [48] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Magueijo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Smolin, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' 88 (2002) 190403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' [49] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Mustafa, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Znojil, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' A: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} +page_content=' 35 (2002) 8929.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE5T4oBgHgl3EQfKQ5P/content/2301.05464v1.pdf'} diff --git a/sNFIT4oBgHgl3EQfyiuq/content/2301.11361v1.pdf b/sNFIT4oBgHgl3EQfyiuq/content/2301.11361v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..0f351ef440a82a14e65aba84f2c6ac8641d027bb --- /dev/null +++ b/sNFIT4oBgHgl3EQfyiuq/content/2301.11361v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c66d4bba288765c771911af7ca4a4399e1db3f6eab894bd677591f49067a3a9e +size 495381 diff --git a/sNFIT4oBgHgl3EQfyiuq/vector_store/index.pkl b/sNFIT4oBgHgl3EQfyiuq/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..724bded2c468867c22afe47c286a700a1bb742a5 --- /dev/null +++ b/sNFIT4oBgHgl3EQfyiuq/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2dff90830a3f20c4653fde5d193f37226a2ac9de5175f96855c86b8a58b75cca 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+++ b/vtAzT4oBgHgl3EQfP_vM/content/tmp_files/2301.01194v1.pdf.txt @@ -0,0 +1,2312 @@ +1 + +Extracting optical parameters of Cu-Mn-Fe +spinel oxide nanoparticles for optimizing air- +stable, high-efficiency solar selective coatings +Xiaoxin Wang*†, Can Xu, † and Jifeng Liu** +Thayer School of Engineering, Dartmouth College, 15 Thayer Drive, Hanover, New +Hampshire 03755, USA +†Co-first authors contributing equally to this work +Corresponding authors +*Xiaoxin.Wang@Dartmouth.Edu (X. Wang) +*Jifeng.Liu@Dartmouth.Edu (J. Liu) +Abstract +High-temperature Cu-Mn-Fe spinel-oxide nanoparticle solar selective absorber coatings are +investigated experimentally and theoretically. A reliable, general approach to evaluate absorption +coefficient spectra from the optical measurements of the nanoparticle-pigmented coatings is +developed based on solving the inverse problem using four-flux-radiative method. The derived +absorption properties of NP materials can be directly applied to predict the solar absorptance, +optimize the nanoparticle-pigmented coatings, and analyze the thermal degradation, which agree +well with the experimental results. The analysis reveals that the Cu-Mn-Fe spinel oxides are +fundamentally indirect bandgap ranging from 1.7 to 2.1 eV, while iron-free CuMn2O4 is a direct +bandgap material with Eg=1.84 eV. With the same coating thickness and nanoparticle load, the +solar absorptance ranks in the order of Mn2O3 < MnFe2O4 < CuFe2O4 < CuFeMnO4 < CuMn2O4. +The optimized spray-coated iron-free CuMn2O4 NP-pigmented coating demonstrates a high solar +absorptance of 97%, a low emittance of 55%, a high optical-to-thermal energy conversion + +2 + +efficiency of ~93.5 % under 1000x solar concentration at 750ºC, and long-term endurance upon +thermal cycling between 750°C and room temperature in air. The optical parameter analysis +approach can be easily extended to other material systems to facilitate the searching and optimizing +high-temperature pigmented-solar selective coatings. +Keywords: solar selective coatings; spinel oxide nanoparticle; complex refractive index; thermal +efficiency; four-flux radiative method +Highlight: +1. Develop a general approach to extract the absorption coefficients and refractive indices of +pigment nanoparticles from the optical measurements based on solving the inverse problem of the +four-flux-radiative method. +2. Investigate and search for highly absorbing, high temperature Cu-Mn-Fe oxide pigmented +solar selective coatings in a systematic way. With the same coating thickness and nanoparticle +load, the pigment material performs in the order Mn2O3 < MnFe2O4 < CuFe2O4 < CuFeMnO4 < +CuMn2O4 in terms of solar absorptance +3 Demonstrate iron-free CuMn2O4 coating with a high solar absorptance of 96.9%, low emittance +of 55 %, and a high thermal efficiency of 93.5% under 1000x solar concentration at 750ºC that +also endures long-term thermal cycling between 750 ºC and room temperature in air. + + + + + + +3 + +1. Introduction +Concentrating solar power (CSP) systems utilize optical components to collect and convert +solar energy to thermal energy and then power heat engines to generate electricity. The widely +used thermal energy storage in CSP systems allows the solar energy to be dispatched on demand,1 +providing a great advantage over other non-dispatchable renewable energy sources such as wind +power and solar photovoltaic (PV) power. In order to reduce the levelized cost of energy (LCOE) +of Generation 3 CSP systems towards 50% power efficiency, solar selective absorber coatings are +required to possess long-term thermal stability at high temperatures ≥ 750 ºC in air.2 Cost analysis +reveals that durable oxidation-resistant solar selective coatings with solar absorptance αsolar ≥ 95% +and thermal emittance ε ≤ 60% can guarantee a reduction of the LCOE of CSP plant up to 12%.3 +Under this guideline, low-cost nanoparticle (NP)-pigmented solar absorbers with high solar +absorptance and moderate spectral selectivity is advantageous over highly selective yet more +costly and less thermally stable multilayer cermet coatings. +A very attractive nanoparticle (NP) pigment candidate for solar selective coatings is the +mixed transition metal spinel oxides with a general formula AB2O4 (A, B = metal) because of their +diverse properties and wide availability in versatile applications as electrodes4, catalysts5, magnetic +materials6,7. Their tunable optical properties and high-temperature stability in air are especially +suitable for solar absorber pigments 8-18. These include Co oxide8,9, Cu-Co oxide10,11, Mn-Co +oxide10, Cu-Mn oxide12, Cu-Ni-Co oxide13, Cu-Co-Mn oxide14, Cu-Cr-Mn oxide 15, and Cu-Mn- +Fe oxide16,17,18. In our previous work, we have demonstrated air-stable MnFe2O4 spinel oxide NP- +pigmented solar selective coatings with a high solar absorptance α ~93% and a thermal emittance +ε ~55% for ~90% optical-to-thermal conversion efficiency (ηtherm) using a small load of solar- +absorbing transition metal oxide nanoparticles (10.5 vol. %). 19 Based on our established + +4 + +quantitative approach and experimental findings, the solar absorptance of pigmented coatings is +realistically determined by the product of coating thickness (d) and NP volume concentration (f), +which is flexible for design and optimization of solar absorbing coatings.20 Furthermore, the +appropriate selection of NP oxides pigments can conveniently tune and maximize the solar +absorptance and solar selectivity, which is inherent to the NP materials, in contrast to strict control +of layer thickness (~nm) to take advantage of the interference effect in the multilayer cermet +coatings.21 +However, there are still a couple of challenges in further optimizing these spinel-oxide NP +solar coatings to approach the theoretical efficiency limit of 98%.20 (1) The fundamental optical +parameters of spinel oxide NPs are largely unavailable. The available data are only limited to some +particular compositions in the form of bulk single crystals or thin films.22, 23 Furthermore, defects, +cation inversion and substitution on A and B sites of the spinel lattice, and non-stoichiometry due +to synthesis methods may lead to a large deviation in the optical property of spinel NPs pigments +(10-50 nm in diameter) from that of the bulk material 22,24,25,26. Thus, it is important to develop a +reliable method to derive wavelength-dependent effective absorption coefficient of the NPs from +the measured optical spectra of the NP-pigmented composites. (2) The origin of the prominent +anomalous sub-band-gap absorption in the near infrared (NIR) regime of the solar spectrum +remains unclear in literature, which is important in order to optimize the solar selectivity. +Mechanisms range from charge transfer between bivalence-metal ions occupying distorted +octahedral or tetrahedral sites (e.g. in MnxFe3-xO4+γ single crystals22) to Urbach tail absorption (e.g. +in CuCoOx)27 to chemical substitution and partially inverted configuration between tetrahedral A +sites and octahedral B sites. 28 Therefore, a more accurate understanding of the electronic + +5 + +transitions in spinel nanoparticles should be developed by studying and comparing more +compounds of the same family. +Recently, we have investigated CuMn2-xCrxO4 spinel oxide NP-pigmented solar selective +coating system and demonstrated high thermal efficiency >94% in air at 750 ºC.29 This initial +success opens a large exploration space to further understand the impact of transition metal cationic +species and their lattice sites on the performance of high-temperature solar selective absorbers. In +this paper, a group of spinel Cu-Mn-Fe oxide NPs are systematically investigated as efficient high- +temperature solar selective absorbers. This is also scientifically motivated by a comparison of Fe +vs. Cr alloying into CuMn2O4 NPs to gain more understanding, considering that Fe and Cr are on +either side of Mn in the periodic table. A reliable, general approach to extract the absorption +coefficient spectra from the optical measurements of the NP-pigmented coatings is developed +based on solving the inverse problem using four-flux-radiative method. These derived effective +scattering and absorption cross sections of synthesized NPs are then directly input into the four- +flux radiative model to optimize the pigmented solar selective coatings. The optimized CuMn2O4 +pigmented solar selective coating on the Inconel substrate demonstrates a high solar absorptance +α =96.9% and a low thermal emittance ε =55.0% at 750 °C under 1000x solar concentration. This +performance represents one of the highest efficiencies (93.5%) under this service condition and +exceeds the requirements for Generation 3 CSP systems. The analysis reveals that iron-free +CuMn2O4 spinel is a direct bandgap material with Eg=1.84 eV, while the spinel group of Cu-Mn- +Fe are fundamentally indirect bandgap between 1.7~2.1 eV. Cyclic thermal stability testing +between 750ºC and room temperature shows that the crystalline phase and cationic distribution of +Cu-Mn-Fe spinel oxides are generally very durable, though some low valence states of transition +metals might be oxidized to high valence states gradually and degrade the solar absorptance. Iron- + +6 + +free CuMn2O4 (synthesized Syn42 NP) exhibits the best long-term thermal sustainability for long- +term thermal cycling, each cycle comprising 12h at 750°C and 12 h cooling to room temperature +in air. The optical spectra curve-fitting and modeling approach developed here to extract the +fundamental optical parameters of spinel NPs is applicable to other NP-pigmented solar coating +systems to further optimize the efficiency. +2. Experimental and Modeling Methods +2.1 NP synthesis and spray coating deposition + Cu-Mn-Fe oxide NPs were synthesized by a co-precipitation method. Designed amounts +of Cu2+, Mn2+ and Fe3+ ions from Cu(NO3)2, Mn(NO3)2 and FeCl3 were mixed with 100 mL +deionized water to form solutions. After mechanically stirring, NaOH solution was added dropwise +to the homogeneous mixture to adjust the pH value to 12 to facilitate the co-precipitation reaction. +The precipitated NP were then collected after centrifugation, and washed repeatedly by deionized +water to remove the excessive ions. Further drying, recrystallization and grinding steps were +followed. As a comparison, commercially available MnFe2O4 NPs (purity: 99.99%, average +diameter: 28 nm) and Mn2O3 NPs (purity: 99.99%, average diameter: 30 nm) were purchased from +U.S. research Nanomaterials Inc. +To fabricate NPs-pigmented silicone solar selective coatings, synthesized NPs were +well dispersed in xylene-diluted silicone matrix Bluesil through ultrasonic bath to form uniform +precursors with different viscosities. The spray coatings were performed using a Houseales Mini +sprayer (10 ml) on quartz substrates and high temperature Ni-based super-alloys Inconel 625 sheet +coupons (1 in. × 1 in. square) heated to 80-120 ºC by a hot plate. Here Ni-based super-alloy Inconel +is chosen due to its good high-temperature mechanical behavior, oxidation resistance, and +corrosion resistance. Multiple sprays were carried out to achieve the optimal coating thickness at + +7 + +a given NP pigment concentration. These samples were annealed at 750 ºC for 24 hours and then +cooled down overnight for the following characterizations. All thermal cycling tests were +performed in a box furnace in air. Each cycling test includes 12 h at 750 °C and 12 h at room +temperature to mimic the day-night cycles in practical applications. +2.2 Characterization techniques + X-ray diffraction (XRD, Rigaku 007 X-ray Diffractometer, Cu Kα1 line, λ=0.15406 nm +operating at 40 kV/300 mA and a scanning rate of 2°/min from 10° to 80°) was conducted to reveal +information about crystal structure, phase weight percentage, average NP size, and cationic +distribution of as-synthesized or annealed NPs in the form of powder or embedded in the coatings. +High-resolution transmission electron microscopy images (TEM) were utilized to characterize the +morphology, size and crystal structure of as-synthesized nanoparticles. Scanning electron +microscopy (SEM, FEI XL-30 ESEM FEG, 20 kV, secondary electron (SE) mode) was performed +to study the surface morphology and coating thickness. Energy dispersive spectroscopy (EDS, +EDAX Si (Li) detector with Genesis software) was carried out to detect the chemical composition +and elemental distribution in pigmented coatings. Raman spectra were recorded at room +temperature using a confocal Raman imaging system and a laser radiation source operating at a +wavelength of 532 nm. + The reflectance spectra in the wavelength range of λ = 0.3 ~ 2.5 μm were obtained by using +a Jasco V-570 ultraviolet/visible/near infrared (UV/Vis/NIR) spectrometer equipped with a Jasco +ISN-470 integrating sphere to capture both specular and diffuse reflection. The reflectance spectra +in the mid infrared (MIR) region (λ=2.5 ~20 μm) for thermal emittance measurement was recorded +with a Jasco 4100 Fourier transformation IR (FTIR) spectrometer equipped with a Pike IR +integrating sphere in the range from 400 to 4000 cm-1. + +8 + +2.3 Four-flux radiative modeling and solution of the inverse problem + Four-flux radiative method is used to model the optical response of the NP-pigmented +coatings and derive the optical properties of nanoparticle material from optical measurements with +a Matlab curve fitting method. The details of four-flux radiative theory can be found in Refs. 19- +20. Firstly, the cross sections of scattering (Csca) and absorption (Cabs) of spherical NPs with a +given average size were calculated using Lorentz-Mie theory (Mieplot30) based on the relative +refractive index of NPs (n0+ik0) to that of the matrix material of the coating (nm+ikm). The +absorption (K) and scattering coefficients (S) of the pigmented coating are calculated by +K=Cabs*f/V and S=Csca*f/V by taking into account the NP volume V and NP volume fraction f. If +coatings consist of NPs with various sizes and crystalline phases, the effective K and S of the +coatings are defined as 𝐾 = ∑ +𝐶𝑎𝑏𝑠,𝑖 ∗ 𝑓𝑖 ∗ 𝑉𝑖 +𝑛 +𝑖=1 + and 𝑆 = ∑ +𝐶𝑠𝑐𝑎,𝑖 ∗ 𝑓𝑖 ∗ 𝑉𝑖 +𝑛 +𝑖=1 +, respectively. Then +the effective K, S of the coatings derived from the experiments or K, S values calculated from Cabs +and Csca, along with the coating thickness, are input to the four-flux radiative models to calculate +the reflectance R and transmittance T. For the coatings on the metal substrate such as Inconel, the +light transited through the coating should be absorbed completely by the metal (T=0), thus only +reflectance R is required. + In order to extract the complex refractive index of the NP material (n+ik) from the measured +R and T by solving the inverse problem, two iterative loops of data fitting are introduced as shown +in Fig. 1. One is based on the four-flux radiative method to derive effective K, S values of the +coatings from the measured R, T, while the other loop uses the analytical Mie scattering to obtain +the new refractive index of the NP material from the K, S. These loops are iterated until self- +consistency is reached between the experimental and theoretical optical spectra. The derived +optical parameters (i.e wavelength-dependent n and k) are then input to the four-flux radiative + +9 + +model to design the spinel oxide NP pigmented solar selective coatings, taking into account +multiple scattering. The optical-to-thermal conversion efficiency therm of the solar receiver is used +as figure of merit (FOM) to optimize solar selective coating design, given by Equation 1 +𝜂𝑡ℎ𝑒𝑟𝑚 = 𝐹𝑂𝑀 = +∫ +(1−𝑅(𝜆))𝐼(𝜆)𝑑𝜆−1 +𝐶[∫ +(1−𝑅(𝜆))𝐵(𝜆,𝑇)𝑑𝜆 +∞ +0 +] +∞ +0 + +∫∞ +0 +𝐼(𝜆)𝑑𝜆 += 𝛼𝑠𝑜𝑙𝑎𝑟 − +𝜀𝜎𝑇4 +𝐶𝐼𝑠𝑜𝑙𝑎𝑟 +(1) +Here 𝐼(𝜆) is the AM 1.5 solar spectral radiance at wavelength 𝜆; 𝐼𝑠𝑜𝑙𝑎𝑟 = 1000 𝑊/𝑚2 is the solar +power density integrated from the spectral radiance 𝐼(𝜆), 𝐵(𝜆, 𝑇) is the spectral blackbody thermal +emission at wavelength 𝜆 and temperature 𝑇, 𝑅(𝜆) is spectral reflectance, calculated with four- +flux model. 𝛼solar is the overall spectrally normalized solar absorptance, 𝜀 is the overall thermal +emittance at temperature T, 𝜎 = 5.67 × 10−8 +𝑊 +𝑚2 +𝐾4 is the Stefan-Boltzmann constant, and 𝐶 is solar +concentration ratio. In this paper, T=750°C=1023 K, and C=1000. The spectral range for the +integrals is 300 nm-16 μm. + +Fig. 1 Flow of extracting the complex refractive index of NPs based on iterative, self-consistent Mie +scattering theory and four-flux radiative method. + + + +Initial NP Material +NP Size +Medium Material +Refractive Index +D +Refractive Index nm, km +no, ko + Mieplot +Four Flux +Radiative +Method +Absorption & Scattering +Absorption & +Calculated +Cross Sections +Scattering Coefficients +Reflectance R & +Cabs, Cscat +K, S +Coating +Transmittance T +NP Volume V & +VolumeFractionf +Thickness t +Mieplot +Compare +Curve Fitting +NewNPMaterial +Measured +Refractive Index +Reflectance R & +n,k +Transmittance T10 + +3. Results and Discussion +3.1 Characterization of NP and NP-pigmented coatings + A group of Cu-Mn-Fe oxide NPs are synthesized by co-precipitation method and +characterized by XRD, TEM and Raman spectroscopy. The crystalline phases and compositions +are dependent on the starting material ratios of Cu:Mn:Fe, as summarized in Table 1 and detailed +in the Supporting Information. As an example, the TEM and XRD data of Syn24 are shown in Fig. +2. The data for other samples are summarized in Figs. S1-S3 of the Supporting Information. +Statistical analyses on TEM images of as-synthesized NPs (Fig. 2a and Fig. S1 (a)-(d)) show it is +reasonable to use an average NP diameter D~50 nm for the optical spectra curve-fitting and +theoretical modeling in the later context, as also indicated in Table 1. Both selected area electron +diffraction patterns (Fig. S1(e)(f)) and XRD spectra (Fig. 2 and Fig. S2) reveal the co-existence of +Mn2O3 and spinel oxide phases in the as-synthesized NPs due to phase separation for +nonstoichiometric samples. In fact, a secondary Mn2O3 phase is inevitably formed along with cubic +spinels when the ratio Cu:Mn ≤1.2 in CuxMn3-xO4 according to phase diagram,31 which also holds +true for Cu-Mn-Fe oxide family here. Furthermore, when XRD (311) peaks of spinel Cu-Mn-Fe +oxide NPs are examined closely (Fig. 2c and Fig. S2(c)), each consists of two peaks for most +annealed samples, which are ascribed to CuFe2O4 (~35.5°) and CuMn2O4 (~35.8°), respectively. +Generally speaking, CuMn2O4 dominates the spinel phase for a starting material ratio Mn:Fe > +4:1, such as Syn31, 33, and 42. Otherwise, a higher or even dominant percentage of CuFe2O4 spinel +phase exists, such as Syn 23-25. Syn29 and 35 have mixed CuMn2O4 and CuFe2O4. One particular +case is Syn26 (Cu:Mn:Fe=1:6:2) with a single dominant (311) peak at 35.6°, matching well with +the standard pattern of cubic CuFeMnO4 (ICDD-PDF No.20-03588) spinel structure32. + +11 + +34.5 +35.0 +35.5 +36.0 +36.5 +(311) +(311) +CuMn2O4 +2 + + + + +CuFe2O4 +(c) + +Fig. 2 Cu-Mn-Fe oxide NPs Syn24: (a) A TEM image of as-synthesized NPs showing a region dominated +by relatively small CuFe2O4 NPs. Larger Mn2O3 NPs from the same synthesis are shown in Fig. S1(b). (b) +XRD spectra of as-synthesized NPs compared to those annealed at 750°C for 24 hours in air. The XRD +data of the Inconel substrate is also shown as a reference. (c) Further deconvolution of (311) peaks of the +annealed samples shown in (b), indicating co-existence of CuFe2O4 and CuMn2O4 spinel oxides. +Table 1. A list of Cu-Mn-Fe oxide NP samples investigated in this study. The last 3 columns apply to the +NPs after annealing for 24h at 750ºC + +NP No. +Cu:Mn:Fe +TEM NP +Size (nm)** +XRD NP-24h + Size (nm) +Spinel Lattice +Constant (Å) +Spinel Weight +Percentage +Syn23 +1:0.5:1.5 +1:0.6:1.7* +41.5±11.9 +30 (sf); +8.387 +100% +Syn24 +1:3:1 +1:2.7:1.2* +49.5±11.3 +32 (sf); 47 (m) +8.382*** +46.3% +Syn25 +1:1:3 +1:1.2:3.2* +49.2±8.5 +24 (sf); 50 (m) +8.382*** +69.0% +Syn26 +1:6:2 +1:6.9:2.7* +53.8±14.1 +28(sfm); 40(m) +8.376 +71.2% +Syn29 +1:4:1 +1:3.9:0.9* +48.0±9.1 +52 (sf); 31 (sm); 54 (m) +8.370/8.291 +30.2% +Syn31 +1:6:1 +- +49 (sm); 58 (m) + 8.289 +43.3% +Syn33 +1:3:0.5 +- +21 (sm); 50 (m) +8.300 +29.8% +Syn35 +1:3:2 +- +34 (sf); 42 (sm); 56 (m) +8.374/8.295 +82.3% +Syn42 +1:2:0 +- +31 (sm); 49 (m) +8.305 +70.9% +Mn2O3 +Purchased +30 +- +- +0% +MnFe2O4 +Purchased +28 +- +- +99.99% + + *EDS measured element ratio of Cu:Fe:Mn; +sf: spinel phase dominated by CuFe2O4; sm: spinel phase dominated by CuMn2O4; +sfm: spinel CuFeMnO4; m: Mn2O3. +** As-synthesized *** Lattice constant of the dominant sf phase. + +(a) +Syn24 +100nm Spinel Cu-Mn-Fe oxide Inconel +After750°c +annealing +as-synthesized +nconel substrate12 + + Raman spectroscopy is used to further confirm the phases identified from XRD analysis, +as shown in Fig. S3. Due to the strong absorption in the visible range, Raman signals from Syn24 +and Syn42 are relatively weak. However, vibrational frequencies of the modes can be identified +by multi-peak fitting. The Raman data provides the evidence that annealed Syn24 has mixed +phases of CuFe2O4 and Mn2O3, and Syn42 has a dominant CuMn2O4 phase, in agreement with the +XRD analysis. + Coatings comprising as-synthesized Cu-Mn-Fe oxide NPs dispersed in silicone matrix are +deposited on quartz or Inconel substrates by a spray-coating method. Coatings are then treated at +750 °C for 24 hours to stabilize the cubic spinel phase of NPs and meanwhile improve the coating +adhesion to the substrate. SEM images of Syn24-pigmented coating on Inconel are shown in +Fig.S4. No cracks or warps are found on the coating surfaces. Coating thickness is ~8.5 µm +estimated from the cross section image in Fig.S4 (c). The volume concentration of NP is ~13% +calculated from the starting weight ratio of NPs and solid content of the silicone resin. +3.2 Extraction of Optical Parameters of NP-pigmented Coatings on Quartz +Following the flow in Fig. 1, the optical properties of synthesized Cu-Mn-Fe oxides NPs and +commercial NPs (Mn2O3 and MnFe2O4) are determined inversely from the measured reflectance +R and transmittance T of the corresponding pigmented coatings on quartz substrate. Fig. 3 shows +the optical data of Syn29, Syn24, and Syn42, representing a trend of increasing weight percentages +of spinel vs. Mn2O3 NPs (30%, 50% and 71%, respectively, as shown in Table 1). In Fig. 3(a) the +measured absorptance spectra (A=1-R-T) of synthesized and commercial NP-pigmented coatings +are nicely reproduced by the four flux radiative modeling using their derived optical parameters. +Clearly, a full consistency is reached through the iteration loops in Fig. 1 for extracting the +wavelength dependent complex refractive indices. Therefore, the approach developed here for + +13 + +solving the inverse problem of the four-flux method is very reliable. The correspondingly +determined effective absorption coefficient of pigmented-coatings and derived refractive indices +(both n and k) of NP materials are shown in Fig. 3(b) and Fig. 3(c), respectively, while Fig. 3(d) +shows the cross sections of scattering (Csca) and absorption (Cabs) for ~50 nm-diameter NPs based +on Mie scattering theory using refractive indices in Fig.3(c). +500 +1000 +1500 +2000 +2500 +0 +20 +40 +60 +80 +100 + Syn42_EXP. + Syn42_CAL. + Syn29_EXP. + Syn29_CAL. + Syn24_EXP. + Syn24_CAL. +Absorptance (%) +Wavelength (nm) + MnFe2O4_EXP. + MnFe2O4_CAL. + Mn2O3_EXP. + Mn2O3_CAL. +(a) +Quartz substrate + +500 +1000 +1500 +2000 +2500 +1E-4 +0.001 +0.01 +0.1 +1 +10 +Coating Absorption Coefficient (/m) +Wavelength (nm) + Syn24 + Syn29 + Syn42 + Mn2O3 + MnFe2O4 +(b) +500 +1000 +1500 +2000 +2500 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Refractive Index +Wavelength (nm) + Syn24_n + Syn24_k + Syn29_n + Syn29_k + Syn42_n + Syn42_k + Mn2O3_n + Mn2O3_k + MnFe2O4_n + MnFe2O4_k +(c) +500 +1000 +1500 +2000 +2500 +1E-20 +1E-19 +1E-18 +1E-17 +1E-16 +1E-15 + Syn24_Csca + Syn24_Cabs +D=50 nm + Syn29_Csca + Syn29_Cabs + Syn42_Csca + Syn42_Cabs + Mn2O3_Csca + Mn2O3_Cabs +Sca & Abs Cross Sections (m2) +Wavelength (nm) + MnFe2O4_Csca + MnFe2O4_Cabs +(d) + +Fig. 3 (a) Experimentally measured and theoretically modeled spectral absorptance of home-synthesized +Syn24, Syn29 and Syn42 and commercial Mn2O3, MnFe2O4 NP-pigmented coatings on quartz substrate, +showing excellent self-consistency is achieved after the iteration loops shown in Fig. 1. The extracted +optical parameters from the iterative fitting are shown in (b)-(d). (b) The correspondingly derived effective +absorption coefficients of the NP-coatings with different NP materials. (c) The corresponding derived +effective refractive indices of NP materials, including both the real part n and the imaginary part k +(extinction coefficient). (d) Calculated scattering and absorption cross sections of NPs with a size of 50 nm +based on the refractive index data in (c). + +14 + +In terms of solar absorption, the absorption curves of commercial Mn2O3 and MnFe2O4 +NP-coatings in Fig. 3(a) roll off fast from wavelength > 600 nm in a similar way, though MnFe2O4- +coating absorbs 5% more solar light than Mn2O3-coating. As spinel phases are incorporated, the +absorption spectrum is clearly extended to longer wavelengths to better cover the solar spectrum, +as shown in the curves for Syn 29, 24, and 42 (with spinel phase weight percentage increasing +from 30% to 50% to 70%). Though Syn24-coating demonstrates a slightly smaller absorption at +the wavelengths λ < 800 nm than Syn29, it absorbs notably more infrared light than Syn29-coating +in the wavelength range of 800~2500 nm. Overall, the iron-free Syn42 sample exhibits the highest +absorption at λ<2000 nm despite of a slightly small absorption than Syn24 at λ> 2000 nm. +As far as the influence of Mn2O3 phase is concerned, we found that the effective absorption +coefficients of NP-pigmented coatings incorporating spinel phases are 5-50x that of the reference +Mn2O3-pigmented coating in Fig. 3 (b). Therefore, Mn2O3 phase make insignificant contribution +to the solar absorption of synthesized NP-coatings, particularly in the wavelength range of +800~2500 nm. In other words, the curves in Fig. 3 (b) actually reflect the absorption trends +and features of Cu-Mn-Fe spinel oxides in the synthesized NPs. The derived absorption +coefficients of spinel NP materials excluding the effect of Mn2O3 will be presented later in Fig. 6. +Fig. 3(c) further shows that the real part of refractive indices across these samples are +almost constant at n~2.3 at λ= 800-2500 nm. The variation in the real refractive indexes in the +UV/Vis region at shorter wavelengths is associated with the strong band-to-band transitions seen +in the absorption spectra (Fig.3(a)). As compared with-Mn2O3 NP pigmented coating, the spinel +NP-containing coatings exhibit notable features in their imaginary parts of refractive indices: an +absorption edge/peak in the UV/vis regime followed by a broad absorption band in the infrared +band. The absorption edges/peaks are shifted with the spinel composition. Clearly, Syn24 + +15 + +(CuFe2O4 dominated), Syn29 (CuFe2O4+CuMn2O4) and Syn42 (CuMn2O4 dominated) are +arranged in an ascending order of redshift in UV/Vis absorption spectra due to the corresponding +bandgaps modifications summarized in Table 2, which will be further detailed in the analyses of +Fig. 4 and Fig. 6 later. + Last but not least, Fig. 3(d) shows that the absorption cross sections of the spinel NPs is 10~100 +times higher than the scattering cross sections for a NP diameter of 50 nm. Additionally, the +absorption cross sections vary between different synthesized NPs, in contrast to the scattering cross +sections insensitive to the NP materials discussed here. This is because the scattering cross-section +is mainly determined by the real part of the refractive index, n, which are similar across different +NPs as shown in Fig. 3(c). Furthermore, these effective absorption and scattering cross sections of +mixed phases in synthesized NPs will be used to optimize these NP-pigmented solar absorbing +coatings in Section 3.5. +3.3 Band-to-band and d-shell optical transitions of spinel NPs +3.3.1 Effective absorption coefficient spectra of NPs comprising both spinel and Mn2O3 +phases: The absorption coefficients of NP materials, independent on NP size, can be calculated as +α=4*pi*k/λ, where λ is the wavelength, and k is the imaginary part of the refractive index. Fig. 4 +reveals the details of bandgap energy determination from the absorption coefficients using Tauc +plots. The direct bandgap and indirect bandgap can be obtained by linear fit of (αE)2 and (αE)1/2 +vs. photon energy E=hv, respectively. α-Mn2O3 has three direct gaps at 1.20 eV, 1.97 eV and 2.96 +eV (Fig. 4(a)), in agreement with the reported 1.20 eV fundamental gap33. Fig. 4(b) and (c) +respectively show the effective direct and indirect band gaps of the NPs (comprising spinel oxides +and Mn2O3) in Syn24, Syn29 and Syn42. As mentioned earlier and further shown in Fig. 4a, the +absorption is mainly contributed by spinel NPs rather than Mn2O3 NPs since the latter has much + +16 + +lower absorption coefficient. Syn42 with dominant cubic spinel CuMn2O4 phase is a direct band +gap material with Eg=1.87 eV. Syn29, Syn24 and commercial MnFe2O4 NPs are fundamentally +indirect, with bandgaps of 1.74 eV, 1.93 eV and 2.62 eV, respectively. All NPs shown in Fig. 4 +also have a higher level direct gap of ≥ 2.7 eV. +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +0.0 +0.2 +0.4 +0.6 +0.8 +Egdir1=1.20 eV +Egdir3=2.96 eV +(E)2 (eV/cm)2 +Photon Energy (eV) + Mn2O3 +Direct Bandgaps +x 1011 +(a) +Egdir2=1.97 eV +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +0.000 +0.005 +0.010 +0.015 +0.020 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +0 +1 +2 +3 +4 +3.30 eV +3.50 eV +1.87 eV +2.73 eV +(E) +2 (eV/cm) +2 +Photon Energy (eV) + Syn24-mixed + Syn29-mixed + Syn42-mixed + MnFe2O4 +Direct Bandgap +3.25 eV +(b) +x 10 +12800 +600 +400 +500 +Wavelength (nm) +300 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +0 +200 +400 +600 +800 +1000 +1200 +2.62 eV +()1/2 (eV/cm)1/2 +Photon Energy (eV) + Syn24-mixed + Syn29-mixed + MnFe2O4 +Indirect Bandgap +1.93 eV +(c) +1.75 eV +15001000 +Wavelength (nm) +500 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.0 +20.0k +40.0k +60.0k +Sub-band Absorption Coefficient (/cm) +Photon Energy (eV) + Syn24 mixed + Syn29 mixed + MnFe2O4 +(d) + +Fig. 4 Tauc plots of (a) direct bandgaps of Mn2O3, (b) direct bandgaps of Cu-Mn-Fe oxide Syn24, Syn29, +Syn42 and MnFe2O4 NP materials and (c) indirect band gaps of Syn24, Syn29, MnFe2O4 NP materials, +respectively. (d) Sub-band absorption coefficient spectra of Syn24, Syn29, MnFe2O4 materials. + + +17 + +Besides band-to-band optical absorption, strong absorption below the indirect bandgap is +observed in Syn24 and Syn29 NPs, similar to commercial pure MnFe2O4 NPs. The sub-bandgap +absorption spectra are retrieved by subtracting the fitted band-to-band absorption spectra from the +overall absorption spectra for different samples, as shown in Fig. 4(d). The redshift of the sub- +band absorption coefficient peak in Fig. 4(d) enables Syn24 and Syn29 coatings to achieve stronger +solar absorption in the wavelength range > 800 nm in Fig. 3(a) compared to MnFe2O4 NP +pigmented coating. As will be further discussed later, these sub-bandgap infrared absorption bands +are due to d-d transitions of different transition metal cations. +3.3.2 Retrieving absorption spectra of spinel oxide NPs: In order to evaluate accurately the +optical properties of Cu-Mn-Fe spinel oxide NPs with different compositions, the influence of +Mn2O3 impurity phase is further excluded based on descriptions in Section 3.3 using the absorption +spectra in Fig. 4(a) and the weight percentages derived from XRD analyses listed in Table 1. Here +we use Syn24, Syn29, and Syn42 as examples to show the trend of transitions from predominantly +CuFe2O4 to mixed CuFe2O4+CuMn2O4 to CuMn2O4. As shown in Fig. 5(a), CuMn2O4 spinel oxide +NPs in Syn42 has a higher absorption cross section at wavelength <800 nm, while CuFe2O4 in +Syn24 exhibits a relatively strong absorption tail at the wavelength >1200 nm. The absorption +cross section curve of mixed CuFe2O4 and CuMn2O4 spinels from Syn29 lie between those of +CuFe2O4 and CuMn2O4. All the synthesized Cu-Mn-Fe spinel oxides including iron-free CuMn2O4 +have larger absorption cross sections (for NP diameter D=50 nm) than MnFe2O4 NP investigated +in our previous work Ref. [19]. Therefore, it is concluded that the pure spinel oxides in the aspect +of solar absorption performance are listed in the ascending order as MnFe2O41.84 eV. +This peak is between the tetrahedral site Cu2+ absorption band peaked at ~0.8 eV and that of Mn3+ +peaked at ~1.2 eV, biasing more towards the latter. 28, 36, 37 On the other hand, the peak positions +of MnFe2O4 are blueshifted to 1.75 eV and 2.35 eV for intervalence Fe2+-Fe3+ transitions. An +additional peak at 2.8 eV is ascribed to oxygen and Fe3+ ions in octahedral sites for MnFe2O4. The +strong sub-bandgap absorption in Cu-Mn-Fe spinel oxide is beneficial to solar absorption, +especially in the near infrared range. +Table 2. Bandgaps of synthesized NPs and commercial NPs, and the solar absorptance of NP- +pigmented coatings on Inconel substrates. + +NP No. + +Spinel Weight +Percentage +Effective Eg (eV)** +Spinel Eg (eV) +αSolar +Egindir +Egdir +Egindir +Egdir +Syn23 + +100% +1.94 +3.17 +1.94 +3.17 +91.0% +Syn24 + +46.3% +1.93 +3.25 +2.08 +3.28 +96.5% +Syn25 + +69.0% +1.88 +3.25 +2.02 +3.25 +89.8% +Syn26 + +71.2% +1.65 +3.16 +1.66 +3.13 +95.4% +Syn29 + +30.2% +1.74 +2.73 +1.75 +2.48 +95.3% +Syn31 + +43.3% +1.59 +3.10 +1.98 +3.11 +93.6% +Syn33 + +29.8% +1.61 +2.94 +1.83 +2.80 +95.7% +Syn35 + +82.3% +1.67 +2.97 +1.75 +2.99 +93.7% +Syn42 + +70.9% +- +1.87&3.50 +- +1.84&3.42 +96.9% +MnFe2O4 + +99.99% +2.62 +3.30 +2.62 +3.30 +88.4% +Mn2O3 + +0% + - 1.20,1.97& 2.96 +- +- +87.8% +*EDS measured element ratio of Cu:Fe:Mn +**The calculation of effective bandgaps takes into account Mn2O3 impurity. + + +20 + +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +0 +200 +400 +600 +800 +1000 +1200 +1400 +1600 +2.08 eV +()1/2 (eV/cm)1/2 +Photon Energy(eV) + Syn24 Spinel + Syn29 Spinel +(a) +1.74 eV +15001000 +Wavelength (nm) +500 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.0 +40.0k +80.0k +120.0k +160.0k +Sub-band Absorption Coefficient (/cm) +Photon Energy (eV) +Sub-band Absorption + Syn29 Spinel + Syn24 Spinel + Syn42 Spinel + MnFe2O4 +(b) + +Fig. 6 (a) Refined indirect bandgaps of Cu-Mn-Fe oxide spinels excluding impurity Mn2O3 phase from the +synthesized Syn24 and Syn29. (b) Deconvolution of sub-band absorption coefficient spectra of spinel +phases of Syn24, Syn29, and MnFe2O4 by Gaussian peak fitting. + +3.3.3 Overall comparison and summary: After successfully determining and comparing +the optical properties of NP spinel materials from their coatings on the quartz substrate for the NPs +in Syn24, 29, and 42, we further extend this method to the NPs from all the solar selective coatings. +The results of bandgaps and solar absorptance are summarized in Table 2 and plotted in Fig. 7. +More details are presented in the Supporting Information, including measured reflectance spectra +and coating absorption spectra in Fig. S5, extracted absorption coefficients of spinel oxide NP +material in Fig. S6, and wavelength-dependent dielectric function of spinel phases in Fig. S7. +Based on the behavior of wavelength-dependent dielectric function, the Cu-Mn-Fe spinel oxide +NPs are classified as Mn-substituted CuFe2O4 group (Syn23, Syn24, Syn25, Syn26, Syn35 ) and +Fe-substituted CuMn2O4 group (Syn29, Syn31, Syn33, Syn42), in agreement with the observations +in XRD analysis (Fig. S2(c)). +All the synthesized Cu-Mn-Fe oxide samples have a higher solar absorptance than that of +MnFe2O4 (88.4%) with the same deposition conditions. The best performance are Syn24 and + +21 + +Syn42 NP-coatings with a solar absorptance of 96.5%, 96.9%, respectively. Further considering +the data we reported recently in Ref. 29 for Cu-Mn-Cr spinel oxide NPs, we can summarize the +following results on modifying the optical properties of CuMn2O4 spinel NPs by Cr and Fe alloying: +(1) As shown in Fig. 7(a), in the case of Cu-Mn-Fe spinel oxide NPS, the indirect bandgap +at ~1.9 eV is almost independent of the Mn/(Mn+Fe) ratio, while the direct bandgap +shows a trend of decrease with increasing Mn composition. This correspondingly +increases the absorption coefficient in the visible light regime for Mn-rich NPs. +(2) As a result of (2), Fig. 7(b) shows that the solar absorptance generally increases with +the Mn/(Mn+Fe) ratio due to the redshift and the correspondingly enhanced direct gap +absorption in the visible regime. The relation is almost linear except for a couple of +outliers, which may be related to the detailed cationic site occupation of those spinel +phases. In contrast, substituting Mn with Cr tends to increase the solar absorptance +from ~97% to ~98% 29. This is an interesting correspondence to the periodic table, +where Cr and Fe are neighbors of Mn on either side and modify the solar absorptance +in opposite ways. +(3) Similarly, alloying with Cr redshifts the d-d transition IR absorption peak from 1.1 eV +to 0.95 eV, while alloying with Fe slightly blueshifts this peak to 1.25 eV, again shifting +the IR absorption band to opposite directions. It could be associated with the change in +ionic radii upon substitution of Mn ions. These features can be applied to finely tune +the IR absorption peak position to further optimize the solar selectivity. + +22 + +0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 + + +Bandgap (eV) +Mn/(Mn+Fe) Ratio + Indirect bandgap + Direct bandgap +(a) +0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 +89 +90 +91 +92 +93 +94 +95 +96 +97 + + +Solar Absorptance Solar (%) +Mn/(Mn+Fe) ratio +(b) + +Fig. 7 (a) Direct and indirect bandgaps vs. Mn/(Mn+Fe) ratio in Cu-Mn-Fe spinel oxide NPs. (b) Solar +absorptance of the NP-pigmented solar coatings as a function of Mn/(Mn+Fe) ratio in the NPs. + +3.4 Computational Optimization of Cu-Mn-Fe Oxide NP-Pigmented Coatings +Based on the optical parameters obtained in Section 3.3, the solar absorptance αsolar of +synthesized NP-pigmented coatings is optimized using the derived effective absorption and +scattering cross sections of NP (D=50 nm) including Mn2O3 impurity phase. For the calculations +of thermal emittance ε and overall efficiency ηtherm, the absorption and scattering cross sections are +obtained by extrapolating the refractive index data into the mid IR regime, and the silicone matrix +absorption in the IR range is also taking into account using reference silicon coating samples +without NPs. As shown previously in Refs. 19 and 29, the latter is the dominant factor in thermal +emittance. Since Syn24 and Syn42 offer the highest solar absorptance with the same volume +concentration and coating thickness, we only focus on Syn24 and Syn42 here with the +corresponding optical data in Fig. 3(d). +Fig. 8 shows color mappings of αsolar and ηtherm vs. NP volume concentration f and coating +thickness d of Syn24 and Syn42 coatings. Clearly, large regimes with excellent fabrication +tolerance are available to offer a high αsolar ≥ 97.3% and ηtherm ≥ 94.1% for Syn24 NP-coatings, + +23 + +and αsolar ≥ 97.7% and ηtherm ≥ 94.8% for Syn42 NP-coatings, respectively. Theoretically predicted +data points corresponding to the experimental parameters of f=13 vol. % and d=8.5 µm are also +indicated with red asterisks in Fig. 8 (a)-(d), in good agreement with the experimentally measured +αsolar and ηtherm listed in Table 2 and shown later in Fig. 11 (αsolar =96.5% , ηtherm = 93.1% for Syn24, +and αsolar =96.9% , ηtherm = 93.6% for Syn42 ). It further confirms that the derived optical properties +of NPs with our approach described in Fig. 1 can successfully optimize the NP-pigmented solar +selective coatings. This approach for inverse problem of four-flux-radiative modeling can also +deepen the understanding of the optical degradation during thermal cycle testing in section 3.5. + +Fig. 8 Theoretically modelled color mappings of solar absorptance and thermal efficiency vs. NP volume +concentration and coating thickness, (a)(b) Syn24 NP-coating, and (c)(d) Syn42 NP-coating. The +determined effective scattering and absorption cross sections of Syn24 and Syn42 are used in the four-flux + +a +Asolar +solar (% +ltherm +15 +15 +95 +90 +90 +96. +85 +.81 +.66 +10 +85 +10 +96. +:29 +93.17 +80 +* +80 +* +95. +78 +92.68 +75 +75 +92.20 +5 +5 +70 +5 +10 +15 +5 +10 +15 +Asolar +αsolar (% +Itherm +15 +15 +IItherm +94 +96 +92 +94 +97. +90 +92 +10 +10 +88 +90 +* +96.93 +88 +86 +96.65 +96.11 +86 +84 +96.38 +93.50 +5 +5 +5 +10 +15 +5 +10 +1524 + +radiative modeling, respectively. The red stars indicate the NP-pigmented coatings with a NP volume +fraction of 13% and a coating thickness of 8.5 µm. +3.5 Cu-Mn-Fe Oxide NP-Pigmented Solar Coating Optical Performance and Thermal +Cycling Tests +3.5.1 Thermal stability testing: We deposited the optimized NP-pigmented coatings +(volume fraction f=13 vol.%, thickness d~8.5 µm) on Inconel for optical measurement and thermal +cycling tests at 750°C in air. Each simulated day-night thermal cycle comprises 12 h at 750 ºC in +air and 12 h cooling to room temperature. The samples are tested for 30-60 day-night cycles. The +NPs used in the coatings include Syn23 (dominated by CuFe2O4), Syn24 (Mn-substituted CuFe2O4 +and 54 wt.% Mn2O3), Syn26 (CuFeMnO4 and 29 wt.% Mn2O3), Syn29 (Mn-substituted CuFe2O4, +Fe-substituted CuMn2O4 and 70 wt.% Mn2O3 ) and Syn42 (CuMn2O4 and 29 wt.% Mn2O3). The +thermal stability of commercial MnFe2O4-coatings can be found in our previous work19. SEM and +X-ray are used to identify the influences of thermal treatment on micro-morphology and crystalline +phases. No cracks, warps or flaking are observed in the coatings on Inconel substrate after long- +term thermal stability test. Negligible change is observed in the microstructure of a typical example +(Syn24 coating) after 60 simulated day-night cycles between 750°C and room temperature in air, +comparing SEM images in Fig. S8 after the thermal cycles with those in Fig. S4 before thermal +cycles. XRD data indicate that the Cu-Mn-Fe oxide spinel phases and Mn2O3 phase are relatively +stable. As shown in Fig. S9(a), Syn23 with nearly 100% Fe-rich spinel oxide start to decompose +into more stable phases, including Fe2O3, after 20 thermal cycles, similar to the behavior of +MnFe2O4 after long-term annealing at 750ºC reported in our previous work19. By comparison, +incorporating CuMn2O4 greatly enhance the spinel oxide phase stability. For example, Fig. S9 (b) +reveals that the change of spinel weight percentage in Syn24 is within 2 wt.% after 60 thermal +cycles, and no phase decomposition was observed. This is also the case for CuMn2O4-dominated + +25 + +spinel oxide NPs in Syn 42. Therefore, increasing Mn contents not only increases solar absorptance +as shown in Fig. 7(b), but also enhances the spinel phase stability compared to their Fe-rich +counterparts. + +Fig. 9 Comparison of reflectance spectra in the solar spectral range after different thermal cycles between +750°C and room temperature in air (a) Syn24-coating on Inconel, and (b) Syn42-coating on Inconel. Each +simulated day-night thermal cycle comprises 12 h at 750 ºC in air and 12 h cooling to room +temperature. +The thermal stability of optical performace can be further revealed in the reflectance +spectra of Cu-Mn-Fe oxide NP- coatings after various thermal cycles. The reflectance spectra of +NP-pigmented coatings in the solar spectrum range after different thermal cycles are shown in Fig. +9 (Syn24 and Syn42) and Fig.S10 (Syn23, Syn26 and Syn42). The reflectance of Syn24-coating +in the solar spectral regime increase largely for the first 10 cycles, and then gradually increase with +more thermal cycles. Correspondingly, αsolar of Syn24-coating drops by 0.9% from 96.7% to +95.8% after the first 10 cyclic thermal tests, and drops 0.7% after another 20 cycles and 0.5% after +30 cycles. The reflectance spectra of Syn42 are stable and αsolar maintains at ~97% vs. the thermal +cycles. The optical degradation of Syn24-coating mechanism will be discussed later. On the other +hand, the typical reflectance spectra in the infrared range are shown in Fig.S11 (Syn24 and Syn42) +for the thermal emittance calculation. The big dip at ~8000 nm in the reflectance curves + +26 + +corresponds to the absorption bands of the silicone resin. It is possible to reduce the thermal +emittance loss and improve the solar absorption selectivity by choosing appropriate better silicone +materials with less thermal emittance. + Accordingly, the solar absorptance, thermal emittance and thermal efficiency as a function +of thermal cycles are summarized in Fig. 10. The trends of thermal efficiency ηtherm are similar to +solar absorptance αsolar, indicating that thermal emittance εtherm loss is less significant under the sun +concentration ratio C=1000 based on Equation 1. In contrast to non-selective Pyromark +(αsolar~97%, εtherm~87%), all synthesized-NP coatings exhibit solar absorption selectivity with the +thermal emittance below 60%, which can be further reduced by choosing better low-emittance +high-temperature silicone resin. Syn42-coating has the best thermal stability, with negligible +degradation in the optical performance after 30 thermal cycles, i.e. αsolar ~97%, εtherm ~55% and +ηtherm ~93.5%. These performances satisfy the requirements of the next-generation, high- +temperature solar selective coatings (αsolar > 95%, and εtherm < 60%) to ensure LCOE reduction. +Coatings consisting of Mn-substituted CuFe2O4 (Syn24), Fe-substituted CuMn2O4 (Syn29) or +CuMnFeO4 (Syn26) show a similar trend of degradation in the solar absorptance and thermal +efficiency with thermal cycles. The thermal efficiency decreases from 93.1% to 90.1% for Syn24 +coatings after 60 thermal cycles, from 92.2% to 89.1% for Syn29-coating after 30 cycles, and +91.8% to 90.6% for Syn26-coating after 20 cycles. By contrast, αsolar and ηtherm of Syn23-coating +with nearly pure Fe-rich CuFe1.7Mn0.6O4 spinel increases from 87.8% to 90.2% due to the +emergence of crystalline rhombohedral Fe2O3 (JCPDS no. 33-0664) phase (Fig. S9(a))38. This +behavior is similar to that of MnFe2O4 NPs, where thermodynamically driven spinel phase +decomposition at 750ºC actually increases rather than decrease the thermal efficiency. Overall, the +thermal stability of Cu-Mn-Fe oxide NP-coatings at 750°C in the air ranks in the order: Fe- + +27 + +substituted CuMn2O4 ≈ Mn-substituted CuFe2O4 ≈ CuMnFeO4 ≪ CuMn2O4 in terms of optical +performances and phase stability. On the other hand, CuFe2O4 tends to decompose after long-term +thermal cycles with ~2% increase in thermal efficiency, although still lower than the case of +CuMn2O4. It also suggests that the iron ions should be tightly linked to the performance +degradation. This assumption will be confirmed with coating absorption coefficients derived from +the reflectance spectra in Fig. 9. +0 +10 +20 +30 +40 +50 +60 +86 +88 +90 +92 +94 +Thermal Efficiency therm (%) +Thermal Test Cycles + Syn23_coating + Syn24_coating + Syn25_coating + Syn29_coating + Syn42_coating +(a) +0 +10 +20 +30 +40 +50 +60 +90 +92 +94 +96 +98 +Solar Absorbance (%) +Thermal Test Cycles + Syn23_Coating + Syn24_Coating + Syn26_Coating + Syn29_Coating + Syn42_Coating +(b) +0 +10 +20 +30 +40 +50 +60 +40 +45 +50 +55 +60 +65 +Thermal Emittance (%) +Thermal Test Cycles + Syn23_Coating + Syn24_Coating + Syn26_Coating + Syn29_Coating + Syn42_Coating +(c) + +Fig. 10 Comparison of optical performance of NP-pigmented coatings as function of thermal test cycles at +750°C in the air (a) thermal efficiency, (a) solar absorptance and (c) thermal emittance. + +3.5.2 Degradation mechanism of Cu-Mn-Fe spinel NP-pigmented solar coatings: In +order to understand the degradation mechanism of Syn24-NP coatings, the effective coating +absorption coefficients are determined with the same approach described in Section 3.3. Fig. 11 +(a) compares the effective NP coating absorption coefficients vs. various thermal treatment cycles. +Some interesting features are identified: (1) With increasing thermal cycles, the effective coating +absorption coefficient is decreased, where the reduction amount is shown in Fig. 11(b). (2) the +reduction is more prominent for the first 10 cycles, i.e., a peak reduction of 0.1/um after 10 cycles, +and additional 0.05/um after another 50 cycles; (3) the reduction spans a broad wavelength range +between 500 nm and 2000 nm. The similarity between the 10 cycles-solar absorption reduction +spectra of Syn24 coating on Inconel and the quartz excludes the substrate influence, such as the + +28 + +oxidation of Inconel substrate. (4) the spectral shape of the reduction in the optical coefficients +coincides with that of the d-d transition optical absorption spectrum of Syn24 NP in Fig. 4(d), +which is replotted in Fig. 11(b) for comparison. Considering that the sub-band absorption peaks +are ascribed to the intervalence transitions between Fe2+ ions and Fe3+ ions in octahedral sites, the +reduction in these peaks indicates the number of Fe ions in octahedral sites decreases. +500 +1000 +1500 +2000 +2500 +0.1 +1 +Coating Absorption Coefficient (/m) +Wavelength (nm) + 0 cycles + 10 cycles + 20 cycles + 30 cycles + 40 cycles + 50 cycles + 60 cycles +Syn24_Coating on Inconel +(a) +500 +1000 +1500 +2000 +2500 +-0.10 +-0.05 +0.00 +0.05 +0.10 +0.15 +Reduction in +Coating Absorption Coefficient (/m) +Wavelength (nm) + Syn24 on Inconel after 60 cycles + Syn24 on Inconel after 10 cycles + Syn24 on Quartz after 10 cycles +(b) +-20.0k +0.0 +20.0k +40.0k + Syn24 Effective +sub-band Abs. Coef. +Sub-Band Absorption Coefficient (/cm) +0.5 +4 3 +2 +1 +Photon Energy (eV) + +Fig. 11 (a) Coating absorption coefficients of Syn24-coating on Inconel after various thermal test +cycles, (b) Reduction in coating absorption coefficients of Syn24-coating on Inconel after 10 +cycles and 60 cycles, and of Syn24-coating on Quartz after 10 cycles. Effective sub-band +absorption coefficient of Syn24 is also included for comparison. + X-ray photoelectron spectroscopy (XPS) analysis reveals that Cu1+, Cu2+, Mn3+, Mn4+ and +Fe3+ ions are simultaneously present in CuFeMnO4 in our previous work in Ref. 29. In the case of +Cu-Mn-Fe spinel oxides, some Fe2+ ions may exist due to partial reduction of Cu2+ to Cu1+ and +oxygen deficiency39. Here we examine the thermal-induced cation-redistribution in the tetrahedral +and octahedral sites by using the value of the intensity ratio between a pair of diffraction lines of +the spinel structure40,41. X-ray peak intensity Ihkl is proportional to the absolute value square of the +structure factors |Fhkl|2, multiplicity factor P for the plane (hkl), and the Lorentz polarization factor +Lp according to Buerger’s equation40. Diffraction peaks (220) and (422) are only related to the +tetrahedrally-coordinated cations, while (440) peak is attributed not only to cations in tetrahedral + +29 + +sites, but also to those in octahedral sites. Therefore, the intensity ratios of I220/I440 and I440/I422 are +very sensitive to the cationic distribution in the spinel structure42. In particular, a higher ratio of +octahedral to tetrahedral site occupation for Fe3+, Fe3+(oh)/Fe3+(th), will lead to a relative increase +in I440 compared to I220 or I422. Although Fig. S9 (b) shows no presence of new phases and +negligible change in spinel percentage for Syn24 upon thermal cycling, the intensity ratio I220/I440 +decreases from ~1.8 to ~1.5, and I440/I422 increases from ~1.8 to ~2.2 for Mn-substituted CuFe2O4 +spinel in the Syn24 coating after 60 thermal cycles (Fig. 12). It indicates a rise in the ratio of +octahedrally-coordinated to tetrahedrally-coordinated Fe3+ ions 38. Therefore, the most reasonable +explanation for the solar absorptance degradation of Cu-Mn-Fe oxide spinels is the oxidation of +Fe2+ ions in the octahedral sites upon thermal cycles, thus increasing the Fe3+(oh)/Fe3+(th) ratio +and suppressing the d-d optical transitions between Fe2+ ions and Fe3+ ions in the octahedral sites +to reduce the optical absorption. This is also consistent with the fact that iron-free CuMn2O4 and +Cu-Mn-Cr oxide NP (Syn42) coatings are both thermally stable upon 750°C/room temperature +thermal cycling in air (Ref. 29). +0 +10 +20 +30 +40 +50 +60 +1.4 +1.6 +1.8 +2.0 +2.2 +2.4 + I220/I440 + smooth line of I220/I440 + I440/I422 + smooth line of I440/I422 +XRD Line Intensity Ratio +Thermal Test Cycles + +Fig. 12 XRD Peak intensity ratios of I220/I440 and I440/I422 as function of thermal test cycles for Mn- +substituted CuFe2O4 spinel in the Syn24-coating. + +30 + +4. Conclusions +Complementing our previous studies on Cu-Mn-Cr spinel NP solar selective coatings 29, in +this paper Cu-Mn-Fe spinel oxide NPs are synthesized by co-precipitation method and +systematically evaluated for high-temperature solar selective coatings. A reliable, general +approach to evaluate the absorption coefficients from the optical measurements of the NP- +pigmented coatings is developed based on the inverse problem of the four-flux-radiative method, +which agrees very well between theoretical modeling and experiment. The derived absorption +properties of NP materials have been directly utilized to elucidate the direct and indirect bandgap +transitions in the visible regime as well as the d-d sub-band absorption in the IR regime, thereby +optimizing the solar absorptance and evaluating the degradation mechanism of optical +performance upon high temperature endurance tests. +Depending on the starting material ratio of Cu:Mn:Fe, the spinel phase may be dominated +by CuFe2O4, CuMn2O4 or a combination of CuFe2O4 and CuMn2O4. The analysis reveals that iron- +free CuMn2O4 spinel is a direct bandgap material with Eg=1.84 eV, while the spinel group of Cu- +Mn-Fe are fundamentally indirect bandgap between 1.7~2.1 eV. The large sub-band d-d transition +absorption peaks for Cu-Mn-Fe oxide spinels are located 1.15~1.30 eV and 2.00~2.05 eV, +corresponding to transitions between Fe2+ ions and Fe3+ ions in octahedral sites. Exactly opposite +to Cr alloying, Fe alloying tends to decrease the solar absorptance by blueshifting the direct gap +transition in the visible regime as well as the d-d transition peak in the near infrared regime. With +the same coating thickness and nanoparticle load, the solar absorptance ranks in the order of Mn2O3 +< MnFe2O4 < CuFe2O4 < CuFeMnO4 < CuMn2O4. On the other hand, the thermal stability at 750°C +in air ranks in the order: Fe-substituted CuMn2O4 < Mn-substituted CuFe2O4 ≈ CuMnFeO4 ≪ +CuMn2O4.. The significant sub-band absorption is suppressed by the oxidation of Fe2+ ions in the + +31 + +octahedral sites upon thermal cyclic tests, causing the degradation in the solar absorptance of Cu- +Mn-Fe oxide NP. +The systematic analysis and computational modeling enables us to optimize iron-free +CuMn2O4 sample to demonstrate a high solar absorptance of 96.9%, low emittance of 55.0%, and +a high thermal efficiency of ~93.5% under 1000x solar concentration at 750ºC in air, in good +agreement with the optimized values calculated with derived optical parameters. Furthermore, +CuMn2O4-NP coating maintains its 93.5% thermal efficiency after long-term thermal cycling +between 750°C and room temperature. This solar coating deign and optimization approach can be +readily extended to other material systems to speed up the searching and optimize high- +temperature pigmented-solar selective coatings. +Author Contributions +The authors make equal contributions to the paper. Xiaoxin Wang conducted the optical +modeling and data analysis, and drafted the manuscript. Can Xu fabricated the NP-pigmented +coatings and performed the thermal endurance testing as well as materials and optical +characterization. Jifeng Liu proposed the original idea of investigating Cu-Mn-Fe oxide NP +pigments and supervised the project. All authors contributed to the final editing of the paper. +Acknowledgement +This project was funded by U.S. Department of Energy, Solar Energy Technologies Office, +under the award numbers DE-EE0007112 and DE-EE0008530. +Supporting information +Supplementary data associated with this article can be found in the online version at : + +32 + +Data Availability +The raw data required to reproduce these findings are available upon request to the +correspondence author at Jifeng.Liu@dartmouth.edu. 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Jadhav, “X-Ray, IR and bulk magnetic properties of Cu1+xMnxFe2-2xO4 +ferrite system”, Journal of Materials Science 37 (2002) 1443-1448. Doi:10.1023/A:1014505620254 + + + +1 + + +Supporting Material +Extracting optical properties of Cu-Mn-Fe spinel +oxide nanoparticles for optimizing air-stable, +high-efficiency solar selective coatings +Xiaoxin Wang*†, Can Xu, † and Jifeng Liu** + +1 Characterization of nanoparticles (NPs) and NP-pigmented coatings +1.1 TEM Data +Fig.S1 shows ypical TEM images of some as-synthesized NPs with additional information +provided by the selected-area electron diffraction（SAED). The average NP sizes are 41.5 nm, +49.5 nm, 49.2 nm, 53.8 nm, and 48.0 nm for as-synthesized Syn23, Syn24, Syn25, Syn26 and +Syn29 in Fig.S1(a)-(d), respectively, with a standard deviation of ~10 nm. As-synthesized Syn25 +and Syn23 share similar SAED patterns presented in Fig.S1(e), in which the rings represent +different diffraction lines of cubic spinel nanocrystals. As-synthesized Syn26 has dominant +diffraction rings of cubic α-Mn2O3 crystals in Fig.S1(f), similar to the case of Syn29. + + + +2 + + + +Fig.S1 TEM images of as-synthesized NPs, (a) Syn23, (b) Syn26, (c) Syn25, and (d) Syn29. +Selected area electron diffraction (SAED) patterns of (e) as-synthesized Syn23, cubic spinel Cu- +Mn-Fe oxide phase, and (f) as-synthesized Syn26, Mn2O3 phase. + +(a)Syn23 +(b) Syn26 +100nm +100 +nm +(c)Syn25 +(d) Syn29 +100nm +100nm(e) Syn23 +(111) +(220) +(311) +1/nm +(222) +(400) +(440)(f) Syn26 +(211) +(321) +(422) +/m +(433)3 + +1.2 Supplemental XRD Data + +Fig. S2 (a) XRD spectra of as-synthesized NPs (b) XRD spectra of NPs annealed at 750°C for 24 hours +in air. (c) further deconvolution of (311) peaks of annealed samples listed in (b). The peaks at ~35.5º and +~35.8º correspond to CuFe2O4 and CuMn2O4, respectively. +The crystalline phases are further identified with the X-ray diffraction technique. Fig. S2(a) +compares XRD data of some as-synthesized nanoparticles. As discussed in the main text, a +secondary Mn2O3 phase is inevitably formed along with cubic spinels when the ratio Cu/Mn ≤1.2 +due +to +thermodynamic +phase +equilibrium. +As +expected, +as-synthesized +Syn23 +(Cu:Mn:Fe=1:0.5:1.5) is a pure spinel, Syn24 (Cu:Mn:Fe=1:3:1) has a small amount of Mn2O3, +while both Syn29 (Cu:Mn:Fe=1:4:1) and Syn26 (Cu:Mn:Fe=1:6:2) have a dominant Mn2O3 phase. +Higher temperature is required to thermodynamically promote the formation of a stable cubic +spinel phase from the co-precipitated nanocrystals, particularly in the cases of low ratio of Cu/Mn +< 0.3. Therefore, all the as-synthesized nanoparticle samples or as-deposited coatings are post- +annealed at 750°C for 24 h to enhance the spinel oxide phases, as shown in Fig. S2(b). The weight +percentages of crystalline phases are quantified by applying a standard intensity correction based +on the reference intensity ratio (RIR) from the sample to corundum (I/Ic, 4.50 for cubic Mn2O3 and +5.13 for cubic CuFe2O4 [S1]), as included in Table 1 of the main text. +Furthermore, when XRD (311) peaks of cubic Cu-Mn-Fe oxides are examined closely (Fig. +S2(c)), it consists of two peaks for most annealed samples, which are ascribed to CuFe2O4 (~35.5°) + +Mn203 +(a) +2 +(b) +Cu-Mn-Feoxide Spinel Mn,O3 +Cu-Mn-Fe oxide +Inconel +(c) +Inconel +2 +(311) +(111) +(220) +Syn2324h +(211) +(411) +(222) +400) +(422) +(511) +> (440) +Syn26_24h +4 +Syn23_24h +Syn26 +Syn2524h +6yn42-24h +Syn29 +Syn2424h +Syp25_24h +Syn24 +T +Syn2924h +Syn31_24h +Syn25 +Syn33_24h +Syn35_24h +Syn23 +Syn3524h +Syn3324h +Syn2424h +Syn29_24h +Syn3124h +Inconel +Syq26_24h +Syn4224h +1 +15 20 25 +30 +3540 +4550 +55 +6065 +20 +30 +40 +50 +60 +34.5 +35.0 +35.5 +36.0 +36.5 +20 () +20(°) +20(°)4 + +and CuMn2O4 (~35.8°), respectively. The Fe-substituted CuMn2O4 or Mn-substituted CuFe2O4 +leads to a slight shift in the diffraction peak positions. The lattice parameters of Cu-Mn-Fe oxide +spinels determined from the diffraction peaks in Fig. S2(b) are listed in Table 1. The value of the +lattice parameter for cubic spinel CuFe2O4 system varies from 8.37~8.39Å, consistent the reported +value of 8.384 Å[s2]. The lattice parameter of CuMn2O4 is in the range of 8.29~8.31Å, less than +that of CuFe2O4. The degree of tetrahedral vs. octahedral inversion or stoichiometry of the cations +may also affect the spinel lattice parameter.s3 +The crystal sizes of the mixed oxide phases are also derived via the Scherrer formula from +the deconvoluted (311) spinel peaks and Mn2O3 (222) peak in Fig. S2(c), respectively. As shown +in Table 1 of the maintex, Mn2O3 NPs exhibit a large particle size ~50 nm, while spinel Cu-Mn- +Fe oxides have a smaller size 30~40 nm. The difference in NP sizes of spinel and α-Mn2O3 phases +can explain the relatively large size deviation (~10 nm) observed in the mixed NPs (Fig. S1). +1.3 Raman Data +Fig. S3 shows the Raman spectra of the Syn24 and Syn42 on quartz taken in the range from +150 cm-1 to 750 cm-1. Five characteristic bands (A1g+Eg+3 F2g) are assigned to the cubic inverse- +spinel copper ferrite3. Specifically, 200 cm-1, 466 cm-1, and 558 cm-1 are originated from three F2g +modes, while 258 cm-1 and 655 cm-1 correspond to Eg and A1g modes, respectively. The appearance +of an extra spinel peak at 390 cm-1 is associated with a breakdown of the momentum conservation +rule due to nanoscale size. Another set of Raman active optical modes are related to cubic +Mn2O3( bixbyite) phase, which are located at 314 cm-1, 490 cm-1, 613 cm-1.4 For Syn42 NPs, the +signals of Mn2O3 are weak and the vibration modes of CuMn2O4 have a slight redshift compared +to CuFe2O4. So the Raman data provides further evidence that annealed Syn24 has mixed phases +of CuFe2O4 and Mn2O3, while Syn42 has a dominant CuMn2O4 phase. + +5 + +150 +300 +450 +600 +750 +0 +100 +200 +300 +400 +500 +600 +700 + m +610 cm-1 + s +639 cm-1 + s +576 cm-1 + m +460 cm-1 + s +490 cm-1 + s +390 cm-1 + m +313 cm-1 + s +258 cm-1 +Intensity(a.u.) +Raman Shift (cm-1) +s: Spinel CuFe2O4 +m: Mn2O3 +Syn24-Coating on Quartz + s +200 cm-1 +(a) +200 +300 +400 +500 +600 +700 +0 +100 +200 +300 +400 +Intensity (a.u.) +Raman Shift (cm-1) +s: Spinel CuMn2O4 +m: Mn2O3 +Syn42-Coating on Quartz +(b) + s +192 cm-1 + s +250 cm-1 + m +300 cm-1 + s +390 cm-1 + s +477 cm-1 + s +567 cm-1 s +636 cm-1 + +Fig.S3 Raman spectra of Syn24 and Syn42 NP-pigmented coating deposited on quartz and +annealed at 750°C for 24 hours in air. +1.4 SEM Images of the Coatings before Thermal Cycling +Fig. S4 (a)-(b) show the SEM images of Syn24-pigmented coating on Inconel. No cracks +or warps are found on the coating surfaces. Coating thickness is ~ 8.5 µm estimated from the tilted +view of cross section image in Fig. S4(c). From the starting weight ratio of NPs and solid content +of the silicone, the volume concentration of NP is calculated as 13%. These two parameters will +be used to determine the optical properties of Cu-Mn-Fe oxides. + +Fig.S4 SEM images of Syn24 NP-pigmented coatings on Inconel substrate after annealing at +750 °C for 24 hours, (a)-(b) surface microstructure, and (c) cross-section. + + + + +a(c) +20μm6 + +2. Extracted Optical Properties of NP-pigmented Solar Coatings on Inconel +After successfully extracting the optical properties of some NP spinel materials from their +coatings on the quartz substrate, we further extend this self-consistent method to the realistic solar +selective coatings on high-temperature alloy substrates such as Inconel. It is also found that the +difference in the optical properties (such as refractive index) extracted from the NP coatings +(Syn24,Syn29 and Syn42) on quartz and their coatings on Inconel are negligible. +Fig. S5(a) summarize the reflectance spectra of all synthesized NP- solar selective coatings +on the high-temperature Inconel substrate. With the same spray coating conditions, the coating +thickness is about 8.5 μm and NP volume concentration is 13%. One prominent feature of +pigmented coatings is that the reflectance or solar absorptance can be tuned by the choice of NP +materials. Based on the solar absorption performance, these coatings can be classified into three +groups. Syn24, Syn33 and Syn42 NPs enable lowest reflectance <10% in the full solar spectrum, +while the reflectance of Syn23 and Syn25 NP-pigmented coatings continue increasing with the +wavelength and reach to >25% at λ=2500 nm. The curves of Syn26, Syn29, Syn31 and Syn35 NP- +coatings lie between two groups above-mentioned, but they exhibit different trends. The +reflectance of Syn35 and Syn31 rolls up slowly from 5% to 10% and 15%, respectively. Syn29 +and Syn26 NP-coatings have a low reflectance ~5% for λ<1200 nm and a gradual increase in +reflectance up to 15% at λ=2500 nm. Overall, all NP-pigmented coatings have 95% absorption for +λ<500 nm but have a pronounced difference in the solar absorption for λ>500 nm. The +distinguished behavior in optical absorption gives rise to various solar absorptance values, as +shown in Table 2. All the synthesized Cu-Mn-Fe oxide samples have a higher solar absorptance +than that of MnFe2O4 (88.4%) with the same deposition conditions. The best performance are +Syn24 and Syn42 NP-coatings with a solar absorptance of 96.5%, 96.9%, respectively. + +7 + +The derived effective absorption coefficients of coatings on Inconel are shown in Fig. +S5(b). Syn23 and Syn 25 NPs have a lower effective coating absorption coefficient, while Syn42 +and Syn29 show the highest value, again indicating that CuMn2O4 is preferred over CuFe2O4 in +terms of solar absorption. Following the same procedure described in Section 3.3, the information +of direct and indirect bandgaps of effective synthesized NP with different compositions is listed in +Table 2. Besides a large direct bandgap ranging from 2.9 eV to 3.3 eV, the synthesized NPs (except +iron-free Syn42 NP) have a fundamental effective indirect band gap (1.59~1.94 eV). +500 +1000 +1500 +2000 +2500 +0 +5 +10 +15 +20 +25 +30 +35 +40 +Reflectance (%) +Wavelength (nm) + Syn23 + Syn24 + Syn25 + Syn26 + Syn29 + Syn31 + Syn33 + Syn35 + Syn42 +Inconel +(a) +` +500 +1000 +1500 +2000 +2500 +0.01 +0.1 +1 +10 +Coating Absorption Coefficient (/m) +Wavelength (nm) + Syn23 + Syn24 + Syn25 + Syn26 + Syn29 + Syn31 + Syn33 + Syn35 + Syn42 +(b) + +Fig. S5 Comparison of (a) measured reflectance spectra and (b) derived effective absorption coefficients of +synthesized NP-pigmented coatings on Inconel substrates. + +Considering a large variation in the spinel weight percentage among those synthesized NPs +listed in Table 2 of the maintext, further efforts are made to extract the optical properties of pure +spinel phases excluding Mn2O3 phase influence. Fig. S6 compares the absorption coefficients of +spinel NP materials, which are also used to obtain the bandgap information (Table 2). The indirect +band gaps of Cu-Mn-Fe oxide spinel phase shows a slight increase, in the range of 1.67~2.08 eV. +Fig.S6 also suggests that with the same coating thickness and nanoparticle load, the solar +absorptance of the pure spinel oxides ranks in the order of MnFe2O4 (purchased) < CuFe2O4 (in + +8 + +Syn23) < CuFeMnO4 (in Syn26) 1.25 eV), +while Fe-substituted CuMn2O4 from Syn33 and Mn-substituted CuFe2O4 from Syn24 have a broad +low-energy absorption peak (photon energy <1.25 eV). The mixed CuFe2O4 and CuMn2O4 spinels +from Syn29 exhibit a higher absorption coefficient almost in the whole solar spectrum than +CuMn2O4 spinel from Syn42. However, the spinel percentage is only 30% in Syn29 vs. 70% in +Syn42, so 95.3% for Syn29 NP-coating vs. 96.9% for Syn42 coating, as listed in Table 1. The +optical transitions contributing to the absorption coefficients of Cu-Mn-Fe oxide spinels are well +presented with the dielectric permittivity ε (=ε1+iε2) as function of photon energy in Fig.S8. +500 +1000 +1500 +2000 +2500 +1k +10k +100k + MnFe2O4 + Syn29 + Syn31 + Syn33 + Syn35 + Syn42 +Spinel NP Material +Absorption Coefficient (/cm) +Wavelength (nm) + Syn23 + Syn24 + Syn25 + Syn26 +Spinel ONLY + +Fig. S6 Absorption coefficients of spinel materials in synthesized NPs and commercial MnFe2O4. +Based on XRD analysis (Fig.S2(c)) the Cu-Mn-Fe oxide spinels are classified as CuFe2O4 +group (Syn23, Syn24, Syn25, Syn26 ) and CuMn2O4 group (Syn31, Syn33, Syn42). This is largely +consistent with the trend of wavelength-dependent dielectric function shown in Fig. S7. Among +NPs with mixed CuFe2O4 and CuMn2O4 spinels, the ε trend of Syn35 spinels is more like CuFe2O4, +while Syn29 spinels are more like dielectric permittivity behavior of CuMn2O4. Two major + +9 + +features in Fig. S7 is the low-energy absorption for photon energy < 2 eV and high-energy +absorption > 2.5 eV. Substitution with Mn in CuFe2O4 or substitution with Fe in CuMn2O4 alters +the composition and inversion status of oxide spinels, thus changing the optical transition intensity. +For example, the spinels of Syn29 and Syn33 have relatively strong low-energy transitions. As +discussed in Section 3.2, the low-energy transitions ≤2 eV are assigned to the intervalance +transtions between ions in octahedral sites, particularly Fe2+ and Fe3+ ions. The high-energy +absorption > 2.5 eV is ascribed to the transitions between oxygen and ions in octahedral sites. +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +5 +6 +7 +8 +9 +10 +11 + +Photon Energy (eV) + Syn23 Spinel + Syn24 Spinel + Syn25 Spinel + Syn26 Spinel + Syn35 Spinel +(a) +CuFe2O4 group + +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +0 +1 +2 +3 +4 +5 + +Photon Energy (eV) + Syn23 Spinel + Syn24 Spinel + Syn25 Spinel + Syn26 Spinel + Syn35 Spinel +(b) +CuFe2O4 Group + +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +0 +2 +4 +6 +8 +10 +12 +14 + +Photon Energy (eV) + Syn29 Spinel + Syn31 Spinel + Syn33 Spinel + Syn42 Spinel +(c) +CuMn2O4 Group +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +1 +2 +3 +4 +5 +6 +7 +8 +2 +Photon Energy (eV) + Syn29 Spinel + Syn31 Spinel + Syn33 Spinel + Syn42 Spinel +(d) +CuMn2O4 Group + +Fig. S7 Dielectric constants of spinel phases of synthesized NPs as function of photon energy, +(a)(b) CuFe2O4 group, (c)(d)CuMn2O4 group. Real part ε1= n2-k2, and imaginary part ε2 =2nk. + +10 + + +3 High-temperature thermal stability of Cu-Fe-Mn oxide NP-pigmented coatings +Fig. S8 reveals the surface microstructure of the coatings after 60 thermal cycles testing. No +cracks , warps or flakes are observed in the coatings. + + +Fig. S8 SEM images of Syn24 NP-pigmented coatings on flat Inconel substrate after 60 thermal +cycles test (720h) at 750°C in the air. + +(a) +500μm +b +100um +C +20um11 + +20 +30 +40 +50 +60 +* +* +* +* +2() + 0 Cycles + 20 Cycles +Syn23-Coating +* +: Fe2O3 +(a) +20 +30 +40 +50 +60 +2(o) + 60 cyles + 40 cyles + 20 cyles + 0 cycles +CuFe2O4 +Mn2O3 + Inconel +Syn24-Coating +(b) + +20 +30 +40 +50 +60 +2() + 0 Cycles + 20 Cycles + 30 Cycles +Syn42-Coating +(c) + +Fig. S9 XRD spectra of (a) Syn23-coating, (b) Syn24-coating, and (c) Syn42-coating on Inconel +substrates after various numbers of thermal cycles between 750 °C and room temperature in air. +500 +1000 +1500 +2000 +2500 +0 +5 +10 +15 +20 +25 +30 +Reflectance(%) +Wavelength (nm) + 24h + cycle10 + cycle 20 + cycle30 +Syn23_Coating on Inconel +(a) +500 +1000 +1500 +2000 +2500 +0 +5 +10 +15 +20 +25 +Reflectance (%) +Wavelength (nm) + cycle 0 + cycle 10 + cycle 20 +Syn26-Coating on Inconel +(b) +500 +1000 +1500 +2000 +2500 +0 +5 +10 +15 +20 +25 +Syn29 coating on Inconel +Reflectance (%) +Wavelength (nm) + 24 h + 10 cycles + 20 cycles + 30 cycles +(c) + +Fig. S10 Comparison of reflectance spectra of the NP-coatings on Inconel after various thermal +test cycles, (a) Syn23 NPs-coating, (b) Syn26 NPs-coating, and (c) Syn29 NPs-coating. +5000 +10000 +15000 +0 +20 +40 +60 +80 +100 + 40 Cyc. + 50 Cyc. + 60 Cyc. +Syn24 NP-Coating +Reflectance (%) +Wavelength (nm) + 0 Cyc. + 10 Cyc. + 20 Cyc. + 30 Cyc. +(a) +5000 +10000 +15000 +0 +20 +40 +60 +80 +100 +Reflectance (%) +Wavelength (nm) + 10 Cycles + 20 Cycles + 30 Cycles + 0 Cycles +(b) +Syn42 NP-Coating + +Fig. S11 Comparison of reflectance spectra in the infrared range for thermal emittance calculation. +(a) Syn24 NP coating, and (b) Syn42 NP coating. + + +12 + +References +S1 J. S. Pinto, L. Cassayre, L. Presmanes, A. Barnabé, Insights on the stability and cationic nonstoichiometry of +CuFeO2 delafossite 58 (2019), 6431-64444. +S2 E. Agouriane1, B. Rabi, A. Essoumhi, A. Razouk, M. Sahlaoui, B. F. O. Costa, M. Sajieddin, Structural and +magnetic properties of CuFe2O4 ferrite nanoparticles synthesized by co-precipitation. J. Mater. Environ. Sci. 7 +(11) (2016) 4116-4120 +s3 H. Furuhashi, M. Inagaki, and S. Naka, Determination of cation distribution in spinels by X-ray diffraction +method, J. Inorg. Nucl. Chem., 3009-3014 (35) 1973 + + + diff --git a/vtAzT4oBgHgl3EQfP_vM/content/tmp_files/load_file.txt b/vtAzT4oBgHgl3EQfP_vM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b528f2e9e87a0836a92ae3dcd301eae71f897dbc --- /dev/null +++ b/vtAzT4oBgHgl3EQfP_vM/content/tmp_files/load_file.txt @@ -0,0 +1,1574 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf,len=1573 +page_content='1 Extracting optical parameters of Cu-Mn-Fe spinel oxide nanoparticles for optimizing air- stable, high-efficiency solar selective coatings Xiaoxin Wang*†, Can Xu, † and Jifeng Liu** Thayer School of Engineering, Dartmouth College, 15 Thayer Drive, Hanover, New Hampshire 03755, USA †Co-first authors contributing equally to this work Corresponding authors Xiaoxin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='Wang@Dartmouth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='Edu (X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Wang) Jifeng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='Liu@Dartmouth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='Edu (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Liu) Abstract High-temperature Cu-Mn-Fe spinel-oxide nanoparticle solar selective absorber coatings are investigated experimentally and theoretically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' A reliable, general approach to evaluate absorption coefficient spectra from the optical measurements of the nanoparticle-pigmented coatings is developed based on solving the inverse problem using four-flux-radiative method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' The derived absorption properties of NP materials can be directly applied to predict the solar absorptance, optimize the nanoparticle-pigmented coatings, and analyze the thermal degradation, which agree well with the experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' The analysis reveals that the Cu-Mn-Fe spinel oxides are fundamentally indirect bandgap ranging from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='7 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='1 eV, while iron-free CuMn2O4 is a direct bandgap material with Eg=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='84 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' With the same coating thickness and nanoparticle load, the solar absorptance ranks in the order of Mn2O3 < MnFe2O4 < CuFe2O4 < CuFeMnO4 < CuMn2O4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' The optimized spray-coated iron-free CuMn2O4 NP-pigmented coating demonstrates a high solar absorptance of 97%, a low emittance of 55%, a high optical-to-thermal energy conversion 2 efficiency of ~93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='5 % under 1000x solar concentration at 750ºC, and long-term endurance upon thermal cycling between 750°C and room temperature in air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' The optical parameter analysis approach can be easily extended to other material systems to facilitate the searching and optimizing high-temperature pigmented-solar selective coatings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Keywords: solar selective coatings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' spinel oxide nanoparticle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' complex refractive index;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' thermal efficiency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' four-flux radiative method Highlight: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Develop a general approach to extract the absorption coefficients and refractive indices of pigment nanoparticles from the optical measurements based on solving the inverse problem of the four-flux-radiative method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Investigate and search for highly absorbing, high temperature Cu-Mn-Fe oxide pigmented solar selective coatings in a systematic way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' With the same coating thickness and nanoparticle load, the pigment material performs in the order Mn2O3 < MnFe2O4 < CuFe2O4 < CuFeMnO4 < CuMn2O4 in terms of solar absorptance 3 Demonstrate iron-free CuMn2O4 coating with a high solar absorptance of 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='9%, low emittance of 55 %, and a high thermal efficiency of 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='5% under 1000x solar concentration at 750ºC that also endures long-term thermal cycling between 750 ºC and room temperature in air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Introduction Concentrating solar power (CSP) systems utilize optical components to collect and convert solar energy to thermal energy and then power heat engines to generate electricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' The widely used thermal energy storage in CSP systems allows the solar energy to be dispatched on demand,1 providing a great advantage over other non-dispatchable renewable energy sources such as wind power and solar photovoltaic (PV) power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' In order to reduce the levelized cost of energy (LCOE) of Generation 3 CSP systems towards 50% power efficiency, solar selective absorber coatings are required to possess long-term thermal stability at high temperatures ≥ 750 ºC in air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='2 Cost analysis reveals that durable oxidation-resistant solar selective coatings with solar absorptance αsolar ≥ 95% and thermal emittance ε ≤ 60% can guarantee a reduction of the LCOE of CSP plant up to 12%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='3 Under this guideline, low-cost nanoparticle (NP)-pigmented solar absorbers with high solar absorptance and moderate spectral selectivity is advantageous over highly selective yet more costly and less thermally stable multilayer cermet coatings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' A very attractive nanoparticle (NP) pigment candidate for solar selective coatings is the mixed transition metal spinel oxides with a general formula AB2O4 (A, B = metal) because of their diverse properties and wide availability in versatile applications as electrodes4, catalysts5, magnetic materials6,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Their tunable optical properties and high-temperature stability in air are especially suitable for solar absorber pigments 8-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' These include Co oxide8,9, Cu-Co oxide10,11, Mn-Co oxide10, Cu-Mn oxide12, Cu-Ni-Co oxide13, Cu-Co-Mn oxide14, Cu-Cr-Mn oxide 15, and Cu-Mn- Fe oxide16,17,18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' In our previous work, we have demonstrated air-stable MnFe2O4 spinel oxide NP- pigmented solar selective coatings with a high solar absorptance α ~93% and a thermal emittance ε ~55% for ~90% optical-to-thermal conversion efficiency (ηtherm) using a small load of solar- absorbing transition metal oxide nanoparticles (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='5 vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' %).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 19 Based on our established 4 quantitative approach and experimental findings, the solar absorptance of pigmented coatings is realistically determined by the product of coating thickness (d) and NP volume concentration (f), which is flexible for design and optimization of solar absorbing coatings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='20 Furthermore, the appropriate selection of NP oxides pigments can conveniently tune and maximize the solar absorptance and solar selectivity, which is inherent to the NP materials, in contrast to strict control of layer thickness (~nm) to take advantage of the interference effect in the multilayer cermet coatings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='21 However, there are still a couple of challenges in further optimizing these spinel-oxide NP solar coatings to approach the theoretical efficiency limit of 98%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='20 (1) The fundamental optical parameters of spinel oxide NPs are largely unavailable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' The available data are only limited to some particular compositions in the form of bulk single crystals or thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='22, 23 Furthermore, defects, cation inversion and substitution on A and B sites of the spinel lattice, and non-stoichiometry due to synthesis methods may lead to a large deviation in the optical property of spinel NPs pigments (10-50 nm in diameter) from that of the bulk material 22,24,25,26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Thus, it is important to develop a reliable method to derive wavelength-dependent effective absorption coefficient of the NPs from the measured optical spectra of the NP-pigmented composites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' (2) The origin of the prominent anomalous sub-band-gap absorption in the near infrared (NIR) regime of the solar spectrum remains unclear in literature, which is important in order to optimize the solar selectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Mechanisms range from charge transfer between bivalence-metal ions occupying distorted octahedral or tetrahedral sites (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' in MnxFe3-xO4+γ single crystals22) to Urbach tail absorption (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' in CuCoOx)27 to chemical substitution and partially inverted configuration between tetrahedral A sites and octahedral B sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 28 Therefore, a more accurate understanding of the electronic 5 transitions in spinel nanoparticles should be developed by studying and comparing more compounds of the same family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Recently, we have investigated CuMn2-xCrxO4 spinel oxide NP-pigmented solar selective coating system and demonstrated high thermal efficiency >94% in air at 750 ºC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='29 This initial success opens a large exploration space to further understand the impact of transition metal cationic species and their lattice sites on the performance of high-temperature solar selective absorbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' In this paper, a group of spinel Cu-Mn-Fe oxide NPs are systematically investigated as efficient high- temperature solar selective absorbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' This is also scientifically motivated by a comparison of Fe vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Cr alloying into CuMn2O4 NPs to gain more understanding, considering that Fe and Cr are on either side of Mn in the periodic table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' A reliable, general approach to extract the absorption coefficient spectra from the optical measurements of the NP-pigmented coatings is developed based on solving the inverse problem using four-flux-radiative method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' These derived effective scattering and absorption cross sections of synthesized NPs are then directly input into the four- flux radiative model to optimize the pigmented solar selective coatings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' The optimized CuMn2O4 pigmented solar selective coating on the Inconel substrate demonstrates a high solar absorptance α =96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='9% and a low thermal emittance ε =55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='0% at 750 °C under 1000x solar concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' This performance represents one of the highest efficiencies (93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='5%) under this service condition and exceeds the requirements for Generation 3 CSP systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' The analysis reveals that iron-free CuMn2O4 spinel is a direct bandgap material with Eg=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='84 eV, while the spinel group of Cu-Mn- Fe are fundamentally indirect bandgap between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='7~2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='1 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Cyclic thermal stability testing between 750ºC and room temperature shows that the crystalline phase and cationic distribution of Cu-Mn-Fe spinel oxides are generally very durable, though some low valence states of transition metals might be oxidized to high valence states gradually and degrade the solar absorptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Iron- 6 free CuMn2O4 (synthesized Syn42 NP) exhibits the best long-term thermal sustainability for long- term thermal cycling, each cycle comprising 12h at 750°C and 12 h cooling to room temperature in air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' The optical spectra curve-fitting and modeling approach developed here to extract the fundamental optical parameters of spinel NPs is applicable to other NP-pigmented solar coating systems to further optimize the efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Experimental and Modeling Methods 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='1 NP synthesis and spray coating deposition Cu-Mn-Fe oxide NPs were synthesized by a co-precipitation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Designed amounts of Cu2+, Mn2+ and Fe3+ ions from Cu(NO3)2, Mn(NO3)2 and FeCl3 were mixed with 100 mL deionized water to form solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' After mechanically stirring, NaOH solution was added dropwise to the homogeneous mixture to adjust the pH value to 12 to facilitate the co-precipitation reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' The precipitated NP were then collected after centrifugation, and washed repeatedly by deionized water to remove the excessive ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Further drying, recrystallization and grinding steps were followed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' As a comparison, commercially available MnFe2O4 NPs (purity: 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='99%, average diameter: 28 nm) and Mn2O3 NPs (purity: 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='99%, average diameter: 30 nm) were purchased from U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' research Nanomaterials Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' To fabricate NPs-pigmented silicone solar selective coatings, synthesized NPs were well dispersed in xylene-diluted silicone matrix Bluesil through ultrasonic bath to form uniform precursors with different viscosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' The spray coatings were performed using a Houseales Mini sprayer (10 ml) on quartz substrates and high temperature Ni-based super-alloys Inconel 625 sheet coupons (1 in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' × 1 in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' square) heated to 80-120 ºC by a hot plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Here Ni-based super-alloy Inconel is chosen due to its good high-temperature mechanical behavior, oxidation resistance, and corrosion resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Multiple sprays were carried out to achieve the optimal coating thickness at 7 a given NP pigment concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' These samples were annealed at 750 ºC for 24 hours and then cooled down overnight for the following characterizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' All thermal cycling tests were performed in a box furnace in air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Each cycling test includes 12 h at 750 °C and 12 h at room temperature to mimic the day-night cycles in practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='2 Characterization techniques X-ray diffraction (XRD, Rigaku 007 X-ray Diffractometer, Cu Kα1 line, λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='15406 nm operating at 40 kV/300 mA and a scanning rate of 2°/min from 10° to 80°) was conducted to reveal information about crystal structure, phase weight percentage, average NP size, and cationic distribution of as-synthesized or annealed NPs in the form of powder or embedded in the coatings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' High-resolution transmission electron microscopy images (TEM) were utilized to characterize the morphology, size and crystal structure of as-synthesized nanoparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Scanning electron microscopy (SEM, FEI XL-30 ESEM FEG, 20 kV, secondary electron (SE) mode) was performed to study the surface morphology and coating thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Energy dispersive spectroscopy (EDS, EDAX Si (Li) detector with Genesis software) was carried out to detect the chemical composition and elemental distribution in pigmented coatings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Raman spectra were recorded at room temperature using a confocal Raman imaging system and a laser radiation source operating at a wavelength of 532 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' The reflectance spectra in the wavelength range of λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='3 ~ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='5 μm were obtained by using a Jasco V-570 ultraviolet/visible/near infrared (UV/Vis/NIR) spectrometer equipped with a Jasco ISN-470 integrating sphere to capture both specular and diffuse reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' The reflectance spectra in the mid infrared (MIR) region (λ=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='5 ~20 μm) for thermal emittance measurement was recorded with a Jasco 4100 Fourier transformation IR (FTIR) spectrometer equipped with a Pike IR integrating sphere in the range from 400 to 4000 cm-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='3 Four-flux radiative modeling and solution of the inverse problem Four-flux radiative method is used to model the optical response of the NP-pigmented coatings and derive the optical properties of nanoparticle material from optical measurements with a Matlab curve fitting method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' The details of four-flux radiative theory can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 19- 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Firstly, the cross sections of scattering (Csca) and absorption (Cabs) of spherical NPs with a given average size were calculated using Lorentz-Mie theory (Mieplot30) based on the relative refractive index of NPs (n0+ik0) to that of the matrix material of the coating (nm+ikm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' The absorption (K) and scattering coefficients (S) of the pigmented coating are calculated by K=Cabs*f/V and S=Csca*f/V by taking into account the NP volume V and NP volume fraction f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' If coatings consist of NPs with various sizes and crystalline phases, the effective K and S of the coatings are defined as 𝐾 = ∑ 𝐶𝑎𝑏𝑠,𝑖 ∗ 𝑓𝑖 ∗ 𝑉𝑖 𝑛 𝑖=1 and 𝑆 = ∑ 𝐶𝑠𝑐𝑎,𝑖 ∗ 𝑓𝑖 ∗ 𝑉𝑖 𝑛 𝑖=1 , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Then the effective K, S of the coatings derived from the experiments or K, S values calculated from Cabs and Csca, along with the coating thickness, are input to the four-flux radiative models to calculate the reflectance R and transmittance T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' For the coatings on the metal substrate such as Inconel, the light transited through the coating should be absorbed completely by the metal (T=0), thus only reflectance R is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' In order to extract the complex refractive index of the NP material (n+ik) from the measured R and T by solving the inverse problem, two iterative loops of data fitting are introduced as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' One is based on the four-flux radiative method to derive effective K, S values of the coatings from the measured R, T, while the other loop uses the analytical Mie scattering to obtain the new refractive index of the NP material from the K, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' These loops are iterated until self- consistency is reached between the experimental and theoretical optical spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' The derived optical parameters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='e wavelength-dependent n and k) are then input to the four-flux radiative 9 model to design the spinel oxide NP pigmented solar selective coatings, taking into account multiple scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' The optical-to-thermal conversion efficiency \uf0e7therm of the solar receiver is used as figure of merit (FOM) to optimize solar selective coating design, given by Equation 1 𝜂𝑡ℎ𝑒𝑟𝑚 = 𝐹𝑂𝑀 = ∫ (1−𝑅(𝜆))𝐼(𝜆)𝑑𝜆−1 𝐶[∫ (1−𝑅(𝜆))𝐵(𝜆,𝑇)𝑑𝜆 ∞ 0 ] ∞ 0 ∫∞ 0 𝐼(𝜆)𝑑𝜆 = 𝛼𝑠𝑜𝑙𝑎𝑟 − 𝜀𝜎𝑇4 𝐶𝐼𝑠𝑜𝑙𝑎𝑟 (1) Here 𝐼(𝜆) is the AM 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='5 solar spectral radiance at wavelength 𝜆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 𝐼𝑠𝑜𝑙𝑎𝑟 = 1000 𝑊/𝑚2 is the solar power density integrated from the spectral radiance 𝐼(𝜆), 𝐵(𝜆, 𝑇) is the spectral blackbody thermal emission at wavelength 𝜆 and temperature 𝑇, 𝑅(𝜆) is spectral reflectance, calculated with four- flux model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 𝛼solar is the overall spectrally normalized solar absorptance, 𝜀 is the overall thermal emittance at temperature T, 𝜎 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='67 × 10−8 𝑊 𝑚2 𝐾4 is the Stefan-Boltzmann constant, and 𝐶 is solar concentration ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' In this paper, T=750°C=1023 K, and C=1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' The spectral range for the integrals is 300 nm-16 μm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 1 Flow of extracting the complex refractive index of NPs based on iterative, self-consistent Mie scattering theory and four-flux radiative method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Initial NP Material NP Size Medium Material Refractive Index D Refractive Index nm, km no, ko Mieplot Four Flux Radiative Method Absorption & Scattering Absorption & Calculated Cross Sections Scattering Coefficients Reflectance R & Cabs, Cscat K, S Coating Transmittance T NP Volume V & VolumeFractionf Thickness t Mieplot Compare Curve Fitting NewNPMaterial Measured Refractive Index Reflectance R & n,k Transmittance T10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Results and Discussion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='1 Characterization of NP and NP-pigmented coatings A group of Cu-Mn-Fe oxide NPs are synthesized by co-precipitation method and characterized by XRD, TEM and Raman spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' The crystalline phases and compositions are dependent on the starting material ratios of Cu:Mn:Fe, as summarized in Table 1 and detailed in the Supporting Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' As an example, the TEM and XRD data of Syn24 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' The data for other samples are summarized in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' S1-S3 of the Supporting Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Statistical analyses on TEM images of as-synthesized NPs (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 2a and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' S1 (a)-(d)) show it is reasonable to use an average NP diameter D~50 nm for the optical spectra curve-fitting and theoretical modeling in the later context, as also indicated in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Both selected area electron diffraction patterns (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' S1(e)(f)) and XRD spectra (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 2 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' S2) reveal the co-existence of Mn2O3 and spinel oxide phases in the as-synthesized NPs due to phase separation for nonstoichiometric samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' In fact, a secondary Mn2O3 phase is inevitably formed along with cubic spinels when the ratio Cu:Mn ≤1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='2 in CuxMn3-xO4 according to phase diagram,31 which also holds true for Cu-Mn-Fe oxide family here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Furthermore, when XRD (311) peaks of spinel Cu-Mn-Fe oxide NPs are examined closely (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 2c and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' S2(c)), each consists of two peaks for most annealed samples, which are ascribed to CuFe2O4 (~35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='5°) and CuMn2O4 (~35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='8°), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Generally speaking, CuMn2O4 dominates the spinel phase for a starting material ratio Mn:Fe > 4:1, such as Syn31, 33, and 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Otherwise, a higher or even dominant percentage of CuFe2O4 spinel phase exists, such as Syn 23-25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Syn29 and 35 have mixed CuMn2O4 and CuFe2O4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' One particular case is Syn26 (Cu:Mn:Fe=1:6:2) with a single dominant (311) peak at 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='6°, matching well with the standard pattern of cubic CuFeMnO4 (ICDD-PDF No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='20-03588) spinel structure32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 11 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='5 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='0 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='5 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='0 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='5 (311) (311) CuMn2O4 2\uf071\uf020\uf028 \uf06f \uf029 CuFe2O4 (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 2 Cu-Mn-Fe oxide NPs Syn24: (a) A TEM image of as-synthesized NPs showing a region dominated by relatively small CuFe2O4 NPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Larger Mn2O3 NPs from the same synthesis are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' S1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' (b) XRD spectra of as-synthesized NPs compared to those annealed at 750°C for 24 hours in air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' The XRD data of the Inconel substrate is also shown as a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' (c) Further deconvolution of (311) peaks of the annealed samples shown in (b), indicating co-existence of CuFe2O4 and CuMn2O4 spinel oxides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' A list of Cu-Mn-Fe oxide NP samples investigated in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' The last 3 columns apply to the NPs after annealing for 24h at 750ºC NP No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Cu:Mn:Fe TEM NP Size (nm)** XRD NP-24h Size (nm) Spinel Lattice Constant (Å) Spinel Weight Percentage Syn23 1:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='5:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='5 1:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='6:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='7* 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='5±11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='9 30 (sf);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='387 100% Syn24 1:3:1 1:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='7:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='2* 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='5±11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='3 32 (sf);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 47 (m) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='382*** 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='3% Syn25 1:1:3 1:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='2:3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='2* 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='2±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='5 24 (sf);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 50 (m) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='382*** 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='0% Syn26 1:6:2 1:6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='9:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='7* 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='8±14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='1 28(sfm);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 40(m) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='376 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='2% Syn29 1:4:1 1:3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='9:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='9* 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='0±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='1 52 (sf);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 31 (sm);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 54 (m) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='370/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='291 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='2% Syn31 1:6:1 49 (sm);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 58 (m) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='289 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='3% Syn33 1:3:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='5 21 (sm);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 50 (m) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='300 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='8% Syn35 1:3:2 34 (sf);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 42 (sm);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 56 (m) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='374/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='295 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='3% Syn42 1:2:0 31 (sm);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 49 (m) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='305 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='9% Mn2O3 Purchased 30 0% MnFe2O4 Purchased 28 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='99% EDS measured element ratio of Cu:Fe:Mn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' sf: spinel phase dominated by CuFe2O4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' sm: spinel phase dominated by CuMn2O4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' sfm: spinel CuFeMnO4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' m: Mn2O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' ** As-synthesized *** Lattice constant of the dominant sf phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' (a) Syn24 100nm Spinel Cu-Mn-Fe oxide Inconel After750°c annealing as-synthesized nconel substrate12 Raman spectroscopy is used to further confirm the phases identified from XRD analysis, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Due to the strong absorption in the visible range, Raman signals from Syn24 and Syn42 are relatively weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' However, vibrational frequencies of the modes can be identified by multi-peak fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' The Raman data provides the evidence that annealed Syn24 has mixed phases of CuFe2O4 and Mn2O3, and Syn42 has a dominant CuMn2O4 phase, in agreement with the XRD analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Coatings comprising as-synthesized Cu-Mn-Fe oxide NPs dispersed in silicone matrix are deposited on quartz or Inconel substrates by a spray-coating method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Coatings are then treated at 750 °C for 24 hours to stabilize the cubic spinel phase of NPs and meanwhile improve the coating adhesion to the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' SEM images of Syn24-pigmented coating on Inconel are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' No cracks or warps are found on the coating surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Coating thickness is ~8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='5 µm estimated from the cross section image in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='S4 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' The volume concentration of NP is ~13% calculated from the starting weight ratio of NPs and solid content of the silicone resin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='2 Extraction of Optical Parameters of NP-pigmented Coatings on Quartz Following the flow in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 1, the optical properties of synthesized Cu-Mn-Fe oxides NPs and commercial NPs (Mn2O3 and MnFe2O4) are determined inversely from the measured reflectance R and transmittance T of the corresponding pigmented coatings on quartz substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 3 shows the optical data of Syn29, Syn24, and Syn42, representing a trend of increasing weight percentages of spinel vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Mn2O3 NPs (30%, 50% and 71%, respectively, as shown in Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 3(a) the measured absorptance spectra (A=1-R-T) of synthesized and commercial NP-pigmented coatings are nicely reproduced by the four flux radiative modeling using their derived optical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Clearly, a full consistency is reached through the iteration loops in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 1 for extracting the wavelength dependent complex refractive indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Therefore, the approach developed here for 13 solving the inverse problem of the four-flux method is very reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' The correspondingly determined effective absorption coefficient of pigmented-coatings and derived refractive indices (both n and k) of NP materials are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 3(b) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 3(c), respectively, while Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 3(d) shows the cross sections of scattering (Csca) and absorption (Cabs) for ~50 nm-diameter NPs based on Mie scattering theory using refractive indices in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 500 1000 1500 2000 2500 0 20 40 60 80 100 Syn42_EXP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Syn42_CAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Syn29_EXP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Syn29_CAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Syn24_EXP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Syn24_CAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Absorptance (%) Wavelength (nm) MnFe2O4_EXP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' MnFe2O4_CAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Mn2O3_EXP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Mn2O3_CAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' (a) Quartz substrate 500 1000 1500 2000 2500 1E-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='1 1 10 Coating Absorption Coefficient (/\uf06dm) Wavelength (nm) Syn24 Syn29 Syn42 Mn2O3 MnFe2O4 (b) 500 1000 1500 2000 2500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='0 Refractive Index Wavelength (nm) Syn24_n Syn24_k Syn29_n Syn29_k Syn42_n Syn42_k Mn2O3_n Mn2O3_k MnFe2O4_n MnFe2O4_k (c) 500 1000 1500 2000 2500 1E-20 1E-19 1E-18 1E-17 1E-16 1E-15 Syn24_Csca Syn24_Cabs D=50 nm Syn29_Csca Syn29_Cabs Syn42_Csca Syn42_Cabs Mn2O3_Csca Mn2O3_Cabs Sca & Abs Cross Sections (m2) Wavelength (nm) MnFe2O4_Csca MnFe2O4_Cabs (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 3 (a) Experimentally measured and theoretically modeled spectral absorptance of home-synthesized Syn24, Syn29 and Syn42 and commercial Mn2O3, MnFe2O4 NP-pigmented coatings on quartz substrate, showing excellent self-consistency is achieved after the iteration loops shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' The extracted optical parameters from the iterative fitting are shown in (b)-(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' (b) The correspondingly derived effective absorption coefficients of the NP-coatings with different NP materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' (c) The corresponding derived effective refractive indices of NP materials, including both the real part n and the imaginary part k (extinction coefficient).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' (d) Calculated scattering and absorption cross sections of NPs with a size of 50 nm based on the refractive index data in (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 14 In terms of solar absorption, the absorption curves of commercial Mn2O3 and MnFe2O4 NP-coatings in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 3(a) roll off fast from wavelength > 600 nm in a similar way, though MnFe2O4- coating absorbs 5% more solar light than Mn2O3-coating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' As spinel phases are incorporated, the absorption spectrum is clearly extended to longer wavelengths to better cover the solar spectrum, as shown in the curves for Syn 29, 24, and 42 (with spinel phase weight percentage increasing from 30% to 50% to 70%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Though Syn24-coating demonstrates a slightly smaller absorption at the wavelengths λ < 800 nm than Syn29, it absorbs notably more infrared light than Syn29-coating in the wavelength range of 800~2500 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Overall, the iron-free Syn42 sample exhibits the highest absorption at λ<2000 nm despite of a slightly small absorption than Syn24 at λ> 2000 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' As far as the influence of Mn2O3 phase is concerned, we found that the effective absorption coefficients of NP-pigmented coatings incorporating spinel phases are 5-50x that of the reference Mn2O3-pigmented coating in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 3 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Therefore, Mn2O3 phase make insignificant contribution to the solar absorption of synthesized NP-coatings, particularly in the wavelength range of 800~2500 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' In other words, the curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 3 (b) actually reflect the absorption trends and features of Cu-Mn-Fe spinel oxides in the synthesized NPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' The derived absorption coefficients of spinel NP materials excluding the effect of Mn2O3 will be presented later in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 3(c) further shows that the real part of refractive indices across these samples are almost constant at n~2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='3 at λ= 800-2500 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' The variation in the real refractive indexes in the UV/Vis region at shorter wavelengths is associated with the strong band-to-band transitions seen in the absorption spectra (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='3(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' As compared with-Mn2O3 NP pigmented coating, the spinel NP-containing coatings exhibit notable features in their imaginary parts of refractive indices: an absorption edge/peak in the UV/vis regime followed by a broad absorption band in the infrared band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' The absorption edges/peaks are shifted with the spinel composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Clearly, Syn24 15 (CuFe2O4 dominated), Syn29 (CuFe2O4+CuMn2O4) and Syn42 (CuMn2O4 dominated) are arranged in an ascending order of redshift in UV/Vis absorption spectra due to the corresponding bandgaps modifications summarized in Table 2, which will be further detailed in the analyses of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 4 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 6 later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Last but not least, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 3(d) shows that the absorption cross sections of the spinel NPs is 10~100 times higher than the scattering cross sections for a NP diameter of 50 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Additionally, the absorption cross sections vary between different synthesized NPs, in contrast to the scattering cross sections insensitive to the NP materials discussed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' This is because the scattering cross-section is mainly determined by the real part of the refractive index, n, which are similar across different NPs as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Furthermore, these effective absorption and scattering cross sections of mixed phases in synthesized NPs will be used to optimize these NP-pigmented solar absorbing coatings in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='3 Band-to-band and d-shell optical transitions of spinel NPs 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='1 Effective absorption coefficient spectra of NPs comprising both spinel and Mn2O3 phases: The absorption coefficients of NP materials, independent on NP size, can be calculated as α=4*pi*k/λ, where λ is the wavelength, and k is the imaginary part of the refractive index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 4 reveals the details of bandgap energy determination from the absorption coefficients using Tauc plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' The direct bandgap and indirect bandgap can be obtained by linear fit of (αE)2 and (αE)1/2 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' photon energy E=hv, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' α-Mn2O3 has three direct gaps at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='20 eV, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='97 eV and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='96 eV (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 4(a)), in agreement with the reported 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='20 eV fundamental gap33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 4(b) and (c) respectively show the effective direct and indirect band gaps of the NPs (comprising spinel oxides and Mn2O3) in Syn24, Syn29 and Syn42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' As mentioned earlier and further shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 4a, the absorption is mainly contributed by spinel NPs rather than Mn2O3 NPs since the latter has much 16 lower absorption coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Syn42 with dominant cubic spinel CuMn2O4 phase is a direct band gap material with Eg=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='87 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Syn29, Syn24 and commercial MnFe2O4 NPs are fundamentally indirect, with bandgaps of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='74 eV, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='93 eV and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='62 eV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' All NPs shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 4 also have a higher level direct gap of ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='7 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='8 Egdir1=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='20 eV Egdir3=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='96 eV (\uf061E)2 (eV/cm)2 Photon Energy (eV) Mn2O3 Direct Bandgaps x 1011 (a) Egdir2=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='97 eV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='5 0 1 2 3 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='30 eV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='50 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='87 eV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='73 eV (\uf061E) 2 (eV/cm) 2 Photon Energy (eV) Syn24-mixed Syn29-mixed Syn42-mixed MnFe2O4 Direct Bandgap 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='25 eV (b) x 10 12800 600 400 500 Wavelength (nm) 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='5 0 200 400 600 800 1000 1200 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='62 eV (\uf061\uf045)1/2 (eV/cm)1/2 Photon Energy (eV) Syn24-mixed Syn29-mixed MnFe2O4 Indirect Bandgap 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='93 eV (c) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='75 eV 15001000 Wavelength (nm) 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='0k 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='0k 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='0k Sub-band Absorption Coefficient (/cm) Photon Energy (eV) Syn24 mixed Syn29 mixed MnFe2O4 (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 4 Tauc plots of (a) direct bandgaps of Mn2O3, (b) direct bandgaps of Cu-Mn-Fe oxide Syn24, Syn29, Syn42 and MnFe2O4 NP materials and (c) indirect band gaps of Syn24, Syn29, MnFe2O4 NP materials, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' (d) Sub-band absorption coefficient spectra of Syn24, Syn29, MnFe2O4 materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 17 Besides band-to-band optical absorption, strong absorption below the indirect bandgap is observed in Syn24 and Syn29 NPs, similar to commercial pure MnFe2O4 NPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' The sub-bandgap absorption spectra are retrieved by subtracting the fitted band-to-band absorption spectra from the overall absorption spectra for different samples, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 4(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' The redshift of the sub- band absorption coefficient peak in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 4(d) enables Syn24 and Syn29 coatings to achieve stronger solar absorption in the wavelength range > 800 nm in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 3(a) compared to MnFe2O4 NP pigmented coating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' As will be further discussed later, these sub-bandgap infrared absorption bands are due to d-d transitions of different transition metal cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='2 Retrieving absorption spectra of spinel oxide NPs: In order to evaluate accurately the optical properties of Cu-Mn-Fe spinel oxide NPs with different compositions, the influence of Mn2O3 impurity phase is further excluded based on descriptions in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content='3 using the absorption spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 4(a) and the weight percentages derived from XRD analyses listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Here we use Syn24, Syn29, and Syn42 as examples to show the trend of transitions from predominantly CuFe2O4 to mixed CuFe2O4+CuMn2O4 to CuMn2O4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' 5(a), CuMn2O4 spinel oxide NPs in Syn42 has a higher absorption cross section at wavelength <800 nm, while CuFe2O4 in Syn24 exhibits a relatively strong absorption tail at the wavelength >1200 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' The absorption cross section curve of mixed CuFe2O4 and CuMn2O4 spinels from Syn29 lie between those of CuFe2O4 and CuMn2O4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' All the synthesized Cu-Mn-Fe spinel oxides including iron-free CuMn2O4 have larger absorption cross sections (for NP diameter D=50 nm) than MnFe2O4 NP investigated in our previous work Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfP_vM/content/2301.01194v1.pdf'} +page_content=' Therefore, it is concluded that the pure spinel oxides in the aspect of solar absorption performance are listed in the ascending order as MnFe2O4